[{"Name":"Linear Inequalities","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Introduction","Duration":"4m 30s","ChapterTopicVideoID":5058,"CourseChapterTopicPlaylistID":3875,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.150","Text":"In this clip, we\u0027ll start talking about inequalities."},{"Start":"00:03.150 ","End":"00:05.565","Text":"It may last for several clips."},{"Start":"00:05.565 ","End":"00:08.850","Text":"There\u0027s 4 kinds of inequalities."},{"Start":"00:08.850 ","End":"00:12.060","Text":"Usually, we write something on each side,"},{"Start":"00:12.060 ","End":"00:14.265","Text":"could be a number or an expression."},{"Start":"00:14.265 ","End":"00:19.350","Text":"I could write something like a less than b,"},{"Start":"00:19.350 ","End":"00:21.435","Text":"a is less than b."},{"Start":"00:21.435 ","End":"00:27.315","Text":"This one, I would pronounce a is less than or equal to b."},{"Start":"00:27.315 ","End":"00:31.680","Text":"Some people also say a is not greater than b,"},{"Start":"00:31.680 ","End":"00:34.110","Text":"but I say less than or equal to."},{"Start":"00:34.110 ","End":"00:38.575","Text":"Then we have a is greater than b."},{"Start":"00:38.575 ","End":"00:42.995","Text":"Here, a greater than or equal to b."},{"Start":"00:42.995 ","End":"00:47.975","Text":"But as I say, some people would say a is not less than b,"},{"Start":"00:47.975 ","End":"00:50.370","Text":"or at least b."},{"Start":"00:50.420 ","End":"00:56.315","Text":"It could be true or it could be false depending on what a and b are."},{"Start":"00:56.315 ","End":"00:59.695","Text":"Let me go back to equalities."},{"Start":"00:59.695 ","End":"01:01.970","Text":"This is not an inequality,"},{"Start":"01:01.970 ","End":"01:03.500","Text":"this is an equality."},{"Start":"01:03.500 ","End":"01:05.585","Text":"I can say a=b."},{"Start":"01:05.585 ","End":"01:08.630","Text":"But just by analogy, I could say,"},{"Start":"01:08.630 ","End":"01:12.055","Text":"for example, I could write 3 things."},{"Start":"01:12.055 ","End":"01:15.730","Text":"I could write 2=5,"},{"Start":"01:15.730 ","End":"01:19.270","Text":"I could write 6=6,"},{"Start":"01:19.270 ","End":"01:28.460","Text":"and I could write 4=2."},{"Start":"01:28.460 ","End":"01:31.700","Text":"Now some of them would be true and some would be false."},{"Start":"01:31.700 ","End":"01:33.810","Text":"This one is false,"},{"Start":"01:33.810 ","End":"01:35.430","Text":"this one is true,"},{"Start":"01:35.430 ","End":"01:37.620","Text":"and this one is false."},{"Start":"01:37.620 ","End":"01:43.085","Text":"By analogy, do this with the others where it\u0027s not immediately clear."},{"Start":"01:43.085 ","End":"01:49.985","Text":"Let us take the same pairs of numbers and this time put a less than symbol between."},{"Start":"01:49.985 ","End":"01:54.110","Text":"In this case, we see that this is true,"},{"Start":"01:54.110 ","End":"01:56.890","Text":"this is false,"},{"Start":"01:56.890 ","End":"01:59.650","Text":"and this is also false."},{"Start":"01:59.650 ","End":"02:03.530","Text":"I want you to think of an inequality just like inequality as being a statement"},{"Start":"02:03.530 ","End":"02:07.220","Text":"you could write between 2 quantities and it could be true or false."},{"Start":"02:07.220 ","End":"02:10.730","Text":"Now let\u0027s go for the less than or equal to."},{"Start":"02:10.730 ","End":"02:13.250","Text":"I\u0027ll write here less than or equal to,"},{"Start":"02:13.250 ","End":"02:14.510","Text":"less than or equal to,"},{"Start":"02:14.510 ","End":"02:16.265","Text":"less than or equal to."},{"Start":"02:16.265 ","End":"02:20.310","Text":"Once again, let\u0027s say which is true, and which is false."},{"Start":"02:20.310 ","End":"02:23.700","Text":"Well, certainly 2 is less than or equal to 5."},{"Start":"02:23.700 ","End":"02:27.325","Text":"For some reason this one confuses people."},{"Start":"02:27.325 ","End":"02:31.430","Text":"6 less than or equal to 6 is definitely true."},{"Start":"02:31.430 ","End":"02:35.015","Text":"This thing is true if this one is less than this one,"},{"Start":"02:35.015 ","End":"02:37.720","Text":"and it\u0027s also true if this one equals this one."},{"Start":"02:37.720 ","End":"02:41.030","Text":"Sure. 6=6. Yes, it\u0027s true,"},{"Start":"02:41.030 ","End":"02:44.030","Text":"but it\u0027s also true is that 6 is less than or equal to 6."},{"Start":"02:44.030 ","End":"02:47.270","Text":"I don\u0027t know why, but some people get confused with that one."},{"Start":"02:47.270 ","End":"02:50.720","Text":"4 less than or equal to 2, no way."},{"Start":"02:50.720 ","End":"02:54.797","Text":"Next one. This one is called greater than."},{"Start":"02:54.797 ","End":"03:00.780","Text":"2 is greater than 5, no."},{"Start":"03:00.780 ","End":"03:05.885","Text":"6 greater than 6 also know 4 greater than 2, yes."},{"Start":"03:05.885 ","End":"03:11.019","Text":"Finally, greater than or equal to, in which case,"},{"Start":"03:11.019 ","End":"03:15.165","Text":"2 is not greater or equal to 5,"},{"Start":"03:15.165 ","End":"03:18.690","Text":"6 is greater than or equal to 6 because it\u0027s"},{"Start":"03:18.690 ","End":"03:23.830","Text":"equal and 4 is greater than or equal to 2 because it\u0027s greater than."},{"Start":"03:24.650 ","End":"03:26.720","Text":"This is an overkill,"},{"Start":"03:26.720 ","End":"03:31.070","Text":"but I\u0027m sure you know now what\u0027s greater than or equal to and what\u0027s less than."},{"Start":"03:31.070 ","End":"03:34.480","Text":"I\u0027ll just point out that these 2,"},{"Start":"03:34.480 ","End":"03:40.645","Text":"this one and this one are called strict inequalities."},{"Start":"03:40.645 ","End":"03:44.150","Text":"These are called non-strict."},{"Start":"03:44.150 ","End":"03:46.940","Text":"I\u0027ve even heard the term loose."},{"Start":"03:46.940 ","End":"03:49.370","Text":"I\u0027m not sure. I\u0027ll just say non-strict."},{"Start":"03:49.370 ","End":"03:53.690","Text":"I\u0027m not sure if loose is legal mathematical term."},{"Start":"03:53.690 ","End":"03:58.030","Text":"Strict when it does not include the possibility of equality."},{"Start":"03:58.030 ","End":"04:02.720","Text":"Non-strict when it allows for the possibility that the 2 could be equal."},{"Start":"04:02.720 ","End":"04:06.605","Text":"Now, up till now, we\u0027ve just dealt with numbers."},{"Start":"04:06.605 ","End":"04:12.455","Text":"What makes it more interesting is that we can do inequalities with algebraic expressions."},{"Start":"04:12.455 ","End":"04:14.435","Text":"Just like we solve equalities,"},{"Start":"04:14.435 ","End":"04:17.300","Text":"we can solve inequalities."},{"Start":"04:17.300 ","End":"04:21.725","Text":"But within equalities, things work a bit differently."},{"Start":"04:21.725 ","End":"04:25.145","Text":"Now I\u0027m going to introduce a few basic rules,"},{"Start":"04:25.145 ","End":"04:27.800","Text":"but I won\u0027t write them all in advance."},{"Start":"04:27.800 ","End":"04:31.950","Text":"I\u0027ll just introduce them through the examples as we need them."}],"Thumbnail":null,"ID":5049},{"Watched":false,"Name":"Basic Rules","Duration":"10m 22s","ChapterTopicVideoID":5050,"CourseChapterTopicPlaylistID":3875,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.015","Text":"I\u0027ll start with some examples and we\u0027ll introduce the basic rules as we need them."},{"Start":"00:06.015 ","End":"00:16.170","Text":"My first example will be 12x plus 9 is bigger than 9x."},{"Start":"00:16.170 ","End":"00:18.960","Text":"Inequalities are very similar to equalities."},{"Start":"00:18.960 ","End":"00:21.540","Text":"For example, if this was an equality,"},{"Start":"00:21.540 ","End":"00:25.920","Text":"one thing I might do would be to subtract 9x from both sides."},{"Start":"00:25.920 ","End":"00:27.810","Text":"This is what I\u0027m going to do here."},{"Start":"00:27.810 ","End":"00:29.820","Text":"If I subtract 9x,"},{"Start":"00:29.820 ","End":"00:34.155","Text":"then I get that 3x."},{"Start":"00:34.155 ","End":"00:37.620","Text":"I was going to say plus 9 bigger than 0,"},{"Start":"00:37.620 ","End":"00:39.360","Text":"but we can do two steps in one."},{"Start":"00:39.360 ","End":"00:42.210","Text":"Let\u0027s also subtract 9 from both sides."},{"Start":"00:42.210 ","End":"00:47.470","Text":"We get that 3x is greater than -9."},{"Start":"00:47.470 ","End":"00:51.410","Text":"This is the same as we were doing in equalities."},{"Start":"00:51.410 ","End":"00:54.425","Text":"But I have to write down that this is okay."},{"Start":"00:54.425 ","End":"00:58.910","Text":"I\u0027m going to write Rule 1 here which says that you can"},{"Start":"00:58.910 ","End":"01:03.350","Text":"add or subtract the same thing from both sides of an inequality."},{"Start":"01:03.350 ","End":"01:06.440","Text":"Here we used the subtraction twice."},{"Start":"01:06.440 ","End":"01:08.810","Text":"We subtracted 9x from both sides,"},{"Start":"01:08.810 ","End":"01:12.215","Text":"and then we subtracted 9 from both sides."},{"Start":"01:12.215 ","End":"01:13.550","Text":"Next thing I want to do,"},{"Start":"01:13.550 ","End":"01:14.750","Text":"if this was an equation,"},{"Start":"01:14.750 ","End":"01:16.625","Text":"I would divide by 3,"},{"Start":"01:16.625 ","End":"01:19.355","Text":"and turns out I can do that here."},{"Start":"01:19.355 ","End":"01:22.145","Text":"Let me just get some more space here."},{"Start":"01:22.145 ","End":"01:30.415","Text":"We can say that x is bigger than -9 divided by 3 is -3."},{"Start":"01:30.415 ","End":"01:34.550","Text":"Notice that I divided by 3 which is a positive number."},{"Start":"01:34.550 ","End":"01:40.520","Text":"This is important to note because Rule 2 says that is you\u0027re"},{"Start":"01:40.520 ","End":"01:47.150","Text":"allowed to multiply both sides of an inequality or divide by a positive number."},{"Start":"01:47.150 ","End":"01:50.390","Text":"I\u0027m going to emphasize positive."},{"Start":"01:50.390 ","End":"01:52.850","Text":"Now suppose instead of this example,"},{"Start":"01:52.850 ","End":"01:54.080","Text":"let\u0027s take this line."},{"Start":"01:54.080 ","End":"01:55.970","Text":"Suppose it was a -3 here."},{"Start":"01:55.970 ","End":"01:57.905","Text":"I\u0027ll do a separate exercise."},{"Start":"01:57.905 ","End":"02:04.250","Text":"I want to solve the inequality, -3x\u003e-9."},{"Start":"02:04.250 ","End":"02:08.180","Text":"Here\u0027s where things go different than with equations."},{"Start":"02:08.180 ","End":"02:13.365","Text":"With the equations, I would just divide both sides by -3."},{"Start":"02:13.365 ","End":"02:16.915","Text":"If I divided both sides by -3,"},{"Start":"02:16.915 ","End":"02:19.165","Text":"then you would get here,"},{"Start":"02:19.165 ","End":"02:25.660","Text":"x and here -9 over -3 would be 3."},{"Start":"02:25.660 ","End":"02:30.860","Text":"However, in the case of multiplying or dividing by a negative,"},{"Start":"02:30.860 ","End":"02:34.835","Text":"what we do is reverse the inequality."},{"Start":"02:34.835 ","End":"02:37.190","Text":"Now what do I mean by reverse?"},{"Start":"02:37.190 ","End":"02:39.245","Text":"I mean if it was greater than,"},{"Start":"02:39.245 ","End":"02:40.580","Text":"you put less than,"},{"Start":"02:40.580 ","End":"02:42.440","Text":"if it was greater or equal to,"},{"Start":"02:42.440 ","End":"02:44.300","Text":"you put less than or equal to,"},{"Start":"02:44.300 ","End":"02:47.320","Text":"the similar thing with the opposite direction."},{"Start":"02:47.320 ","End":"02:49.580","Text":"I\u0027ll write that as a rule."},{"Start":"02:49.580 ","End":"02:53.990","Text":"This Rule 3 says that if you multiply or divide both sides"},{"Start":"02:53.990 ","End":"02:58.400","Text":"of an inequality by the same negative number,"},{"Start":"02:58.400 ","End":"03:00.860","Text":"then you have to reverse."},{"Start":"03:00.860 ","End":"03:05.330","Text":"I\u0027ll emphasize that the direction of the inequality,"},{"Start":"03:05.330 ","End":"03:08.270","Text":"this is the most important thing to remember because this is where"},{"Start":"03:08.270 ","End":"03:11.345","Text":"it differs from equalities,"},{"Start":"03:11.345 ","End":"03:14.465","Text":"inequalities if you multiply or divide by a negative,"},{"Start":"03:14.465 ","End":"03:17.090","Text":"you reverse instead of greater than,"},{"Start":"03:17.090 ","End":"03:19.505","Text":"you get less than."},{"Start":"03:19.505 ","End":"03:21.290","Text":"It\u0027s easy to prove,"},{"Start":"03:21.290 ","End":"03:22.400","Text":"but I\u0027m not going to do it."},{"Start":"03:22.400 ","End":"03:25.715","Text":"I\u0027ll just show you with a couple of examples."},{"Start":"03:25.715 ","End":"03:29.510","Text":"If I take, for example that x=2,"},{"Start":"03:29.510 ","End":"03:35.310","Text":"then 2 is less than 3 and certainly -3x,"},{"Start":"03:35.310 ","End":"03:40.770","Text":"which is -6 is bigger than -9."},{"Start":"03:40.770 ","End":"03:42.930","Text":"Sure. 6 is less than 9,"},{"Start":"03:42.930 ","End":"03:45.650","Text":"but -6 is bigger than -9."},{"Start":"03:45.650 ","End":"03:48.805","Text":"It\u0027s further to the right on the number line."},{"Start":"03:48.805 ","End":"03:50.900","Text":"If instead of this 2,"},{"Start":"03:50.900 ","End":"03:52.895","Text":"I took another example."},{"Start":"03:52.895 ","End":"03:57.154","Text":"Let\u0027s say 4 where this does not hold true,"},{"Start":"03:57.154 ","End":"04:00.898","Text":"then -3 times 4 is -12,"},{"Start":"04:00.898 ","End":"04:04.545","Text":"is also not bigger than -9."},{"Start":"04:04.545 ","End":"04:08.240","Text":"It\u0027s the opposite. If this is true,"},{"Start":"04:08.240 ","End":"04:09.980","Text":"then this is true and vice versa."},{"Start":"04:09.980 ","End":"04:11.870","Text":"We can see by example."},{"Start":"04:11.870 ","End":"04:13.685","Text":"I don\u0027t need this."},{"Start":"04:13.685 ","End":"04:16.430","Text":"We just accept this as it is."},{"Start":"04:16.430 ","End":"04:20.990","Text":"That\u0027s what you have to worry about and most mistakes are made from not"},{"Start":"04:20.990 ","End":"04:25.970","Text":"following Rule 3 that it\u0027s important that if you multiply or divide by a negative,"},{"Start":"04:25.970 ","End":"04:28.640","Text":"you have to reverse the inequality."},{"Start":"04:28.640 ","End":"04:32.700","Text":"Now there\u0027s some rules about exponents."},{"Start":"04:32.700 ","End":"04:35.475","Text":"Write the word exponents,"},{"Start":"04:35.475 ","End":"04:40.565","Text":"and then I\u0027ll just write the rules and later we\u0027ll demonstrate how to use them."},{"Start":"04:40.565 ","End":"04:43.130","Text":"Rule number 4 says,"},{"Start":"04:43.130 ","End":"04:48.710","Text":"I need some more space here that I have to add the exponents."},{"Start":"04:48.710 ","End":"04:54.630","Text":"I\u0027m going to assume them to be integer whole numbers positive or negative."},{"Start":"04:54.630 ","End":"04:57.800","Text":"Now Rule 4 is,"},{"Start":"04:57.800 ","End":"05:02.240","Text":"you can raise both sides of an inequality by"},{"Start":"05:02.240 ","End":"05:08.465","Text":"the same power provided that that number is odd."},{"Start":"05:08.465 ","End":"05:12.980","Text":"That sounds strange and I\u0027ll explain in a moment why its needs to be odd."},{"Start":"05:12.980 ","End":"05:16.520","Text":"As one small correction I have to make when I said integer,"},{"Start":"05:16.520 ","End":"05:18.395","Text":"I meant positive integer."},{"Start":"05:18.395 ","End":"05:21.560","Text":"I\u0027ll give you some examples why the odd is important."},{"Start":"05:21.560 ","End":"05:29.435","Text":"For example, -3 is less than -2."},{"Start":"05:29.435 ","End":"05:35.985","Text":"Now if I cube them and I take -3^3,"},{"Start":"05:35.985 ","End":"05:39.784","Text":"and I take -2^3,"},{"Start":"05:39.784 ","End":"05:43.125","Text":"this is -27 and this is -8,"},{"Start":"05:43.125 ","End":"05:45.785","Text":"and it\u0027s still less than."},{"Start":"05:45.785 ","End":"05:54.750","Text":"But if I look at -3 to an even power like 2 and -2^2."},{"Start":"05:54.850 ","End":"06:02.505","Text":"This is 9 and this is 4 and 9 is bigger than 4,"},{"Start":"06:02.505 ","End":"06:05.205","Text":"so this is true,"},{"Start":"06:05.205 ","End":"06:07.125","Text":"but this is not true."},{"Start":"06:07.125 ","End":"06:10.955","Text":"That\u0027s because of the odd, even thing."},{"Start":"06:10.955 ","End":"06:14.645","Text":"Now if we only worked with positive numbers,"},{"Start":"06:14.645 ","End":"06:23.735","Text":"then it would be true for both odd and even and that\u0027s going to be the next rule."},{"Start":"06:23.735 ","End":"06:28.700","Text":"But before that, I also want to explain why I added the word positive here."},{"Start":"06:28.700 ","End":"06:32.810","Text":"Because again, suppose I take this -3 and -2."},{"Start":"06:32.810 ","End":"06:37.370","Text":"This time I raised to the ^ not 3, not 2,"},{"Start":"06:37.370 ","End":"06:40.755","Text":"but -1 and the -2,"},{"Start":"06:40.755 ","End":"06:42.975","Text":"I raise the ^ -1."},{"Start":"06:42.975 ","End":"06:45.755","Text":"Now this -1 is an odd number."},{"Start":"06:45.755 ","End":"06:48.950","Text":"Now I\u0027ll explain why the positive is important."},{"Start":"06:48.950 ","End":"06:51.575","Text":"Let\u0027s take, for example,"},{"Start":"06:51.575 ","End":"06:55.715","Text":"2 is less than 3 certainly."},{"Start":"06:55.715 ","End":"07:03.909","Text":"But if I take 2^-1 and 3^-1,"},{"Start":"07:03.909 ","End":"07:06.945","Text":"this is 1/2, this is 1/3."},{"Start":"07:06.945 ","End":"07:10.260","Text":"But 1/2 is bigger than 1/3."},{"Start":"07:10.260 ","End":"07:16.055","Text":"It doesn\u0027t work with -1 even though -1 is odd,"},{"Start":"07:16.055 ","End":"07:18.110","Text":"but it\u0027s not positive."},{"Start":"07:18.110 ","End":"07:21.860","Text":"Note that if I raise these 2 to an even power,"},{"Start":"07:21.860 ","End":"07:27.595","Text":"2^4 and 3^4 is 16 and 81,"},{"Start":"07:27.595 ","End":"07:30.455","Text":"that\u0027s still is less than."},{"Start":"07:30.455 ","End":"07:33.320","Text":"The next rule is going to be that if I have"},{"Start":"07:33.320 ","End":"07:37.145","Text":"just positive numbers on both sides of an inequality,"},{"Start":"07:37.145 ","End":"07:40.940","Text":"then I can raise them to odd or even powers and that will be fine."},{"Start":"07:40.940 ","End":"07:42.965","Text":"Let me just clear this up."},{"Start":"07:42.965 ","End":"07:44.540","Text":"Well, just before I erase this,"},{"Start":"07:44.540 ","End":"07:50.780","Text":"let me note that if I took one of these to be 0,"},{"Start":"07:50.780 ","End":"07:55.805","Text":"for example, 0 is less than 3,"},{"Start":"07:55.805 ","End":"07:59.330","Text":"and then I raise it to an exponent odd or even."},{"Start":"07:59.330 ","End":"08:02.410","Text":"Let\u0027s say the same 4 then 0^4,"},{"Start":"08:02.410 ","End":"08:07.455","Text":"and 3^4, they still compare the same way."},{"Start":"08:07.455 ","End":"08:09.740","Text":"What Rule 5 will say,"},{"Start":"08:09.740 ","End":"08:11.150","Text":"and I\u0027ll write it in a moment,"},{"Start":"08:11.150 ","End":"08:14.580","Text":"is that if you\u0027re dealing with positive or 0, in other words,"},{"Start":"08:14.580 ","End":"08:17.060","Text":"non-negative on both sides,"},{"Start":"08:17.060 ","End":"08:22.350","Text":"then you can raise to any positive power odd or even."},{"Start":"08:22.350 ","End":"08:24.900","Text":"I\u0027ll erase and write."},{"Start":"08:24.900 ","End":"08:28.760","Text":"Here I\u0027ve written it up that if both sides of an inequality"},{"Start":"08:28.760 ","End":"08:32.600","Text":"are non-negative, which includes positive,"},{"Start":"08:32.600 ","End":"08:35.914","Text":"but it also applies if one of them is 0,"},{"Start":"08:35.914 ","End":"08:41.285","Text":"then you can raise both of them to the same power, odd or even."},{"Start":"08:41.285 ","End":"08:45.905","Text":"But I\u0027m still only talking about positive integers."},{"Start":"08:45.905 ","End":"08:48.575","Text":"That\u0027s assumed in all of these."},{"Start":"08:48.575 ","End":"08:52.535","Text":"This should be highlighted that it\u0027s very important."},{"Start":"08:52.535 ","End":"08:56.105","Text":"There is one more that I\u0027d like to write."},{"Start":"08:56.105 ","End":"08:58.220","Text":"I\u0027m not sure if we\u0027ll use it or not,"},{"Start":"08:58.220 ","End":"09:00.320","Text":"but I\u0027d like to write it anyway."},{"Start":"09:00.320 ","End":"09:04.520","Text":"It\u0027s a generalization of 4 and 5."},{"Start":"09:04.520 ","End":"09:13.805","Text":"What I wanted to say is that 4 and 5 apply also for roots as well as powers."},{"Start":"09:13.805 ","End":"09:18.760","Text":"I don\u0027t know if to call this Rule 6 or to add rules 4a and 5a."},{"Start":"09:18.760 ","End":"09:24.380","Text":"Anyway, what I mean is that if I have an inequality,"},{"Start":"09:24.380 ","End":"09:25.850","Text":"I can take an odd root,"},{"Start":"09:25.850 ","End":"09:29.285","Text":"like the cube root of both sides or the fifth root."},{"Start":"09:29.285 ","End":"09:31.910","Text":"If both sides are non-negative,"},{"Start":"09:31.910 ","End":"09:34.970","Text":"then I can take the square root,"},{"Start":"09:34.970 ","End":"09:36.620","Text":"cube root, fourth root, fifth root,"},{"Start":"09:36.620 ","End":"09:39.425","Text":"whatever root I want of both sides."},{"Start":"09:39.425 ","End":"09:41.720","Text":"Actually there is one more."},{"Start":"09:41.720 ","End":"09:44.290","Text":"It\u0027s so trivial, I almost didn\u0027t mention it."},{"Start":"09:44.290 ","End":"09:47.100","Text":"Let\u0027s call it Number 0."},{"Start":"09:47.100 ","End":"09:49.805","Text":"I\u0027ll just write it in shorthand."},{"Start":"09:49.805 ","End":"09:54.290","Text":"What I wrote is switch sides and reverse the inequality."},{"Start":"09:54.290 ","End":"09:59.885","Text":"What I mean is that if a is less than b,"},{"Start":"09:59.885 ","End":"10:05.165","Text":"I can switch sides and the inequality and say b is bigger than a."},{"Start":"10:05.165 ","End":"10:09.815","Text":"Or for example, if c is bigger or equal to d,"},{"Start":"10:09.815 ","End":"10:17.330","Text":"I can switch the c and the d and say d is less than or equal to c. This is too obvious,"},{"Start":"10:17.330 ","End":"10:18.890","Text":"but I wrote it anyway."},{"Start":"10:18.890 ","End":"10:20.930","Text":"Okay, after the break,"},{"Start":"10:20.930 ","End":"10:23.280","Text":"we\u0027ll do some examples."}],"Thumbnail":null,"ID":5050},{"Watched":false,"Name":"Examples","Duration":"6m 28s","ChapterTopicVideoID":5051,"CourseChapterTopicPlaylistID":3875,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.610","Text":"We\u0027re continuing with inequalities."},{"Start":"00:02.610 ","End":"00:07.845","Text":"We\u0027ve just presented the rules and now let\u0027s do an example."},{"Start":"00:07.845 ","End":"00:10.380","Text":"Here\u0027s the example."},{"Start":"00:10.380 ","End":"00:12.390","Text":"I wrote it out already."},{"Start":"00:12.390 ","End":"00:18.960","Text":"Notice that this is the non-strict inequality less than or equal to."},{"Start":"00:18.960 ","End":"00:23.070","Text":"We\u0027ll just proceed pretty much like with equations,"},{"Start":"00:23.070 ","End":"00:25.935","Text":"but we have to remember the limitations of the rules."},{"Start":"00:25.935 ","End":"00:30.015","Text":"The first thing we can do is expand brackets."},{"Start":"00:30.015 ","End":"00:32.460","Text":"What we get is, let\u0027s see,"},{"Start":"00:32.460 ","End":"00:34.152","Text":"x times x is x^2,"},{"Start":"00:34.152 ","End":"00:36.390","Text":"x times 5 is 5x,"},{"Start":"00:36.390 ","End":"00:41.310","Text":"then minus 3x plus 15,"},{"Start":"00:41.310 ","End":"00:46.187","Text":"less than or equal to 2x minus 1,"},{"Start":"00:46.187 ","End":"00:51.340","Text":"minus 4x plus x^2."},{"Start":"00:51.340 ","End":"00:53.330","Text":"Now, as with inequalities,"},{"Start":"00:53.330 ","End":"00:58.490","Text":"we can bring stuff over to the other side, additions and subtractions."},{"Start":"00:58.490 ","End":"01:00.845","Text":"Not the multiplication and division part,"},{"Start":"01:00.845 ","End":"01:04.670","Text":"but bringing things over to the other side and changing the sign is like"},{"Start":"01:04.670 ","End":"01:09.075","Text":"additional subtraction and that\u0027s allowed in any inequality."},{"Start":"01:09.075 ","End":"01:15.660","Text":"If we collect all terms containing x and x^2,"},{"Start":"01:15.660 ","End":"01:17.040","Text":"let\u0027s say on the left,"},{"Start":"01:17.040 ","End":"01:18.944","Text":"and numbers on the right,"},{"Start":"01:18.944 ","End":"01:21.245","Text":"let\u0027s see what we get."},{"Start":"01:21.245 ","End":"01:25.325","Text":"We get x^2 minus x^2,"},{"Start":"01:25.325 ","End":"01:27.200","Text":"and that just cancels out."},{"Start":"01:27.200 ","End":"01:30.470","Text":"I\u0027ll just emphasize that this cancels with this."},{"Start":"01:30.470 ","End":"01:34.400","Text":"You could say at this point we subtracted x^2 from both sides."},{"Start":"01:34.400 ","End":"01:37.320","Text":"As for x\u0027s, well,"},{"Start":"01:37.320 ","End":"01:43.880","Text":"we have 5 minus 3 on this side and then another minus 2 plus 4."},{"Start":"01:43.880 ","End":"01:46.985","Text":"If I compute that 5 minus 3 is 2,"},{"Start":"01:46.985 ","End":"01:49.700","Text":"less 2 is 0 plus 4."},{"Start":"01:49.700 ","End":"01:52.905","Text":"We\u0027ve got 4x on the left,"},{"Start":"01:52.905 ","End":"01:54.885","Text":"and on the right,"},{"Start":"01:54.885 ","End":"02:04.145","Text":"here we have minus 1 and then we have 15 comes over as minus 15, so minus 16."},{"Start":"02:04.145 ","End":"02:07.115","Text":"Now we\u0027re going to divide by 4."},{"Start":"02:07.115 ","End":"02:11.840","Text":"I note that this 4 is positive."},{"Start":"02:11.840 ","End":"02:13.550","Text":"This is important to note."},{"Start":"02:13.550 ","End":"02:18.065","Text":"Otherwise, I\u0027d have to reverse the direction of the inequality."},{"Start":"02:18.065 ","End":"02:22.910","Text":"In this case, we can just divide by 4 and we get x less than"},{"Start":"02:22.910 ","End":"02:27.335","Text":"or equal to minus 16 over 4 is minus 4,"},{"Start":"02:27.335 ","End":"02:30.245","Text":"and that\u0027s the answer."},{"Start":"02:30.245 ","End":"02:32.240","Text":"A straightforward example."},{"Start":"02:32.240 ","End":"02:36.785","Text":"Notice that pretty much this was linear in the sense that,"},{"Start":"02:36.785 ","End":"02:40.085","Text":"yeah, we had x^2 for a while but it dropped out."},{"Start":"02:40.085 ","End":"02:44.810","Text":"First, we\u0027ll be doing mostly linear inequalities."},{"Start":"02:44.810 ","End":"02:46.730","Text":"I\u0027d like to do 1 more example."},{"Start":"02:46.730 ","End":"02:49.280","Text":"Let\u0027s just make this plural, examples."},{"Start":"02:49.280 ","End":"02:51.185","Text":"The next one will be,"},{"Start":"02:51.185 ","End":"02:53.210","Text":"here we have fractions,"},{"Start":"02:53.210 ","End":"02:59.375","Text":"so we have to multiply both sides by a common denominator."},{"Start":"02:59.375 ","End":"03:05.050","Text":"The least common denominator looks like 30."},{"Start":"03:05.050 ","End":"03:09.990","Text":"3 and 10 they need 30 and 5 already goes into that."},{"Start":"03:09.990 ","End":"03:14.090","Text":"Let\u0027s multiply both sides by 30,"},{"Start":"03:14.090 ","End":"03:19.490","Text":"but notice that 30 is a positive number,"},{"Start":"03:19.490 ","End":"03:23.915","Text":"just like I noted in the previous example that 4 was positive,"},{"Start":"03:23.915 ","End":"03:25.370","Text":"when you multiply or divide,"},{"Start":"03:25.370 ","End":"03:29.180","Text":"you really have to remember to check that."},{"Start":"03:29.180 ","End":"03:32.540","Text":"It\u0027s positive, we keep the direction."},{"Start":"03:32.540 ","End":"03:35.240","Text":"In each case we look for the missing factor in"},{"Start":"03:35.240 ","End":"03:38.240","Text":"the denominator then multiply to the numerator."},{"Start":"03:38.240 ","End":"03:41.540","Text":"3 goes into 30 10 times,"},{"Start":"03:41.540 ","End":"03:44.240","Text":"and I\u0027ll just write that here for the time being,"},{"Start":"03:44.240 ","End":"03:46.745","Text":"10 into 30 goes 3 times,"},{"Start":"03:46.745 ","End":"03:49.750","Text":"5 into 30 goes 6 times,"},{"Start":"03:49.750 ","End":"03:54.115","Text":"and 7 is just 7 over 1 goes in the full 30 times."},{"Start":"03:54.115 ","End":"04:04.860","Text":"Now we get 10 times x plus 4 minus 3 times x plus 3"},{"Start":"04:04.860 ","End":"04:10.110","Text":"minus 6 times 3x minus"},{"Start":"04:10.110 ","End":"04:17.575","Text":"5 is less than 30 times 7."},{"Start":"04:17.575 ","End":"04:20.855","Text":"Let me continue over here."},{"Start":"04:20.855 ","End":"04:23.900","Text":"We were running out of space."},{"Start":"04:23.900 ","End":"04:30.915","Text":"10x plus 40 minus 3x minus 9,"},{"Start":"04:30.915 ","End":"04:39.949","Text":"minus 18x plus 30 and is less than 210 linear."},{"Start":"04:39.949 ","End":"04:42.530","Text":"Let\u0027s keep the x\u0027s on the left,"},{"Start":"04:42.530 ","End":"04:44.075","Text":"numbers on the right."},{"Start":"04:44.075 ","End":"04:48.670","Text":"This is just addition and subtraction so it\u0027s okay for inequalities too."},{"Start":"04:48.670 ","End":"04:54.000","Text":"Let\u0027s see, the x\u0027s I\u0027m talking about now."},{"Start":"04:54.000 ","End":"04:57.405","Text":"10 minus 3 minus"},{"Start":"04:57.405 ","End":"05:05.055","Text":"18 is 10 minus 21,"},{"Start":"05:05.055 ","End":"05:12.030","Text":"which is minus 11x less than, let\u0027s see,"},{"Start":"05:12.030 ","End":"05:18.240","Text":"210 minus 40, that\u0027s 170"},{"Start":"05:18.240 ","End":"05:25.820","Text":"plus 9 is 179 plus 30,"},{"Start":"05:25.820 ","End":"05:29.855","Text":"I make the 209."},{"Start":"05:29.855 ","End":"05:33.140","Text":"Now at this point when we want to isolate x,"},{"Start":"05:33.140 ","End":"05:37.040","Text":"we have to divide by minus 11,"},{"Start":"05:37.040 ","End":"05:40.650","Text":"and minus 11 is certainly negative."},{"Start":"05:40.650 ","End":"05:45.860","Text":"This time we\u0027re going to have to reverse the direction."},{"Start":"05:45.860 ","End":"05:47.630","Text":"Instead of less than,"},{"Start":"05:47.630 ","End":"05:49.865","Text":"we\u0027re going to get a greater than."},{"Start":"05:49.865 ","End":"05:53.215","Text":"The minus 11 when I divide by it,"},{"Start":"05:53.215 ","End":"05:59.945","Text":"leaves me x and 209 divided by minus 11 will be minus something."},{"Start":"05:59.945 ","End":"06:02.940","Text":"Let\u0027s see, divides evenly,"},{"Start":"06:02.940 ","End":"06:09.680","Text":"minus 19 and x bigger than minus 19 is the solution."},{"Start":"06:09.680 ","End":"06:13.865","Text":"I guess I forgot to underline the solution in the previous one."},{"Start":"06:13.865 ","End":"06:16.280","Text":"Light note, never mind."},{"Start":"06:16.280 ","End":"06:18.500","Text":"That\u0027s the second example,"},{"Start":"06:18.500 ","End":"06:24.380","Text":"and there are solved example exercises besides the tutorial."},{"Start":"06:24.380 ","End":"06:25.985","Text":"I\u0027ll settle for this."},{"Start":"06:25.985 ","End":"06:28.620","Text":"Meanwhile, we\u0027re done."}],"Thumbnail":null,"ID":5051},{"Watched":false,"Name":"Special Cases","Duration":"2m 58s","ChapterTopicVideoID":5059,"CourseChapterTopicPlaylistID":3875,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.865","Text":"We\u0027re still on the topic of inequalities."},{"Start":"00:02.865 ","End":"00:05.685","Text":"I wanted to point out some,"},{"Start":"00:05.685 ","End":"00:07.080","Text":"I don\u0027t know what to call them,"},{"Start":"00:07.080 ","End":"00:09.720","Text":"extreme examples or special cases."},{"Start":"00:09.720 ","End":"00:12.900","Text":"I\u0027ll call them special cases."},{"Start":"00:12.900 ","End":"00:15.330","Text":"I\u0027m going to give 2 examples."},{"Start":"00:15.330 ","End":"00:18.750","Text":"Each one is extreme in an opposite direction."},{"Start":"00:18.750 ","End":"00:23.085","Text":"Here\u0027s one example looking innocent enough."},{"Start":"00:23.085 ","End":"00:25.170","Text":"Let\u0027s try and solve it."},{"Start":"00:25.170 ","End":"00:30.390","Text":"The first thing to do would be to open up the brackets on the right so we"},{"Start":"00:30.390 ","End":"00:36.383","Text":"get here 4x minus 18,"},{"Start":"00:36.383 ","End":"00:38.400","Text":"I meant this to be 3x,"},{"Start":"00:38.400 ","End":"00:42.105","Text":"not too late, plus 3x."},{"Start":"00:42.105 ","End":"00:46.640","Text":"Now we put the x\u0027s on the left numbers on the right,"},{"Start":"00:46.640 ","End":"00:47.990","Text":"just like with equations."},{"Start":"00:47.990 ","End":"00:51.145","Text":"We can do that because it\u0027s addition and subtraction."},{"Start":"00:51.145 ","End":"00:58.785","Text":"What we get is 7 minus 4 minus 3 is 0x, we get nothing."},{"Start":"00:58.785 ","End":"01:02.120","Text":"It\u0027s just 0 on the left and on the right,"},{"Start":"01:02.120 ","End":"01:07.685","Text":"we get minus 18 plus 15 is minus 3."},{"Start":"01:07.685 ","End":"01:14.360","Text":"The x has disappeared completely and all we\u0027re left with is a numerical statement,"},{"Start":"01:14.360 ","End":"01:17.290","Text":"0 less than 3,"},{"Start":"01:17.290 ","End":"01:21.020","Text":"which happens to be a false statement."},{"Start":"01:21.020 ","End":"01:23.465","Text":"Now when this happens,"},{"Start":"01:23.465 ","End":"01:28.190","Text":"and you get the variable dropping out and you get a false statement,"},{"Start":"01:28.190 ","End":"01:31.100","Text":"it means that there is no solution,"},{"Start":"01:31.100 ","End":"01:34.445","Text":"there is no x for which this inequality works."},{"Start":"01:34.445 ","End":"01:39.790","Text":"We might write something in words like no x is a solution to the inequality."},{"Start":"01:39.790 ","End":"01:45.390","Text":"Now I\u0027m going to show you another peculiar example, a special case."},{"Start":"01:45.390 ","End":"01:51.495","Text":"Here it is 6 times x over 2 plus 1 bigger or equal to 3x plus 4."},{"Start":"01:51.495 ","End":"01:53.415","Text":"You open the brackets."},{"Start":"01:53.415 ","End":"01:59.145","Text":"6 times x over 2 is 3x plus 6,"},{"Start":"01:59.145 ","End":"02:03.390","Text":"bigger or equal to 3x plus 4,"},{"Start":"02:03.390 ","End":"02:06.540","Text":"bring the x\u0027s to the left, numbers to the right."},{"Start":"02:06.540 ","End":"02:11.855","Text":"Once again the x\u0027s disappear 3x minus 3x is nothing."},{"Start":"02:11.855 ","End":"02:17.985","Text":"0, bigger or equal to 4 minus 6 is minus 2."},{"Start":"02:17.985 ","End":"02:20.205","Text":"Once again, we end up with,"},{"Start":"02:20.205 ","End":"02:23.040","Text":"no x just a numerical statement,"},{"Start":"02:23.040 ","End":"02:28.640","Text":"0 bigger or equal to minus 2 and this time,"},{"Start":"02:28.640 ","End":"02:32.210","Text":"as opposed to the previous example it happens to be true."},{"Start":"02:32.210 ","End":"02:36.439","Text":"When this happens in x drops out to negate a true statement,"},{"Start":"02:36.439 ","End":"02:40.280","Text":"then it turns out that every x is a solution."},{"Start":"02:40.280 ","End":"02:42.230","Text":"You would write some words to that effect,"},{"Start":"02:42.230 ","End":"02:47.825","Text":"or sometimes we just abbreviate and say the answer is all x or something like that."},{"Start":"02:47.825 ","End":"02:52.685","Text":"Anyway, I wanted you to be aware of extreme examples."},{"Start":"02:52.685 ","End":"02:58.540","Text":"No solution and everything is a solution. That\u0027s it."}],"Thumbnail":null,"ID":5052},{"Watched":false,"Name":"Combining with AND OR","Duration":"12m 4s","ChapterTopicVideoID":5049,"CourseChapterTopicPlaylistID":3875,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.540","Text":"Continuing with inequalities, in this clip,"},{"Start":"00:03.540 ","End":"00:10.200","Text":"we\u0027ll learn how to combine inequalities using the keywords AND and OR."},{"Start":"00:10.200 ","End":"00:14.550","Text":"These are sometimes called Boolean operators for those of you who studied logic."},{"Start":"00:14.550 ","End":"00:17.040","Text":"But let\u0027s just give an example."},{"Start":"00:17.040 ","End":"00:20.295","Text":"I\u0027ll use the AND example."},{"Start":"00:20.295 ","End":"00:22.529","Text":"Here the inequalities,"},{"Start":"00:22.529 ","End":"00:27.960","Text":"the first one is 6x-7\u003cx+13."},{"Start":"00:27.960 ","End":"00:32.410","Text":"The second one is 5x-3\u003e/=2."},{"Start":"00:32.410 ","End":"00:35.600","Text":"I\u0027ve got the word and between them."},{"Start":"00:35.600 ","End":"00:37.310","Text":"What this means,"},{"Start":"00:37.310 ","End":"00:41.329","Text":"to solve this system of inequalities,"},{"Start":"00:41.329 ","End":"00:45.665","Text":"is to find the x\u0027s for which both statements are true."},{"Start":"00:45.665 ","End":"00:48.830","Text":"For each given x, this could be true or false,"},{"Start":"00:48.830 ","End":"00:51.110","Text":"and for the other inequality."},{"Start":"00:51.110 ","End":"00:54.565","Text":"I want both of the inequalities to hold."},{"Start":"00:54.565 ","End":"00:56.360","Text":"That\u0027s what AND really means."},{"Start":"00:56.360 ","End":"00:57.845","Text":"When you have 2 statements,"},{"Start":"00:57.845 ","End":"00:59.480","Text":"and you put AND between them,"},{"Start":"00:59.480 ","End":"01:01.730","Text":"it means that both of them have to be true."},{"Start":"01:01.730 ","End":"01:04.400","Text":"The way to go about it,"},{"Start":"01:04.400 ","End":"01:07.950","Text":"is to first simplify each one separately."},{"Start":"01:07.950 ","End":"01:09.590","Text":"Let\u0027s work on the first one."},{"Start":"01:09.590 ","End":"01:13.475","Text":"Bring the x\u0027s over to the left and the numbers to the right."},{"Start":"01:13.475 ","End":"01:19.300","Text":"I\u0027ve got 6x-x=5x."},{"Start":"01:19.300 ","End":"01:24.710","Text":"On the right, I have 13+7=20."},{"Start":"01:24.710 ","End":"01:28.875","Text":"Now, I divide both sides by 5,"},{"Start":"01:28.875 ","End":"01:32.400","Text":"and 5 is a positive number, note."},{"Start":"01:32.400 ","End":"01:34.335","Text":"So I\u0027m allowed to do that."},{"Start":"01:34.335 ","End":"01:39.330","Text":"x\u003c4, 20/5 is 4."},{"Start":"01:39.330 ","End":"01:41.600","Text":"As for the other one, same thing."},{"Start":"01:41.600 ","End":"01:45.920","Text":"I bring numbers to the right and leave the x\u0027s on the left."},{"Start":"01:45.920 ","End":"01:52.880","Text":"So I\u0027ve got 5x\u003e/=2+3, is 5."},{"Start":"01:52.880 ","End":"01:55.575","Text":"I got 5 again, it\u0027s still positive."},{"Start":"01:55.575 ","End":"02:01.352","Text":"Divide by 5 and I get x\u003e/= 1."},{"Start":"02:01.352 ","End":"02:07.910","Text":"Now I have to see what does it mean to take this and this."},{"Start":"02:07.910 ","End":"02:12.050","Text":"I think it\u0027s best with a sketch or diagram,"},{"Start":"02:12.050 ","End":"02:13.760","Text":"although it\u0027s not necessary."},{"Start":"02:13.760 ","End":"02:16.435","Text":"Let me show you how I would sketch this."},{"Start":"02:16.435 ","End":"02:21.980","Text":"Here I did part of the sketch, a number line."},{"Start":"02:21.980 ","End":"02:25.429","Text":"The arrow just indicates it goes on endlessly,"},{"Start":"02:25.429 ","End":"02:29.900","Text":"and this is the positive direction of x. I should even label"},{"Start":"02:29.900 ","End":"02:36.136","Text":"it x. I drew the points 1 and 4 here,"},{"Start":"02:36.136 ","End":"02:39.050","Text":"because these are the numbers that appear."},{"Start":"02:39.050 ","End":"02:43.570","Text":"The way I represent x\u003c4,"},{"Start":"02:43.570 ","End":"02:46.835","Text":"is I go from 4 downwards."},{"Start":"02:46.835 ","End":"02:51.448","Text":"But I also put a little hollow circle here,"},{"Start":"02:51.448 ","End":"02:54.350","Text":"and a solid circle here."},{"Start":"02:54.350 ","End":"03:00.410","Text":"This is the convention that the 4 here is not included."},{"Start":"03:00.410 ","End":"03:01.745","Text":"It\u0027s less than 4,"},{"Start":"03:01.745 ","End":"03:03.785","Text":"but it does not include 4."},{"Start":"03:03.785 ","End":"03:07.315","Text":"Here we have bigger or equal to 1,"},{"Start":"03:07.315 ","End":"03:11.710","Text":"bigger than 1, but also including 1."},{"Start":"03:11.710 ","End":"03:17.180","Text":"Each one of these is the picture of each inequality separately."},{"Start":"03:17.180 ","End":"03:18.515","Text":"Now, to do an AND,"},{"Start":"03:18.515 ","End":"03:24.470","Text":"I have to see which values of x are covered by both of these."},{"Start":"03:24.470 ","End":"03:28.175","Text":"That turns out to be just the overlap,"},{"Start":"03:28.175 ","End":"03:31.190","Text":"which is this part here."},{"Start":"03:31.190 ","End":"03:33.290","Text":"But to be precise,"},{"Start":"03:33.290 ","End":"03:38.195","Text":"what I\u0027ll do, is I\u0027ll put a solid dot here."},{"Start":"03:38.195 ","End":"03:41.885","Text":"This color is going to indicate the and."},{"Start":"03:41.885 ","End":"03:47.210","Text":"I put a hollow dot here and everything in-between,"},{"Start":"03:47.210 ","End":"03:55.320","Text":"and this line segment would represent the overlap, this and this."},{"Start":"03:55.360 ","End":"03:57.785","Text":"Well, let\u0027s see what it would be."},{"Start":"03:57.785 ","End":"04:00.390","Text":"The way to describe this,"},{"Start":"04:00.390 ","End":"04:05.540","Text":"would be to say everything between 1 and 4 inclusive."},{"Start":"04:05.540 ","End":"04:09.230","Text":"Now, we haven\u0027t talked about double inequalities,"},{"Start":"04:09.230 ","End":"04:17.670","Text":"but I could describe it as 1\u003c/=x\u003c4,"},{"Start":"04:17.750 ","End":"04:21.465","Text":"or x between 1 and 4,"},{"Start":"04:21.465 ","End":"04:23.025","Text":"including the 1,"},{"Start":"04:23.025 ","End":"04:24.580","Text":"not including the 4."},{"Start":"04:24.580 ","End":"04:28.010","Text":"This is called a double inequality and we\u0027ll talk more about those later."},{"Start":"04:28.010 ","End":"04:32.060","Text":"But I\u0027m sure you\u0027ve probably seen this sort of thing already."},{"Start":"04:32.060 ","End":"04:35.780","Text":"This one is x\u003c4,"},{"Start":"04:35.780 ","End":"04:41.600","Text":"this one is x\u003c/=1,"},{"Start":"04:41.600 ","End":"04:46.430","Text":"and this one, x\u003c4\u003e/=1."},{"Start":"04:46.430 ","End":"04:49.024","Text":"If you are going to do it without the sketch,"},{"Start":"04:49.024 ","End":"04:55.100","Text":"then what I would say is to rewrite this instead of saying x\u003e/=1,"},{"Start":"04:55.100 ","End":"04:57.950","Text":"I can say 1\u003c/=x."},{"Start":"04:57.950 ","End":"05:00.830","Text":"That was one of the rules that you can change"},{"Start":"05:00.830 ","End":"05:04.385","Text":"the order as long as you change the direction of the inequality."},{"Start":"05:04.385 ","End":"05:06.320","Text":"Then when you see together,"},{"Start":"05:06.320 ","End":"05:11.155","Text":"let\u0027s say I took 1\u003c/=x and,"},{"Start":"05:11.155 ","End":"05:16.190","Text":"I\u0027ll deliberately write this x under the x, x\u003c4,"},{"Start":"05:16.190 ","End":"05:23.395","Text":"then you can see that these 2 together give you 1\u003c/=x, x\u003c4."},{"Start":"05:23.395 ","End":"05:25.640","Text":"This is actually an and in itself."},{"Start":"05:25.640 ","End":"05:30.910","Text":"This means that this part is true and the second part is true also."},{"Start":"05:30.910 ","End":"05:33.505","Text":"That\u0027s an example with AND."},{"Start":"05:33.505 ","End":"05:38.320","Text":"I mentioned the term double inequality and I\u0027d like to write that down."},{"Start":"05:38.320 ","End":"05:41.495","Text":"I want to call this exercise number 1."},{"Start":"05:41.495 ","End":"05:46.800","Text":"I\u0027m going to take exercise number 2 to be a double inequality."},{"Start":"05:49.910 ","End":"05:57.390","Text":"So 1\u003c/=5-2x\u003c7, it\u0027s a double inequality."},{"Start":"05:57.390 ","End":"05:59.735","Text":"Here\u0027s an inequality, here\u0027s an inequality."},{"Start":"05:59.735 ","End":"06:03.110","Text":"Of course, you can\u0027t mix certain kinds."},{"Start":"06:03.110 ","End":"06:06.170","Text":"I mean, I can\u0027t mix a less than with a bigger"},{"Start":"06:06.170 ","End":"06:10.130","Text":"than or less than or equal to with a bigger than or equal to."},{"Start":"06:10.130 ","End":"06:12.410","Text":"I can only mix in the same direction."},{"Start":"06:12.410 ","End":"06:16.025","Text":"Less than with less than or equal to or less than."},{"Start":"06:16.025 ","End":"06:17.570","Text":"Anyway, you get my drift."},{"Start":"06:17.570 ","End":"06:22.955","Text":"It wouldn\u0027t make sense to have them pointing in the opposite direction anyway."},{"Start":"06:22.955 ","End":"06:26.815","Text":"This, of course, is like having an and."},{"Start":"06:26.815 ","End":"06:32.100","Text":"It\u0027s like I have 1\u003c/=5-2x"},{"Start":"06:32.330 ","End":"06:37.490","Text":"and 5-2x\u003c7."},{"Start":"06:37.490 ","End":"06:45.650","Text":"Now, I took this kind of example where x only appears in the middle between the 2,"},{"Start":"06:45.650 ","End":"06:47.930","Text":"then it\u0027s easier to solve."},{"Start":"06:47.930 ","End":"06:54.680","Text":"You don\u0027t have to do all these diagrams and so on and you don\u0027t have to solve separately."},{"Start":"06:54.680 ","End":"06:59.615","Text":"If you only have x in the middle and we\u0027re talking about linear,"},{"Start":"06:59.615 ","End":"07:01.895","Text":"then what you can do is as follows."},{"Start":"07:01.895 ","End":"07:03.935","Text":"First, subtract 5,"},{"Start":"07:03.935 ","End":"07:08.870","Text":"but you have to subtract 5 from all 3 sides or whatever you want to call them."},{"Start":"07:08.870 ","End":"07:18.670","Text":"I\u0027ve got -2x\u003c7-5 is 2 and 1-5=-4."},{"Start":"07:18.670 ","End":"07:23.540","Text":"I subtract 5 from all because you can add or subtract from inequalities."},{"Start":"07:23.540 ","End":"07:30.480","Text":"Then I see that I have -2 and it\u0027s negative,"},{"Start":"07:30.480 ","End":"07:35.660","Text":"so I\u0027m going to divide all 3 sides or 3 bits by -2 and then I have"},{"Start":"07:35.660 ","End":"07:41.000","Text":"to reverse the direction of the inequalities."},{"Start":"07:41.000 ","End":"07:45.480","Text":"We get that x is going to be bigger than,"},{"Start":"07:45.480 ","End":"07:47.910","Text":"and here bigger than or equal to,"},{"Start":"07:47.910 ","End":"07:52.935","Text":"then 2 divided by -2 is -1,"},{"Start":"07:52.935 ","End":"07:57.930","Text":"and -4 divided by -2 is just 2."},{"Start":"07:57.930 ","End":"08:00.390","Text":"This would be the solution."},{"Start":"08:00.390 ","End":"08:02.135","Text":"It\u0027s perfectly fine."},{"Start":"08:02.135 ","End":"08:07.040","Text":"I\u0027m just more used to having them go from left to right."},{"Start":"08:07.040 ","End":"08:15.550","Text":"So indulge me and allow me to write this as -1\u003cx\u003c/=2."},{"Start":"08:15.550 ","End":"08:17.465","Text":"That\u0027s the answer."},{"Start":"08:17.465 ","End":"08:20.015","Text":"Although as I say, this was perfectly fine too,"},{"Start":"08:20.015 ","End":"08:22.520","Text":"and we did this without a sketch."},{"Start":"08:22.520 ","End":"08:24.020","Text":"Continuing."},{"Start":"08:24.020 ","End":"08:26.405","Text":"Let us take an example with OR,"},{"Start":"08:26.405 ","End":"08:29.170","Text":"that will be number 3."},{"Start":"08:29.170 ","End":"08:36.815","Text":"We\u0027ll take x\u003e/=3 or -4\u003cx\u003c1."},{"Start":"08:36.815 ","End":"08:42.530","Text":"The second one is a double inequality."},{"Start":"08:42.530 ","End":"08:45.680","Text":"Now, even without drawing a sketch,"},{"Start":"08:45.680 ","End":"08:50.105","Text":"you can see that these 2 inequalities have no overlap."},{"Start":"08:50.105 ","End":"08:52.610","Text":"There\u0027s no x that satisfies both."},{"Start":"08:52.610 ","End":"08:57.875","Text":"I mean, x can\u0027t be bigger or equal to 3 and at the same time be less than 1."},{"Start":"08:57.875 ","End":"08:59.660","Text":"When there\u0027s no overlap,"},{"Start":"08:59.660 ","End":"09:01.415","Text":"there is nothing to be done."},{"Start":"09:01.415 ","End":"09:04.525","Text":"You just leave the answer as is."},{"Start":"09:04.525 ","End":"09:06.150","Text":"We can illustrate it."},{"Start":"09:06.150 ","End":"09:08.820","Text":"Put in the points 3,"},{"Start":"09:08.820 ","End":"09:10.513","Text":"1, and -4,"},{"Start":"09:10.513 ","End":"09:15.280","Text":"so that this corresponds to the x bigger or equal to 3."},{"Start":"09:15.280 ","End":"09:20.115","Text":"I put a solid dot here to show that the 3 is included."},{"Start":"09:20.115 ","End":"09:27.950","Text":"This bit with 2 hollow dots gives us -4\u003cx\u003c1."},{"Start":"09:27.950 ","End":"09:32.330","Text":"But saying OR just means we take these 2 bits"},{"Start":"09:32.330 ","End":"09:36.800","Text":"together and this is the picture of the OR of them,"},{"Start":"09:36.800 ","End":"09:39.530","Text":"but there\u0027s nothing to combine or simplify,"},{"Start":"09:39.530 ","End":"09:41.420","Text":"It just stays as is."},{"Start":"09:41.420 ","End":"09:45.155","Text":"Let me take an example which is more interesting."},{"Start":"09:45.155 ","End":"09:47.180","Text":"Number 4,"},{"Start":"09:47.180 ","End":"09:54.880","Text":"x\u003c5 or"},{"Start":"09:54.880 ","End":"09:58.730","Text":"2\u003c/=x\u003c8."},{"Start":"09:58.730 ","End":"10:01.460","Text":"This time, there is an overlap."},{"Start":"10:01.460 ","End":"10:05.360","Text":"For example, if you take x=3,"},{"Start":"10:05.360 ","End":"10:11.555","Text":"it will be in this inequality and it will also satisfy this inequality."},{"Start":"10:11.555 ","End":"10:14.710","Text":"Let\u0027s draw a sketch of this."},{"Start":"10:14.710 ","End":"10:17.520","Text":"Here\u0027s the 2, 5,"},{"Start":"10:17.520 ","End":"10:20.000","Text":"and 8, which are the numbers that appear."},{"Start":"10:20.000 ","End":"10:24.680","Text":"The first one, x\u003c5 is this bit here."},{"Start":"10:24.680 ","End":"10:30.260","Text":"I need to put a hollow dot here, and that\u0027s x\u003c5."},{"Start":"10:30.260 ","End":"10:31.790","Text":"Then this bit here,"},{"Start":"10:31.790 ","End":"10:34.640","Text":"I need a solid dot here,"},{"Start":"10:34.640 ","End":"10:43.510","Text":"because the 2 is included and hollow here because the 8 isn\u0027t, and that\u0027s 2\u003c/=x\u003c8."},{"Start":"10:44.600 ","End":"10:50.765","Text":"The union is any value that\u0027s in this or in this."},{"Start":"10:50.765 ","End":"10:52.880","Text":"Meaning I take the combination."},{"Start":"10:52.880 ","End":"10:54.065","Text":"It\u0027s as if I don\u0027t know,"},{"Start":"10:54.065 ","End":"10:57.995","Text":"glue them together or I like to think of it sometimes like this is like"},{"Start":"10:57.995 ","End":"11:02.495","Text":"a roof and the sun is shining down and where is the shadow going to be."},{"Start":"11:02.495 ","End":"11:05.705","Text":"Well, it\u0027s going to be from here up to here."},{"Start":"11:05.705 ","End":"11:07.520","Text":"Whatever image works for you."},{"Start":"11:07.520 ","End":"11:09.260","Text":"Anyway, this is what you get,"},{"Start":"11:09.260 ","End":"11:11.840","Text":"is this line here, which goes on."},{"Start":"11:11.840 ","End":"11:14.810","Text":"When I put an arrow, I just mean it goes on indefinitely,"},{"Start":"11:14.810 ","End":"11:16.940","Text":"in this case to minus infinity."},{"Start":"11:16.940 ","End":"11:21.460","Text":"Here it goes up to the same place here."},{"Start":"11:21.460 ","End":"11:24.760","Text":"This is the solution of the OR."},{"Start":"11:24.760 ","End":"11:29.495","Text":"If we look at the picture, it\u0027s just x\u003c8."},{"Start":"11:29.495 ","End":"11:32.450","Text":"Then you write it down."},{"Start":"11:32.450 ","End":"11:35.465","Text":"The answer is x\u003c8."},{"Start":"11:35.465 ","End":"11:41.780","Text":"Now there\u0027s lots of combinations that one could do here with AND\u0027s and OR\u0027s."},{"Start":"11:41.780 ","End":"11:48.500","Text":"They\u0027re pretty much covered in the example exercises after the tutorial."},{"Start":"11:48.500 ","End":"11:51.440","Text":"I suggest you go and look there."},{"Start":"11:51.440 ","End":"11:56.300","Text":"There are quite a few solved examples there with all kinds of combinations."},{"Start":"11:56.300 ","End":"11:58.340","Text":"I\u0027ll just say that meanwhile,"},{"Start":"11:58.340 ","End":"12:04.500","Text":"we\u0027re done with this subject of AND/OR for inequalities."}],"Thumbnail":null,"ID":5053},{"Watched":false,"Name":"Exercise 1","Duration":"5m 53s","ChapterTopicVideoID":5052,"CourseChapterTopicPlaylistID":3875,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.800","Text":"In this exercise, we have a bunch of inequalities to solve, 5 of them."},{"Start":"00:04.800 ","End":"00:07.215","Text":"Greater than here, here, here."},{"Start":"00:07.215 ","End":"00:09.735","Text":"Less than, greater or equal to,"},{"Start":"00:09.735 ","End":"00:13.845","Text":"and I\u0027ve rewritten them just so we get some more space for the solutions."},{"Start":"00:13.845 ","End":"00:16.140","Text":"Let\u0027s begin with the first one."},{"Start":"00:16.140 ","End":"00:20.565","Text":"There\u0027s things we can do with inequalities like add numbers to both sides."},{"Start":"00:20.565 ","End":"00:24.330","Text":"The whole concept of bringing numbers to the right and"},{"Start":"00:24.330 ","End":"00:28.215","Text":"letters to the left still works because it\u0027s based on addition and subtraction."},{"Start":"00:28.215 ","End":"00:38.370","Text":"I can bring the 26 over to the other side and say that 45x is bigger than 109 plus 26,"},{"Start":"00:38.370 ","End":"00:46.905","Text":"which happens to equal 135 and then dividing both sides by 3."},{"Start":"00:46.905 ","End":"00:48.750","Text":"When I divide by 3,"},{"Start":"00:48.750 ","End":"00:51.450","Text":"I make note that 3 is positive."},{"Start":"00:51.450 ","End":"00:54.830","Text":"I\u0027ve always emphasize it like this, but first one,"},{"Start":"00:54.830 ","End":"00:59.450","Text":"I will emphasize that it is positive and that\u0027s important that we can"},{"Start":"00:59.450 ","End":"01:05.105","Text":"multiply or divide by a positive number but for negatives, something else happens."},{"Start":"01:05.105 ","End":"01:13.535","Text":"Then we just say that x is bigger than 135/45 and that actually divides neatly,"},{"Start":"01:13.535 ","End":"01:15.245","Text":"it goes in 3 times."},{"Start":"01:15.245 ","End":"01:19.295","Text":"The solution is x is bigger than 3,"},{"Start":"01:19.295 ","End":"01:21.005","Text":"and that\u0027s part a."},{"Start":"01:21.005 ","End":"01:23.540","Text":"On to part b, once again,"},{"Start":"01:23.540 ","End":"01:29.330","Text":"we\u0027ll move the numbers to the right and the letters,"},{"Start":"01:29.330 ","End":"01:31.630","Text":"the variable to the left."},{"Start":"01:31.630 ","End":"01:37.305","Text":"What I get is 7x minus 3x is 4x,"},{"Start":"01:37.305 ","End":"01:45.095","Text":"11 plus 5 is 16 and the inequality keeps the same direction."},{"Start":"01:45.095 ","End":"01:52.400","Text":"Now we divide, take both sides and divide them by 4, which is positive."},{"Start":"01:52.400 ","End":"01:56.990","Text":"So we get that x is bigger than 4."},{"Start":"01:56.990 ","End":"01:59.095","Text":"Onto the next one,"},{"Start":"01:59.095 ","End":"02:05.495","Text":"6x is bigger than this and we can open the brackets, expand."},{"Start":"02:05.495 ","End":"02:10.640","Text":"This is equality so we can certainly replace this by something that\u0027s equal to it,"},{"Start":"02:10.640 ","End":"02:16.110","Text":"which is 6x minus 2."},{"Start":"02:16.110 ","End":"02:21.560","Text":"Now, we can subtract 6x from both sides."},{"Start":"02:21.560 ","End":"02:27.865","Text":"If we do that, we get 0 is bigger than minus 2."},{"Start":"02:27.865 ","End":"02:31.820","Text":"This is peculiar because x has completely disappeared."},{"Start":"02:31.820 ","End":"02:36.380","Text":"When that happens, we check if this is a true statement or not."},{"Start":"02:36.380 ","End":"02:41.540","Text":"If it\u0027s true, then this inequality holds for all x and if it\u0027s not true,"},{"Start":"02:41.540 ","End":"02:43.084","Text":"then we say no solution."},{"Start":"02:43.084 ","End":"02:49.325","Text":"In this case, this is true because 0 is indeed bigger than minus 2."},{"Start":"02:49.325 ","End":"02:55.279","Text":"When this happens, we say that the inequality holds for all x."},{"Start":"02:55.279 ","End":"02:57.530","Text":"Every x is a solution."},{"Start":"02:57.530 ","End":"03:00.485","Text":"You could check it on your own."},{"Start":"03:00.485 ","End":"03:03.560","Text":"Just give me any value of x through 0."},{"Start":"03:03.560 ","End":"03:05.300","Text":"May I put x=0,"},{"Start":"03:05.300 ","End":"03:08.730","Text":"I get 0 is bigger than minus 3."},{"Start":"03:08.730 ","End":"03:11.085","Text":"At the same thing, I put x=1,"},{"Start":"03:11.085 ","End":"03:16.980","Text":"I get 6 is bigger than twice 2 which is 6 bigger than 4, also true."},{"Start":"03:16.980 ","End":"03:18.755","Text":"It works for every x."},{"Start":"03:18.755 ","End":"03:21.455","Text":"If you got here something that was false,"},{"Start":"03:21.455 ","End":"03:23.930","Text":"then it would be no solution."},{"Start":"03:23.930 ","End":"03:25.730","Text":"Maybe we\u0027ll have one of those later."},{"Start":"03:25.730 ","End":"03:30.425","Text":"Meanwhile, let\u0027s go on to part d and just underline the answer."},{"Start":"03:30.425 ","End":"03:35.480","Text":"What we have here is a less than, same rules apply."},{"Start":"03:35.480 ","End":"03:38.880","Text":"Let\u0027s just first of all, expand brackets."},{"Start":"03:38.880 ","End":"03:44.865","Text":"We have 3x minus 36 plus 3x."},{"Start":"03:44.865 ","End":"03:47.840","Text":"Then let us on the left numbers on the right."},{"Start":"03:47.840 ","End":"03:54.955","Text":"So 5x minus 3x minus 3x is 5x minus 6x."},{"Start":"03:54.955 ","End":"03:57.965","Text":"I\u0027ll just write it out. Be easier that way,"},{"Start":"03:57.965 ","End":"03:59.975","Text":"do everything in our heads,"},{"Start":"03:59.975 ","End":"04:05.505","Text":"less than minus 36, minus 15."},{"Start":"04:05.505 ","End":"04:07.440","Text":"Now let\u0027s collect them."},{"Start":"04:07.440 ","End":"04:09.930","Text":"5 minus 3, minus 3 is minus 1."},{"Start":"04:09.930 ","End":"04:13.065","Text":"So I have minus 1x is less than,"},{"Start":"04:13.065 ","End":"04:15.630","Text":"2 minuses, so it\u0027s like a plus."},{"Start":"04:15.630 ","End":"04:20.340","Text":"At first. 36 and 15 is 51,"},{"Start":"04:20.340 ","End":"04:22.590","Text":"so this is minus 51."},{"Start":"04:22.590 ","End":"04:24.155","Text":"Now here\u0027s the thing."},{"Start":"04:24.155 ","End":"04:28.500","Text":"We now want to divide by minus 1."},{"Start":"04:28.500 ","End":"04:35.375","Text":"Minus 1 is negative as opposed to the first exercise why I emphasized that it\u0027s positive."},{"Start":"04:35.375 ","End":"04:38.690","Text":"When we multiply or divide by a negative,"},{"Start":"04:38.690 ","End":"04:40.535","Text":"we switch the inequality."},{"Start":"04:40.535 ","End":"04:42.110","Text":"Instead of less than,"},{"Start":"04:42.110 ","End":"04:44.075","Text":"I write bigger than."},{"Start":"04:44.075 ","End":"04:46.290","Text":"Then divide this by minus 1,"},{"Start":"04:46.290 ","End":"04:48.420","Text":"this is x divide this by minus 1,"},{"Start":"04:48.420 ","End":"04:49.860","Text":"so it\u0027s 51."},{"Start":"04:49.860 ","End":"04:51.930","Text":"Here\u0027s our answer."},{"Start":"04:51.930 ","End":"04:54.570","Text":"Now the last one,"},{"Start":"04:54.570 ","End":"05:01.250","Text":"part e. Start by expanding brackets 2x minus 10."},{"Start":"05:01.250 ","End":"05:07.085","Text":"This time we have a bigger or equal to 1/2 times 4x is 2x."},{"Start":"05:07.085 ","End":"05:10.010","Text":"1/2 times 6 is 3."},{"Start":"05:10.010 ","End":"05:14.165","Text":"Now we bring the variables to the left."},{"Start":"05:14.165 ","End":"05:17.810","Text":"We get 2x minus 2x is 0."},{"Start":"05:17.810 ","End":"05:19.580","Text":"The numbers to the right,"},{"Start":"05:19.580 ","End":"05:22.995","Text":"3 plus 10 is 13."},{"Start":"05:22.995 ","End":"05:24.950","Text":"Is this a true statement?"},{"Start":"05:24.950 ","End":"05:27.440","Text":"Is 0 bigger or equal to 13?"},{"Start":"05:27.440 ","End":"05:30.815","Text":"Well, the answer is no, it\u0027s neither bigger than or equal to."},{"Start":"05:30.815 ","End":"05:36.605","Text":"So this is a false statement and when you get a false statement with no Xs,"},{"Start":"05:36.605 ","End":"05:41.150","Text":"that means that there is no x that satisfies the inequality."},{"Start":"05:41.150 ","End":"05:49.460","Text":"No x, or I can write no solution to the inequality or something like that."},{"Start":"05:49.460 ","End":"05:54.060","Text":"So no solution and that\u0027s it."}],"Thumbnail":null,"ID":5054},{"Watched":false,"Name":"Exercise 2","Duration":"10m 5s","ChapterTopicVideoID":5053,"CourseChapterTopicPlaylistID":3875,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.690","Text":"Here we have another bunch of linear inequalities to solve and just to get more space,"},{"Start":"00:06.690 ","End":"00:09.945","Text":"I copied them out again below, here we are."},{"Start":"00:09.945 ","End":"00:12.135","Text":"Let\u0027s start with the first one."},{"Start":"00:12.135 ","End":"00:14.880","Text":"There is something strange here because there\u0027s going to be"},{"Start":"00:14.880 ","End":"00:18.690","Text":"an x^2 and I did say linear and how does that work out?"},{"Start":"00:18.690 ","End":"00:20.205","Text":"Well, we shall see."},{"Start":"00:20.205 ","End":"00:23.355","Text":"Let\u0027s start by expanding brackets."},{"Start":"00:23.355 ","End":"00:30.105","Text":"What we get is x^2 minus 4x plus 4."},{"Start":"00:30.105 ","End":"00:38.040","Text":"Perhaps I should state that I\u0027m going to be using the formula that a minus b"},{"Start":"00:38.040 ","End":"00:47.350","Text":"squared equals a^2 minus 2ab plus b^2."},{"Start":"00:47.350 ","End":"00:50.625","Text":"Actually instead of writing it again with a plus,"},{"Start":"00:50.625 ","End":"00:53.270","Text":"let\u0027s just say that if it\u0027s a plus here,"},{"Start":"00:53.270 ","End":"00:57.035","Text":"then it\u0027s also a plus over here."},{"Start":"00:57.035 ","End":"00:58.850","Text":"The middle term is a minus."},{"Start":"00:58.850 ","End":"01:01.790","Text":"I mean if this is a minus, the middle terms are minus, this is plus,"},{"Start":"01:01.790 ","End":"01:05.375","Text":"this one is a plus, and so it\u0027ll be good for this and for this."},{"Start":"01:05.375 ","End":"01:08.870","Text":"Continuing, we have another plus 4,"},{"Start":"01:08.870 ","End":"01:10.475","Text":"that\u0027s from the one that was here,"},{"Start":"01:10.475 ","End":"01:11.840","Text":"and then less than,"},{"Start":"01:11.840 ","End":"01:14.495","Text":"on the other side of the inequality now I\u0027m going to use it,"},{"Start":"01:14.495 ","End":"01:16.010","Text":"same thing but with a plus,"},{"Start":"01:16.010 ","End":"01:22.275","Text":"so it\u0027s x^2 plus 4x plus 4 and then plus 20."},{"Start":"01:22.275 ","End":"01:26.060","Text":"Notice that x^2 cancels."},{"Start":"01:26.060 ","End":"01:30.185","Text":"Strictly speaking, I should say subtract x^2 from both sides,"},{"Start":"01:30.185 ","End":"01:31.490","Text":"and then in inequality,"},{"Start":"01:31.490 ","End":"01:34.145","Text":"we\u0027re allowed to subtract the same thing from both sides."},{"Start":"01:34.145 ","End":"01:36.065","Text":"Now it\u0027s obviously going to be linear,"},{"Start":"01:36.065 ","End":"01:41.240","Text":"so let\u0027s move the x\u0027s to the left and the numbers to the right."},{"Start":"01:41.240 ","End":"01:48.920","Text":"On the left we have minus 4x and then another minus 4x."},{"Start":"01:48.920 ","End":"01:52.820","Text":"On the right, maybe before that,"},{"Start":"01:52.820 ","End":"01:57.965","Text":"I\u0027m tempted to just cancel and see something else which is the same on both sides."},{"Start":"01:57.965 ","End":"02:03.395","Text":"This is less than 20 and then minus 4."},{"Start":"02:03.395 ","End":"02:05.270","Text":"Let\u0027s see what we have now."},{"Start":"02:05.270 ","End":"02:12.860","Text":"This is minus 8x and this is less than 16."},{"Start":"02:12.860 ","End":"02:19.475","Text":"Now the thing is that we have to divide both sides by minus 8."},{"Start":"02:19.475 ","End":"02:22.550","Text":"But if you remember when it\u0027s negative"},{"Start":"02:22.550 ","End":"02:25.490","Text":"and I\u0027ll just emphasize it this time, not every time,"},{"Start":"02:25.490 ","End":"02:28.625","Text":"but it\u0027s negative, which means that when we do that,"},{"Start":"02:28.625 ","End":"02:31.240","Text":"we reverse the direction of the inequality,"},{"Start":"02:31.240 ","End":"02:32.540","Text":"so instead of less than,"},{"Start":"02:32.540 ","End":"02:34.085","Text":"it\u0027s going to be greater than."},{"Start":"02:34.085 ","End":"02:37.273","Text":"Dividing by minus 8 gives us x here,"},{"Start":"02:37.273 ","End":"02:41.940","Text":"and here we get minus 2 plus over minus is minus,"},{"Start":"02:41.940 ","End":"02:44.500","Text":"and this is the solution."},{"Start":"02:44.500 ","End":"02:46.310","Text":"On to the next."},{"Start":"02:46.310 ","End":"02:51.725","Text":"Here we have a fraction and we\u0027ll find a common denominator."},{"Start":"02:51.725 ","End":"02:54.935","Text":"Common denominator for 2 and 3 is 6."},{"Start":"02:54.935 ","End":"02:59.410","Text":"We multiply both sides by 6 and then practice what we do,"},{"Start":"02:59.410 ","End":"03:02.295","Text":"is we say 2 goes into 6,"},{"Start":"03:02.295 ","End":"03:07.095","Text":"3 times and 3 goes into 6 twice."},{"Start":"03:07.095 ","End":"03:09.560","Text":"We just multiply this by this and this by this."},{"Start":"03:09.560 ","End":"03:13.760","Text":"The important thing is that this is positive,"},{"Start":"03:13.760 ","End":"03:16.940","Text":"because if not, then you couldn\u0027t do that."},{"Start":"03:16.940 ","End":"03:19.355","Text":"Even the 2 and the 3 are positive,"},{"Start":"03:19.355 ","End":"03:25.210","Text":"but that\u0027s less important. I\u0027ll just write it out."},{"Start":"03:25.210 ","End":"03:30.210","Text":"3 times 8x minus 4 is"},{"Start":"03:30.210 ","End":"03:37.725","Text":"less than 2 times 9 times x plus 1."},{"Start":"03:37.725 ","End":"03:39.600","Text":"Let\u0027s expand brackets."},{"Start":"03:39.600 ","End":"03:42.915","Text":"3 times 8x is 24x,"},{"Start":"03:42.915 ","End":"03:45.195","Text":"3 times 4 is 12,"},{"Start":"03:45.195 ","End":"03:49.650","Text":"but it\u0027s minus less than 2 times 9 is 18,"},{"Start":"03:49.650 ","End":"03:55.115","Text":"so it\u0027s 18 times x plus one, 18x plus 18."},{"Start":"03:55.115 ","End":"03:58.100","Text":"Now x is to the left numbers to the right."},{"Start":"03:58.100 ","End":"04:05.380","Text":"24 minus 18 is 6x,18 plus 12 is 30."},{"Start":"04:05.380 ","End":"04:07.830","Text":"Then we\u0027re going to divide by 6,"},{"Start":"04:07.830 ","End":"04:09.135","Text":"which is positive,"},{"Start":"04:09.135 ","End":"04:13.830","Text":"so we\u0027re left with x less than 5."},{"Start":"04:13.830 ","End":"04:16.004","Text":"On to the next."},{"Start":"04:16.004 ","End":"04:20.475","Text":"That would be Part C. Expand brackets first."},{"Start":"04:20.475 ","End":"04:28.080","Text":"4 times 6x is 24x minus 32 less than 8"},{"Start":"04:28.080 ","End":"04:35.910","Text":"times 3, 24x minus 32."},{"Start":"04:35.910 ","End":"04:37.505","Text":"When you look at that,"},{"Start":"04:37.505 ","End":"04:42.180","Text":"we have exactly the same on both sides."},{"Start":"04:42.180 ","End":"04:44.660","Text":"If we bring everything,"},{"Start":"04:44.660 ","End":"04:47.030","Text":"x\u0027s to the left, we get 0,"},{"Start":"04:47.030 ","End":"04:48.410","Text":"numbers to the right,"},{"Start":"04:48.410 ","End":"04:49.870","Text":"we get 0,"},{"Start":"04:49.870 ","End":"04:52.040","Text":"0 less than 0."},{"Start":"04:52.040 ","End":"04:55.625","Text":"Now, this happens to be false."},{"Start":"04:55.625 ","End":"05:00.770","Text":"If we get rid of x and we get something that\u0027s false,"},{"Start":"05:00.770 ","End":"05:03.950","Text":"then the answer is no solution,"},{"Start":"05:03.950 ","End":"05:08.735","Text":"no x or just no solution to the inequality."},{"Start":"05:08.735 ","End":"05:11.150","Text":"If we got something that was true,"},{"Start":"05:11.150 ","End":"05:13.670","Text":"then it would mean that every x is a solution,"},{"Start":"05:13.670 ","End":"05:14.810","Text":"which is not the case here,"},{"Start":"05:14.810 ","End":"05:17.030","Text":"but maybe in one of the next ones."},{"Start":"05:17.030 ","End":"05:19.194","Text":"That\u0027s no solution."},{"Start":"05:19.194 ","End":"05:23.270","Text":"Next, D. Fractions again,"},{"Start":"05:23.270 ","End":"05:25.670","Text":"common denominator is 12,"},{"Start":"05:25.670 ","End":"05:28.880","Text":"which is positive, so we don\u0027t have to worry about the direction."},{"Start":"05:28.880 ","End":"05:32.300","Text":"3 into 12 goes 4 times."},{"Start":"05:32.300 ","End":"05:38.805","Text":"4 into 12 goes 3 times and here this is like over 1,"},{"Start":"05:38.805 ","End":"05:42.845","Text":"so we just have to multiply this by the whole 12."},{"Start":"05:42.845 ","End":"05:47.775","Text":"Let\u0027s expand, 4 times x minus 6,"},{"Start":"05:47.775 ","End":"05:51.595","Text":"minus 3 times x minus 4,"},{"Start":"05:51.595 ","End":"05:56.630","Text":"bigger or equal to 12 times 12 minus x gives us,"},{"Start":"05:56.630 ","End":"06:04.280","Text":"4x minus 24 minus 3x plus 12."},{"Start":"06:04.280 ","End":"06:11.980","Text":"Note the plus bigger or equal to a 144 minus 12x."},{"Start":"06:11.980 ","End":"06:14.690","Text":"Although the inequalities including greater or equal to,"},{"Start":"06:14.690 ","End":"06:17.480","Text":"you can add and subtract and switch sides,"},{"Start":"06:17.480 ","End":"06:19.985","Text":"no problem, it\u0027s only with multiplication and division."},{"Start":"06:19.985 ","End":"06:23.775","Text":"Let\u0027s move all the x\u0027s to the left and what do we have?"},{"Start":"06:23.775 ","End":"06:25.695","Text":"Let us just do this mentally."},{"Start":"06:25.695 ","End":"06:30.525","Text":"4x minus 3x is x plus 12x,"},{"Start":"06:30.525 ","End":"06:33.915","Text":"that makes that 13x."},{"Start":"06:33.915 ","End":"06:35.835","Text":"See numbers on the right."},{"Start":"06:35.835 ","End":"06:41.835","Text":"I have 144 plus 24 minus 12,"},{"Start":"06:41.835 ","End":"06:45.855","Text":"I make that a 156."},{"Start":"06:45.855 ","End":"06:51.170","Text":"Then if we divide both sides by 13,"},{"Start":"06:51.170 ","End":"06:54.710","Text":"which is a positive number, important to stress,"},{"Start":"06:54.710 ","End":"06:59.430","Text":"then we keep the same inequality divided by 13,"},{"Start":"06:59.710 ","End":"07:05.960","Text":"this goes in exactly 12 times and this is our solution."},{"Start":"07:05.960 ","End":"07:09.035","Text":"Now we come to the last one,"},{"Start":"07:09.035 ","End":"07:11.853","Text":"which also has fractions,"},{"Start":"07:11.853 ","End":"07:15.265","Text":"and we need to find a common denominator."},{"Start":"07:15.265 ","End":"07:19.909","Text":"Always you can multiply the denominators and get a common denominator,"},{"Start":"07:19.909 ","End":"07:22.910","Text":"but that gives you big numbers sometimes better to look for"},{"Start":"07:22.910 ","End":"07:26.480","Text":"a lower common denominator because 5 goes into 10,"},{"Start":"07:26.480 ","End":"07:29.950","Text":"I can ignore the 5 and just find the denominator for 10 and 3,"},{"Start":"07:29.950 ","End":"07:32.435","Text":"in short 30 should do the trick."},{"Start":"07:32.435 ","End":"07:35.285","Text":"We put it over everything with 30."},{"Start":"07:35.285 ","End":"07:37.370","Text":"30 over 10 will give us 3."},{"Start":"07:37.370 ","End":"07:40.760","Text":"We have to multiply here by 3, 5 into 30,"},{"Start":"07:40.760 ","End":"07:45.550","Text":"you\u0027ve 6 times, 3 into 30 goes 10 times."},{"Start":"07:45.550 ","End":"07:49.680","Text":"I\u0027m overlapping a bit but that\u0027s okay."},{"Start":"07:49.680 ","End":"07:56.450","Text":"7 just gets multiplied by the whole 30 because it\u0027s 7 over 1."},{"Start":"07:56.450 ","End":"07:58.570","Text":"Anyway, so let\u0027s see."},{"Start":"07:58.570 ","End":"08:02.915","Text":"3 times 7 minus x,"},{"Start":"08:02.915 ","End":"08:04.010","Text":"there\u0027s a minus here,"},{"Start":"08:04.010 ","End":"08:05.528","Text":"so minus here,"},{"Start":"08:05.528 ","End":"08:11.560","Text":"and then 6 times 3x minus one"},{"Start":"08:11.560 ","End":"08:20.425","Text":"plus 10 times x plus 4 is less than 30 times 7."},{"Start":"08:20.425 ","End":"08:22.054","Text":"I forgot to stress,"},{"Start":"08:22.054 ","End":"08:24.530","Text":"30 is a positive number."},{"Start":"08:24.530 ","End":"08:26.330","Text":"Should have said that before."},{"Start":"08:26.330 ","End":"08:27.640","Text":"That way it\u0027s okay."},{"Start":"08:27.640 ","End":"08:30.049","Text":"We keep the inequality direction."},{"Start":"08:30.049 ","End":"08:36.560","Text":"Let\u0027s see, multiplying out 21 minus 3x minus"},{"Start":"08:36.560 ","End":"08:47.985","Text":"18x plus 6 plus 10x plus 40 is less than 210."},{"Start":"08:47.985 ","End":"08:52.765","Text":"Once again, we put the x\u0027s to the left, numbers to the right."},{"Start":"08:52.765 ","End":"08:53.940","Text":"Do you know what?"},{"Start":"08:53.940 ","End":"08:56.190","Text":"I\u0027ll just mark the x\u0027s so I can see them better."},{"Start":"08:56.190 ","End":"08:57.240","Text":"Here I have an x,"},{"Start":"08:57.240 ","End":"08:58.650","Text":"here I have an x,"},{"Start":"08:58.650 ","End":"09:01.700","Text":"here I have an x, and nowhere else."},{"Start":"09:01.700 ","End":"09:04.505","Text":"Lets see minus 3 minus 18,"},{"Start":"09:04.505 ","End":"09:06.695","Text":"is minus 21,"},{"Start":"09:06.695 ","End":"09:09.935","Text":"plus 10 is minus 11."},{"Start":"09:09.935 ","End":"09:13.910","Text":"I have minus 11x less than,"},{"Start":"09:13.910 ","End":"09:19.910","Text":"let\u0027s see for numbers 210 then minus 40,"},{"Start":"09:19.910 ","End":"09:24.655","Text":"that\u0027s 170, minus 6,"},{"Start":"09:24.655 ","End":"09:32.690","Text":"is 164, and minus 21 is,"},{"Start":"09:32.690 ","End":"09:36.680","Text":"I make it a 143."},{"Start":"09:36.680 ","End":"09:40.595","Text":"We want to divide by minus 11,"},{"Start":"09:40.595 ","End":"09:42.836","Text":"which is negative,"},{"Start":"09:42.836 ","End":"09:44.795","Text":"and so because it\u0027s negative,"},{"Start":"09:44.795 ","End":"09:48.095","Text":"we reverse the direction of the inequality."},{"Start":"09:48.095 ","End":"09:49.740","Text":"Divide this by minus 11,"},{"Start":"09:49.740 ","End":"09:55.310","Text":"we have just x. Divide this by minus 11 plus over minus is minus,"},{"Start":"09:55.310 ","End":"10:00.030","Text":"and it actually goes in exactly 13 times."},{"Start":"10:00.030 ","End":"10:03.740","Text":"This is the solution x bigger than minus 13,"},{"Start":"10:03.740 ","End":"10:06.480","Text":"and that\u0027s the last one. We\u0027re done."}],"Thumbnail":null,"ID":5055},{"Watched":false,"Name":"Exercise 3","Duration":"5m 55s","ChapterTopicVideoID":5054,"CourseChapterTopicPlaylistID":3875,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.490","Text":"In this exercise we have, let\u0027s see,"},{"Start":"00:02.490 ","End":"00:06.299","Text":"4 linear inequalities to solve."},{"Start":"00:06.299 ","End":"00:11.865","Text":"And I\u0027ve rewritten them below to get some more space."},{"Start":"00:11.865 ","End":"00:14.535","Text":"Let\u0027s start with the first."},{"Start":"00:14.535 ","End":"00:17.835","Text":"We\u0027ll open brackets, expand."},{"Start":"00:17.835 ","End":"00:22.470","Text":"We get 5x minus 15,"},{"Start":"00:22.470 ","End":"00:30.295","Text":"less than 2x minus 20 plus 3x."},{"Start":"00:30.295 ","End":"00:33.830","Text":"Let\u0027s collect terms x\u0027s on the left,"},{"Start":"00:33.830 ","End":"00:34.970","Text":"numbers on the right."},{"Start":"00:34.970 ","End":"00:37.055","Text":"We\u0027re allowed to do this with inequalities"},{"Start":"00:37.055 ","End":"00:39.830","Text":"because we\u0027re adding and subtracting from both sides,"},{"Start":"00:39.830 ","End":"00:41.525","Text":"which we can always do."},{"Start":"00:41.525 ","End":"00:46.470","Text":"What we get is 5x minus"},{"Start":"00:46.470 ","End":"00:57.180","Text":"2x minus 3x is less than minus 20 plus 15."},{"Start":"00:57.180 ","End":"01:01.470","Text":"Let\u0027s see. Aha, 5 minus 2 minus 3 is 0."},{"Start":"01:01.470 ","End":"01:11.740","Text":"So we get 0x or just 0 is less than minus 20 plus 15 is minus 5,"},{"Start":"01:11.740 ","End":"01:19.900","Text":"x doesn\u0027t appear here so this thing is either true or false and of course it\u0027s false."},{"Start":"01:25.820 ","End":"01:29.410","Text":"When we get a statement that is false without the x\u0027s,"},{"Start":"01:29.410 ","End":"01:32.425","Text":"it just means that there is no solution."},{"Start":"01:32.425 ","End":"01:40.660","Text":"We just write the word no solution or no x satisfies this inequality and that\u0027s it."},{"Start":"01:40.660 ","End":"01:43.205","Text":"On to the next one."},{"Start":"01:43.205 ","End":"01:48.789","Text":"In this one, we begin as usual by doing a bit of algebra,"},{"Start":"01:48.789 ","End":"01:51.020","Text":"manipulating each side,"},{"Start":"01:51.020 ","End":"01:53.755","Text":"2x minus 10,"},{"Start":"01:53.755 ","End":"01:55.865","Text":"bigger or equal to,"},{"Start":"01:55.865 ","End":"02:03.275","Text":"multiply it out a half times 4 is 2,a half times 6 is 3."},{"Start":"02:03.275 ","End":"02:05.540","Text":"We haven\u0027t done anything to each side."},{"Start":"02:05.540 ","End":"02:09.199","Text":"We just put things that are equal to it so there\u0027s no need to worry."},{"Start":"02:09.199 ","End":"02:13.490","Text":"Now we move variables to the left,"},{"Start":"02:13.490 ","End":"02:15.250","Text":"numbers to the right."},{"Start":"02:15.250 ","End":"02:22.240","Text":"2x minus 2x is bigger or equal to 10 plus 3."},{"Start":"02:22.240 ","End":"02:26.630","Text":"So we\u0027ve got that 0,"},{"Start":"02:26.630 ","End":"02:31.320","Text":"bigger or equal to 13."},{"Start":"02:31.780 ","End":"02:34.760","Text":"There is no x here."},{"Start":"02:34.760 ","End":"02:39.200","Text":"And when there\u0027s no x, we just check if this is a true statement."},{"Start":"02:39.200 ","End":"02:45.645","Text":"Or as is the case here, this is false."},{"Start":"02:45.645 ","End":"02:48.304","Text":"Because this is false,"},{"Start":"02:48.304 ","End":"02:53.780","Text":"then the original inequality has no solution."},{"Start":"02:53.780 ","End":"02:58.710","Text":"There is no x for which this inequality will hold."},{"Start":"02:59.180 ","End":"03:03.600","Text":"For example, suppose you put x=0,"},{"Start":"03:03.600 ","End":"03:06.330","Text":"you would see minus 10 is bigger or"},{"Start":"03:06.330 ","End":"03:06.720","Text":"equal"},{"Start":"03:06.720 ","End":"03:08.020","Text":"to"},{"Start":"03:15.890 ","End":"03:17.655","Text":"3,"},{"Start":"03:17.655 ","End":"03:21.520","Text":"which is certainly false."},{"Start":"03:21.520 ","End":"03:26.030","Text":"No x that you put in here will make this thing work."},{"Start":"03:26.030 ","End":"03:28.420","Text":"Onto the next one."},{"Start":"03:28.420 ","End":"03:32.750","Text":"Number C. As usual,"},{"Start":"03:32.750 ","End":"03:35.390","Text":"do a bit of algebra to simplify each side."},{"Start":"03:35.390 ","End":"03:42.320","Text":"So here we get 6 times x over 3 is 2x 6 times one is 6."},{"Start":"03:42.320 ","End":"03:47.705","Text":"It\u0027s a bigger or equal to type of inequality 2x plus 3,"},{"Start":"03:47.705 ","End":"03:49.955","Text":"bring the x\u0027s to the left,"},{"Start":"03:49.955 ","End":"03:51.845","Text":"numbers to the right,"},{"Start":"03:51.845 ","End":"03:58.660","Text":"we get 0 bigger or equal to minus 3."},{"Start":"03:58.660 ","End":"04:01.475","Text":"Once again, x has dropped out."},{"Start":"04:01.475 ","End":"04:04.415","Text":"But this time it\u0027s a true statement."},{"Start":"04:04.415 ","End":"04:08.960","Text":"This is true, 0 is bigger or equal to minus 3."},{"Start":"04:08.960 ","End":"04:10.340","Text":"In fact, it\u0027s bigger than,"},{"Start":"04:10.340 ","End":"04:12.350","Text":"but this thing covers both cases."},{"Start":"04:12.350 ","End":"04:13.640","Text":"If it was equal, that would be good."},{"Start":"04:13.640 ","End":"04:15.080","Text":"If it\u0027s bigger than it\u0027s good,"},{"Start":"04:15.080 ","End":"04:17.630","Text":"this is true, and when this happens,"},{"Start":"04:17.630 ","End":"04:23.505","Text":"then we say that this inequality holds for all x,"},{"Start":"04:23.505 ","End":"04:26.090","Text":"every x you put in here any value,"},{"Start":"04:26.090 ","End":"04:28.860","Text":"the inequality will come out true."},{"Start":"04:30.540 ","End":"04:34.650","Text":"Last one. Let\u0027s see,"},{"Start":"04:34.650 ","End":"04:41.845","Text":"63 times 2x minus 6 over 2."},{"Start":"04:41.845 ","End":"04:45.455","Text":"I don\u0027t have to divide 63 by 2."},{"Start":"04:45.455 ","End":"04:49.880","Text":"I can take 2 out of the numerator here and get rid of fractions that way."},{"Start":"04:49.880 ","End":"04:51.110","Text":"Let me just do that bit."},{"Start":"04:51.110 ","End":"04:52.400","Text":"This Is 63,"},{"Start":"04:52.400 ","End":"04:54.875","Text":"now if I take 2 out of here,"},{"Start":"04:54.875 ","End":"04:57.215","Text":"I\u0027ll get twice x minus 3."},{"Start":"04:57.215 ","End":"05:05.145","Text":"So this bit is just x minus 3 plus 7 minus x bigger than 0."},{"Start":"05:05.145 ","End":"05:10.060","Text":"Yeah, that was the easiest way to get rid of this fraction."},{"Start":"05:10.720 ","End":"05:19.515","Text":"Now simplify, what\u0027s the brackets."},{"Start":"05:19.515 ","End":"05:21.860","Text":"X minus 3 plus 7 minus x."},{"Start":"05:21.860 ","End":"05:30.650","Text":"Well, the x cancels and minus 3 plus 7 is 4 so we have 63 times 4 bigger than 0."},{"Start":"05:30.650 ","End":"05:33.320","Text":"I didn\u0027t have to multiply this out to know that it\u0027s"},{"Start":"05:33.320 ","End":"05:35.750","Text":"true because positive times positive is certainly"},{"Start":"05:35.750 ","End":"05:41.070","Text":"positive so here this is a true statement without x,"},{"Start":"05:41.070 ","End":"05:44.225","Text":"and when it comes out true as in the previous exercise,"},{"Start":"05:44.225 ","End":"05:47.960","Text":"that means that the inequality is true for all x."},{"Start":"05:47.960 ","End":"05:51.515","Text":"Every x is a solution to this inequality."},{"Start":"05:51.515 ","End":"05:53.120","Text":"And this is the last part."},{"Start":"05:53.120 ","End":"05:55.140","Text":"So we are done."}],"Thumbnail":null,"ID":5056},{"Watched":false,"Name":"Exercise 4","Duration":"13m 54s","ChapterTopicVideoID":5055,"CourseChapterTopicPlaylistID":3875,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.350 ","End":"00:05.160","Text":"In this exercise, there are 5 different inequalities,"},{"Start":"00:05.160 ","End":"00:13.690","Text":"but each of them is made up from 2 separate inequalities combined with the word and."},{"Start":"00:14.000 ","End":"00:18.465","Text":"Whenever we have the word and in an inequality,"},{"Start":"00:18.465 ","End":"00:24.075","Text":"it implies that we solve each 1 separately and then take the intersection,"},{"Start":"00:24.075 ","End":"00:27.000","Text":"written this way sometimes or the word"},{"Start":"00:27.000 ","End":"00:32.760","Text":"intersection of the 2 intervals that we get from each 1 separately."},{"Start":"00:32.760 ","End":"00:36.790","Text":"Let\u0027s start with the first example a."},{"Start":"00:37.640 ","End":"00:41.720","Text":"By the way, you should review intersection of"},{"Start":"00:41.720 ","End":"00:46.625","Text":"intervals if you forgotten it already or something."},{"Start":"00:46.625 ","End":"00:53.435","Text":"First 1, 6x minus 8 less than 2x plus 12 bring the x\u0027s to the left."},{"Start":"00:53.435 ","End":"00:56.135","Text":"For x, bring the numbers to the right."},{"Start":"00:56.135 ","End":"01:01.970","Text":"20 divided by 4x less than 5."},{"Start":"01:01.970 ","End":"01:03.910","Text":"That\u0027s the first 1."},{"Start":"01:03.910 ","End":"01:06.000","Text":"Now the second 1,"},{"Start":"01:06.000 ","End":"01:09.270","Text":"7 minus 6 bigger than 1,"},{"Start":"01:09.270 ","End":"01:13.350","Text":"7x is bigger than 7 divide by 7."},{"Start":"01:13.350 ","End":"01:15.840","Text":"X is bigger than 1."},{"Start":"01:15.840 ","End":"01:20.595","Text":"Now these 2 are combined with and."},{"Start":"01:20.595 ","End":"01:24.050","Text":"I need this and this."},{"Start":"01:24.050 ","End":"01:27.550","Text":"You can do it with a sketch or without a sketch."},{"Start":"01:27.550 ","End":"01:33.910","Text":"Without a sketch, you would just say x less than 5 and x bigger than 1."},{"Start":"01:33.910 ","End":"01:38.095","Text":"We would just get that x is between 5 and 1,"},{"Start":"01:38.095 ","End":"01:40.945","Text":"less than 5 and bigger than 1."},{"Start":"01:40.945 ","End":"01:44.325","Text":"We can write it as a double inequality."},{"Start":"01:44.325 ","End":"01:47.160","Text":"This thing is called a double inequality."},{"Start":"01:47.160 ","End":"01:49.035","Text":"This is the answer."},{"Start":"01:49.035 ","End":"01:55.510","Text":"But it\u0027s often a customary to draw a little sketch."},{"Start":"01:55.510 ","End":"01:58.240","Text":"I start with a number line."},{"Start":"01:58.240 ","End":"02:00.775","Text":"Numbers increase to the right of course,"},{"Start":"02:00.775 ","End":"02:05.170","Text":"they don\u0027t even have to put the 0 in just the numbers that are interesting,"},{"Start":"02:05.170 ","End":"02:08.040","Text":"which is 5 and 1."},{"Start":"02:08.040 ","End":"02:10.725","Text":"It doesn\u0027t have to be to any scale."},{"Start":"02:10.725 ","End":"02:15.330","Text":"The important thing is that the 5 should be to the right of 1."},{"Start":"02:15.330 ","End":"02:17.645","Text":"They should be in the right order at least."},{"Start":"02:17.645 ","End":"02:23.195","Text":"Now, the picture for the inequality x less than 5,"},{"Start":"02:23.195 ","End":"02:27.780","Text":"the standard way of doing it is to draw"},{"Start":"02:28.870 ","End":"02:37.320","Text":"a hollow circle somewhere above the 5 and less than means we\u0027re going in this direction."},{"Start":"02:37.840 ","End":"02:41.015","Text":"This goes on endlessly,"},{"Start":"02:41.015 ","End":"02:43.534","Text":"could say to minus infinity."},{"Start":"02:43.534 ","End":"02:48.920","Text":"The other 1, x bigger than 1 means we start from"},{"Start":"02:48.920 ","End":"02:56.345","Text":"1 and now it\u0027s also a hollow circle."},{"Start":"02:56.345 ","End":"03:01.010","Text":"Take 2. This 1 is also a hollow circle,"},{"Start":"03:01.010 ","End":"03:04.805","Text":"which means that the 1 is not included because 1 isn\u0027t included."},{"Start":"03:04.805 ","End":"03:08.400","Text":"Bigger than 1 means to the right of 1."},{"Start":"03:09.820 ","End":"03:13.055","Text":"I\u0027ll put a little arrow here to show that it goes on without"},{"Start":"03:13.055 ","End":"03:15.845","Text":"end all the way to infinity if you want."},{"Start":"03:15.845 ","End":"03:19.625","Text":"Now the intersection of these 2 intervals,"},{"Start":"03:19.625 ","End":"03:25.310","Text":"we get the intersection from the keyword and is to find whatever they have in common."},{"Start":"03:25.310 ","End":"03:27.140","Text":"The overlap, if you like."},{"Start":"03:27.140 ","End":"03:34.070","Text":"The overlap, would be this part here."},{"Start":"03:34.070 ","End":"03:39.560","Text":"If we look at it. This part here is from 1-5,"},{"Start":"03:39.560 ","End":"03:43.260","Text":"but not including the 1 on not including the 5."},{"Start":"03:43.360 ","End":"03:48.245","Text":"Hollow circle here and a hollow circle here."},{"Start":"03:48.245 ","End":"03:50.360","Text":"Join them with a line."},{"Start":"03:50.360 ","End":"03:57.295","Text":"This represents clearly 1 less than x, less than 5."},{"Start":"03:57.295 ","End":"04:01.100","Text":"Sometimes, if you\u0027ve studied interval notation,"},{"Start":"04:01.100 ","End":"04:07.740","Text":"we would write it as the interval from 1-5."},{"Start":"04:09.800 ","End":"04:14.560","Text":"Take 2 need to edit out the last bit."},{"Start":"04:14.560 ","End":"04:17.825","Text":"Once we found the sketch of the solution,"},{"Start":"04:17.825 ","End":"04:22.880","Text":"then we write it in mathematical terms, which is this here."},{"Start":"04:22.880 ","End":"04:23.900","Text":"But if we didn\u0027t have this,"},{"Start":"04:23.900 ","End":"04:25.640","Text":"we just say, okay, here\u0027s 1."},{"Start":"04:25.640 ","End":"04:28.140","Text":"Here is the larger 1,"},{"Start":"04:28.140 ","End":"04:30.680","Text":"5. Does it include?"},{"Start":"04:30.680 ","End":"04:32.990","Text":"No, so it\u0027s less than otherwise it would be"},{"Start":"04:32.990 ","End":"04:36.020","Text":"less than or equal to if it included and so on."},{"Start":"04:36.020 ","End":"04:39.560","Text":"From this picture, we write it this way."},{"Start":"04:39.560 ","End":"04:45.930","Text":"But like I said, some people don\u0027t need the picture or an easy cases you don\u0027t."},{"Start":"04:46.010 ","End":"04:53.190","Text":"Saves doing a sketch bigger than 1 and less than 5 means between 1 and 5."},{"Start":"04:53.190 ","End":"04:57.130","Text":"Enough said about that onto part B."},{"Start":"04:57.340 ","End":"05:01.095","Text":"Which 1 separately here I have"},{"Start":"05:01.095 ","End":"05:09.440","Text":"2x plus 8 minus 3 bigger than minus 7."},{"Start":"05:09.440 ","End":"05:11.390","Text":"X is to the left."},{"Start":"05:11.390 ","End":"05:18.445","Text":"There\u0027s just 2x here numbers to the right minus 7."},{"Start":"05:18.445 ","End":"05:23.145","Text":"Minus 8 is minus 15 plus 3,"},{"Start":"05:23.145 ","End":"05:26.420","Text":"minus 12, then divide by 2,"},{"Start":"05:26.420 ","End":"05:28.565","Text":"which is a positive number."},{"Start":"05:28.565 ","End":"05:34.370","Text":"It\u0027s okay. X is bigger than minus 6, the first 1."},{"Start":"05:34.370 ","End":"05:36.630","Text":"Now the next 1."},{"Start":"05:36.770 ","End":"05:44.105","Text":"5x plus 6 the same on the left and on the right."},{"Start":"05:44.105 ","End":"05:48.840","Text":"9 minus 3 is 6."},{"Start":"05:49.180 ","End":"05:56.020","Text":"Minus, minus is plus x. X\u0027s to the left."},{"Start":"05:56.020 ","End":"05:59.170","Text":"For x numbers to the right,"},{"Start":"05:59.170 ","End":"06:04.180","Text":"6 minus 6 is 0 divide by 4 which is positive."},{"Start":"06:04.180 ","End":"06:07.880","Text":"X less than or equal to 0."},{"Start":"06:09.350 ","End":"06:13.210","Text":"I\u0027ll use the picture method again,"},{"Start":"06:13.210 ","End":"06:16.610","Text":"although as I said, we can do it without."},{"Start":"06:18.020 ","End":"06:22.435","Text":"I\u0027m going to recycle this picture here and reuse it."},{"Start":"06:22.435 ","End":"06:25.780","Text":"First of all, I have to change the numbers here."},{"Start":"06:25.780 ","End":"06:31.705","Text":"The numbers I have in the right order minus 6 and 0."},{"Start":"06:31.705 ","End":"06:37.075","Text":"The other difference is that here there is a less than or equal to."},{"Start":"06:37.075 ","End":"06:39.620","Text":"This includes the 0,"},{"Start":"06:39.620 ","End":"06:42.940","Text":"this becomes a solid circle."},{"Start":"06:42.940 ","End":"06:45.245","Text":"Also on the overlap,"},{"Start":"06:45.245 ","End":"06:47.660","Text":"this will have to be solid also."},{"Start":"06:47.660 ","End":"06:49.625","Text":"Now from this picture,"},{"Start":"06:49.625 ","End":"06:54.830","Text":"this interval becomes minus 6 and then"},{"Start":"06:54.830 ","End":"07:00.470","Text":"less than without the equals because here it\u0027s hollow circle than x,"},{"Start":"07:00.470 ","End":"07:05.260","Text":"then less than or equal to because it includes the 0."},{"Start":"07:05.260 ","End":"07:08.110","Text":"That\u0027s the answer."},{"Start":"07:08.320 ","End":"07:10.850","Text":"If you can get by without a picture,"},{"Start":"07:10.850 ","End":"07:16.955","Text":"you say, it\u0027s bigger than minus 6 I mean and less than or equal to 0."},{"Start":"07:16.955 ","End":"07:19.045","Text":"It\u0027s just a double inequality."},{"Start":"07:19.045 ","End":"07:24.170","Text":"Next 1 is part C. Let\u0027s see."},{"Start":"07:24.170 ","End":"07:28.190","Text":"We first of all, solve each inequality separately."},{"Start":"07:28.190 ","End":"07:37.135","Text":"The first inequality gives 10x less than 4x minus 12."},{"Start":"07:37.135 ","End":"07:47.570","Text":"6x is less than minus 12 and then divide by 6x less than minus 2."},{"Start":"07:47.570 ","End":"07:49.655","Text":"That\u0027s the left."},{"Start":"07:49.655 ","End":"07:51.965","Text":"I mean, that\u0027s the first 1."},{"Start":"07:51.965 ","End":"07:59.605","Text":"The second 1 is 3x minus x plus 8,"},{"Start":"07:59.605 ","End":"08:04.945","Text":"bigger than minus 13 minus x. X is on the left,"},{"Start":"08:04.945 ","End":"08:07.800","Text":"3x minus x plus x."},{"Start":"08:07.800 ","End":"08:12.360","Text":"That\u0027s just 3x bigger than,"},{"Start":"08:12.360 ","End":"08:15.600","Text":"let\u0027s see, minus 13,"},{"Start":"08:15.600 ","End":"08:24.570","Text":"minus 8 is minus 21 divide by 3, which is positive."},{"Start":"08:24.570 ","End":"08:29.155","Text":"X is bigger than minus 7."},{"Start":"08:29.155 ","End":"08:33.085","Text":"Now I have to combine these 2 with"},{"Start":"08:33.085 ","End":"08:40.100","Text":"and got this and this."},{"Start":"08:40.100 ","End":"08:43.020","Text":"We draw a diagram."},{"Start":"08:43.020 ","End":"08:47.650","Text":"I reuse the first diagram and all I have to do is change the numbers"},{"Start":"08:47.650 ","End":"08:51.920","Text":"down here to minus 7 and minus 2."},{"Start":"08:51.920 ","End":"08:54.130","Text":"You just have to worry to get them in the right order."},{"Start":"08:54.130 ","End":"08:57.310","Text":"Of course, minus 7 is to the left of minus 2,"},{"Start":"08:57.310 ","End":"08:59.815","Text":"and the result is the same."},{"Start":"08:59.815 ","End":"09:03.860","Text":"The inequalities do not include that hollow circles."},{"Start":"09:03.860 ","End":"09:07.325","Text":"This is the overlap between these 2."},{"Start":"09:07.325 ","End":"09:09.640","Text":"I mean, this is the x less than minus 2,"},{"Start":"09:09.640 ","End":"09:11.740","Text":"this is the x bigger than minus 7."},{"Start":"09:11.740 ","End":"09:15.310","Text":"This is x between minus 7 and minus 2,"},{"Start":"09:15.310 ","End":"09:20.309","Text":"which we write as minus 7 less than x,"},{"Start":"09:20.309 ","End":"09:21.830","Text":"less than minus 2."},{"Start":"09:21.830 ","End":"09:28.060","Text":"A double inequalities is called sometimes when there\u0027s 2 inequality signs."},{"Start":"09:31.640 ","End":"09:36.010","Text":"There\u0027s 5 parts are here."},{"Start":"09:36.470 ","End":"09:39.850","Text":"Go on to part D. Again,"},{"Start":"09:39.850 ","End":"09:42.500","Text":"we solve each 1 separately."},{"Start":"09:42.500 ","End":"09:45.110","Text":"I see that here there\u0027s nothing to expand."},{"Start":"09:45.110 ","End":"09:48.050","Text":"We just have to put x\u0027s on the left numbers on the right."},{"Start":"09:48.050 ","End":"09:57.390","Text":"In this 1, it\u0027s 6x minus 2x is 4x plus 3x is 7x on this side."},{"Start":"09:57.390 ","End":"10:04.970","Text":"On this side, 20 plus 15 is 35,"},{"Start":"10:04.970 ","End":"10:06.110","Text":"dividing by 7,"},{"Start":"10:06.110 ","End":"10:09.320","Text":"which is positive, we get x less than 5."},{"Start":"10:09.320 ","End":"10:14.080","Text":"Here, x\u0027s on the left, numbers on the right."},{"Start":"10:14.080 ","End":"10:17.500","Text":"3x minus 3x is 0,"},{"Start":"10:17.500 ","End":"10:22.830","Text":"bigger or equal to 0, 131 minus 131."},{"Start":"10:22.830 ","End":"10:28.340","Text":"Now, when you get something without x,"},{"Start":"10:28.340 ","End":"10:32.040","Text":"this happens to be true."},{"Start":"10:32.230 ","End":"10:35.285","Text":"Because 0 is bigger or equal to 0."},{"Start":"10:35.285 ","End":"10:37.160","Text":"It\u0027s not bigger than it\u0027s equal to,"},{"Start":"10:37.160 ","End":"10:38.525","Text":"but either 1 will do."},{"Start":"10:38.525 ","End":"10:40.505","Text":"When you get this without x."},{"Start":"10:40.505 ","End":"10:43.880","Text":"It\u0027s true for all x."},{"Start":"10:43.880 ","End":"10:46.840","Text":"The answer is all x."},{"Start":"10:46.840 ","End":"10:51.820","Text":"Then there\u0027s no need for a picture."},{"Start":"10:51.940 ","End":"10:59.390","Text":"Basically, this is true for all x if we want to say that x is less than 5."},{"Start":"10:59.390 ","End":"11:02.570","Text":"Here all x means no restriction on x."},{"Start":"11:02.570 ","End":"11:06.685","Text":"Basically means like there\u0027s no restriction."},{"Start":"11:06.685 ","End":"11:12.055","Text":"It just means that x is less than 5."},{"Start":"11:12.055 ","End":"11:14.285","Text":"This doesn\u0027t add anything."},{"Start":"11:14.285 ","End":"11:15.800","Text":"Every x satisfies this,"},{"Start":"11:15.800 ","End":"11:20.000","Text":"there\u0027s no need to do an intersection with anything."},{"Start":"11:20.000 ","End":"11:23.279","Text":"It\u0027s possible to draw a picture."},{"Start":"11:23.730 ","End":"11:27.580","Text":"I copied parts of the picture from part 1."},{"Start":"11:27.580 ","End":"11:30.280","Text":"This is the x less than 5."},{"Start":"11:30.280 ","End":"11:39.220","Text":"The way we, we show all x is to draw a line from infinity to minus infinity."},{"Start":"11:39.220 ","End":"11:41.530","Text":"This is the old x."},{"Start":"11:41.530 ","End":"11:45.320","Text":"When we take the overlap."},{"Start":"11:46.320 ","End":"11:51.280","Text":"Then the overlap is over of this."},{"Start":"11:51.280 ","End":"11:55.555","Text":"They both go on indefinitely to minus infinity."},{"Start":"11:55.555 ","End":"11:59.665","Text":"The overlap is exactly the top 1."},{"Start":"11:59.665 ","End":"12:06.345","Text":"It\u0027s just this and everything lower row to the left of it,"},{"Start":"12:06.345 ","End":"12:09.260","Text":"which is exactly the same as this bit."},{"Start":"12:09.260 ","End":"12:11.930","Text":"It\u0027s just the x less than 5."},{"Start":"12:11.930 ","End":"12:13.790","Text":"We can draw a picture here,"},{"Start":"12:13.790 ","End":"12:16.400","Text":"but I don\u0027t know if it helps or not."},{"Start":"12:16.400 ","End":"12:19.280","Text":"Whenever you have all x, it\u0027s best just to say no restriction,"},{"Start":"12:19.280 ","End":"12:21.065","Text":"just throw it out."},{"Start":"12:21.065 ","End":"12:26.310","Text":"When it\u0027s an and. The case of or it\u0027s something else."},{"Start":"12:27.700 ","End":"12:33.800","Text":"Onto final part. In this 1,"},{"Start":"12:33.800 ","End":"12:36.290","Text":"if we solve this inequality,"},{"Start":"12:36.290 ","End":"12:43.500","Text":"we see that the x\u0027s just cancel 4x minus 4x is 0 and 3 plus 2 is 5,"},{"Start":"12:43.500 ","End":"12:46.590","Text":"we get 0 bigger than 5."},{"Start":"12:46.590 ","End":"12:48.635","Text":"Again, the x\u0027s dropped out."},{"Start":"12:48.635 ","End":"12:54.770","Text":"But in contrast to the previous exercise here also x dropped out,"},{"Start":"12:54.770 ","End":"12:56.434","Text":"but we got a true statement."},{"Start":"12:56.434 ","End":"12:59.975","Text":"This most certainly is a false statement."},{"Start":"12:59.975 ","End":"13:04.110","Text":"Certainly 0 is not bigger than 5."},{"Start":"13:04.130 ","End":"13:08.930","Text":"Actually, it doesn\u0027t even matter what happens here."},{"Start":"13:08.930 ","End":"13:12.710","Text":"Because as soon as we have a false statement,"},{"Start":"13:12.710 ","End":"13:15.515","Text":"there\u0027s no x that\u0027s going to satisfy this."},{"Start":"13:15.515 ","End":"13:22.024","Text":"I could write no such x or no x or no solution to this 1."},{"Start":"13:22.024 ","End":"13:24.200","Text":"There\u0027s no solution to this 1."},{"Start":"13:24.200 ","End":"13:27.475","Text":"We can\u0027t something that\u0027s a solution to both of them,"},{"Start":"13:27.475 ","End":"13:30.815","Text":"can\u0027t have an x that satisfies this and this,"},{"Start":"13:30.815 ","End":"13:34.010","Text":"because it doesn\u0027t satisfy this, it just sold."},{"Start":"13:34.010 ","End":"13:37.070","Text":"The answer stays that there is no x."},{"Start":"13:37.070 ","End":"13:44.430","Text":"Sometimes we say the solution set is the empty set, sometimes you\u0027ll see."},{"Start":"13:45.190 ","End":"13:48.155","Text":"Anyway, we just say there\u0027s no solution."},{"Start":"13:48.155 ","End":"13:50.870","Text":"The fact that you\u0027re better than other condition is not going to suddenly"},{"Start":"13:50.870 ","End":"13:54.420","Text":"create a solution. That\u0027s it."}],"Thumbnail":null,"ID":5057},{"Watched":false,"Name":"Exercise 5","Duration":"12m 44s","ChapterTopicVideoID":5056,"CourseChapterTopicPlaylistID":3875,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.740","Text":"In this exercise, we have a bunch of double inequalities."},{"Start":"00:04.740 ","End":"00:07.048","Text":"Each one is something less than something,"},{"Start":"00:07.048 ","End":"00:10.155","Text":"less than something, or a different inequality."},{"Start":"00:10.155 ","End":"00:15.165","Text":"The idea is to break it up into two inequalities."},{"Start":"00:15.165 ","End":"00:22.404","Text":"In general, if we say something like a is less than b less than c,"},{"Start":"00:22.404 ","End":"00:28.710","Text":"then this means that a is less than b"},{"Start":"00:28.710 ","End":"00:32.910","Text":"and b is less than c. This is"},{"Start":"00:32.910 ","End":"00:37.620","Text":"the general strategy that we\u0027re going to use in these exercises."},{"Start":"00:37.620 ","End":"00:40.270","Text":"Let\u0027s start."},{"Start":"00:41.090 ","End":"00:44.385","Text":"In the first one,"},{"Start":"00:44.385 ","End":"00:51.510","Text":"we can rewrite it as 3 less than x plus"},{"Start":"00:51.510 ","End":"00:58.920","Text":"1 and x plus 1 less than 5."},{"Start":"00:58.920 ","End":"01:05.730","Text":"This one if we subtract 1 we get 2 less than x,"},{"Start":"01:05.730 ","End":"01:08.620","Text":"which means that x is bigger than 2."},{"Start":"01:08.620 ","End":"01:10.820","Text":"We could write it the other way round."},{"Start":"01:10.820 ","End":"01:17.810","Text":"This one gives us that x is less than 4 by subtracting 1."},{"Start":"01:17.810 ","End":"01:24.930","Text":"What we have is x bigger than 2 and x less than 4."},{"Start":"01:24.930 ","End":"01:27.975","Text":"When we combine them,"},{"Start":"01:27.975 ","End":"01:30.330","Text":"I\u0027m not going to draw the diagram like previously,"},{"Start":"01:30.330 ","End":"01:34.500","Text":"we can see that this means that, in fact,"},{"Start":"01:34.500 ","End":"01:36.015","Text":"I\u0027ll use this form here,"},{"Start":"01:36.015 ","End":"01:39.510","Text":"2 less than x less than 4."},{"Start":"01:39.510 ","End":"01:42.790","Text":"X is between 2 and 4, not inclusive."},{"Start":"01:42.790 ","End":"01:46.070","Text":"Sometimes, as in this case,"},{"Start":"01:46.070 ","End":"01:50.356","Text":"there is a shortcut way but it usually doesn\u0027t work,"},{"Start":"01:50.356 ","End":"01:51.455","Text":"but here it does."},{"Start":"01:51.455 ","End":"01:54.980","Text":"We can work on all sides or three sides of the equation,"},{"Start":"01:54.980 ","End":"01:57.140","Text":"so to speak, simultaneously."},{"Start":"01:57.140 ","End":"02:01.875","Text":"We could say 3 less than x plus 1 less than 5."},{"Start":"02:01.875 ","End":"02:07.305","Text":"Then we can subtract 1 from all three sides."},{"Start":"02:07.305 ","End":"02:09.430","Text":"Subtracting 1 we get 2,"},{"Start":"02:09.430 ","End":"02:12.615","Text":"subtracting 1 from here we just get x,"},{"Start":"02:12.615 ","End":"02:15.180","Text":"and subtracting 1 we get 4."},{"Start":"02:15.180 ","End":"02:17.720","Text":"This gives us the answer right away,"},{"Start":"02:17.720 ","End":"02:19.510","Text":"but this only sometimes works."},{"Start":"02:19.510 ","End":"02:25.355","Text":"For example, if we have just x in the middle plus something,"},{"Start":"02:25.355 ","End":"02:27.380","Text":"can\u0027t say exactly when,"},{"Start":"02:27.380 ","End":"02:28.610","Text":"but in certain occasions,"},{"Start":"02:28.610 ","End":"02:31.060","Text":"it\u0027s clear that we can work on all three of them together."},{"Start":"02:31.060 ","End":"02:35.540","Text":"In the remaining ones I\u0027ve checked it doesn\u0027t work,"},{"Start":"02:35.540 ","End":"02:40.730","Text":"so we\u0027re going to have to break each one from b to e into two separate inequalities."},{"Start":"02:40.730 ","End":"02:49.205","Text":"Let\u0027s start with b. I write 6x minus 38 less than or equal to x minus 3"},{"Start":"02:49.205 ","End":"02:54.420","Text":"and x minus 3 less than or equal"},{"Start":"02:54.420 ","End":"03:00.445","Text":"to 5x plus 7 and solve each one separately."},{"Start":"03:00.445 ","End":"03:02.505","Text":"Bringing the xs to the left,"},{"Start":"03:02.505 ","End":"03:05.610","Text":"it\u0027s 5x and the numbers to the right,"},{"Start":"03:05.610 ","End":"03:10.530","Text":"38 minus 3 is 35,"},{"Start":"03:10.530 ","End":"03:14.295","Text":"and then divide by 5 which is a positive number."},{"Start":"03:14.295 ","End":"03:17.530","Text":"X is less than or equal to 7."},{"Start":"03:17.530 ","End":"03:19.775","Text":"That\u0027s the first one. The second one,"},{"Start":"03:19.775 ","End":"03:22.070","Text":"bring the xs to the left."},{"Start":"03:22.070 ","End":"03:30.550","Text":"I\u0027ve got minus 4x less than or equal to adding the 3 to the other side."},{"Start":"03:30.550 ","End":"03:32.320","Text":"I mean minus becomes plus,"},{"Start":"03:32.320 ","End":"03:38.280","Text":"so this is 10 and then dividing by minus 4."},{"Start":"03:38.280 ","End":"03:41.920","Text":"Here because I\u0027m dividing by a negative number,"},{"Start":"03:41.920 ","End":"03:50.880","Text":"I have to change the sign of the inequality so x is greater or equal to 10 over minus 4,"},{"Start":"03:50.880 ","End":"03:53.500","Text":"that\u0027s what\u0027s minus, and 10 over 4 is 2."},{"Start":"03:53.500 ","End":"03:58.055","Text":"5. Then combining this,"},{"Start":"03:58.055 ","End":"04:03.310","Text":"this, and this we don\u0027t need a diagram."},{"Start":"04:03.310 ","End":"04:11.020","Text":"I think it\u0027s clear that because negative 2.5 is less than 7,"},{"Start":"04:11.020 ","End":"04:14.950","Text":"so we\u0027re okay if we just say that minus"},{"Start":"04:14.950 ","End":"04:21.605","Text":"2.5 less than or equal to x is less than or equal to 7."},{"Start":"04:21.605 ","End":"04:23.230","Text":"X is between the two."},{"Start":"04:23.230 ","End":"04:24.940","Text":"Actually, this is what this means,"},{"Start":"04:24.940 ","End":"04:29.530","Text":"that x is less than or equal to 7 and x is bigger or equal to minus 2.5."},{"Start":"04:29.530 ","End":"04:31.685","Text":"No diagram needed."},{"Start":"04:31.685 ","End":"04:34.470","Text":"Let\u0027s go on to the next one."},{"Start":"04:34.470 ","End":"04:36.730","Text":"I\u0027ve got to underline the answer there."},{"Start":"04:36.730 ","End":"04:41.845","Text":"That\u0027s that. Then we can go on to part c and split it up into two inequalities."},{"Start":"04:41.845 ","End":"04:50.910","Text":"6x minus 1 plus 3 less than 7x plus 14 and the other one,"},{"Start":"04:50.910 ","End":"05:01.040","Text":"the same 7x plus 14 less than 4 times 2x minus 1 plus 12."},{"Start":"05:01.040 ","End":"05:06.860","Text":"Now we do the work of simplifying and collecting 6x minus"},{"Start":"05:06.860 ","End":"05:13.605","Text":"6 plus 3 less than 7x plus 14."},{"Start":"05:13.605 ","End":"05:14.900","Text":"We\u0027ve expanded the brackets."},{"Start":"05:14.900 ","End":"05:17.090","Text":"Now let\u0027s collect xs on the left."},{"Start":"05:17.090 ","End":"05:23.040","Text":"6x minus 7x is minus x and on the right,"},{"Start":"05:23.040 ","End":"05:29.790","Text":"14 plus 6 minus 3 is 17."},{"Start":"05:29.790 ","End":"05:36.570","Text":"Now we\u0027re going to divide by minus 1."},{"Start":"05:36.570 ","End":"05:40.438","Text":"We need to reverse the sign of the inequality,"},{"Start":"05:40.438 ","End":"05:44.570","Text":"and we get x is bigger than minus 17."},{"Start":"05:44.570 ","End":"05:48.453","Text":"The other one,"},{"Start":"05:48.453 ","End":"05:52.334","Text":"7x plus 14 less than,"},{"Start":"05:52.334 ","End":"05:53.717","Text":"here we have,"},{"Start":"05:53.717 ","End":"05:58.230","Text":"8x minus 4 plus 12."},{"Start":"05:58.230 ","End":"06:00.225","Text":"X is on the left."},{"Start":"06:00.225 ","End":"06:06.490","Text":"Again it looks like minus x,"},{"Start":"06:06.490 ","End":"06:12.120","Text":"7 minus 8 is less than, well, let\u0027s see,"},{"Start":"06:12.120 ","End":"06:17.550","Text":"minus 14 minus 4 plus 12,"},{"Start":"06:17.550 ","End":"06:20.710","Text":"I make that minus 6."},{"Start":"06:21.080 ","End":"06:28.782","Text":"We divide by minus 1 again,"},{"Start":"06:28.782 ","End":"06:36.765","Text":"then we get that the sign changes and it\u0027s x bigger than 6."},{"Start":"06:36.765 ","End":"06:47.246","Text":"Now, if we say that x is bigger than 6 and x is bigger than minus 17,"},{"Start":"06:47.246 ","End":"06:50.360","Text":"basically, this is redundant because if it\u0027s"},{"Start":"06:50.360 ","End":"06:54.050","Text":"bigger than 6 it\u0027s automatically bigger than minus 17,"},{"Start":"06:54.050 ","End":"06:56.705","Text":"so this becomes unnecessary."},{"Start":"06:56.705 ","End":"06:59.555","Text":"Actually, this is just the answer,"},{"Start":"06:59.555 ","End":"07:04.925","Text":"x is bigger than 6 because this already covers this."},{"Start":"07:04.925 ","End":"07:09.470","Text":"That\u0027s the answer to part c. Now let\u0027s go on"},{"Start":"07:09.470 ","End":"07:15.710","Text":"to d and e. First d. As usual,"},{"Start":"07:15.710 ","End":"07:18.739","Text":"we\u0027re going to split it up into two inequalities."},{"Start":"07:18.739 ","End":"07:27.805","Text":"But one thing we can do as a whole is to get rid of a numerical denominator."},{"Start":"07:27.805 ","End":"07:33.510","Text":"I suggest multiplying all three sides,"},{"Start":"07:33.510 ","End":"07:35.685","Text":"we call it this, this, and this by,"},{"Start":"07:35.685 ","End":"07:37.920","Text":"let\u0027s see, 3 times 5 is 15."},{"Start":"07:37.920 ","End":"07:44.105","Text":"If we multiply everything by 15 then we have to put a 15 here."},{"Start":"07:44.105 ","End":"07:49.050","Text":"Here we need a 5 which is what\u0027s missing and here we\u0027re missing a 3."},{"Start":"07:49.050 ","End":"07:53.110","Text":"We can get another double inequality."},{"Start":"07:53.110 ","End":"08:00.335","Text":"15 times 6 is 90 less than 5 times 2x plus 10."},{"Start":"08:00.335 ","End":"08:02.035","Text":"I\u0027ll multiply it out already."},{"Start":"08:02.035 ","End":"08:09.640","Text":"That gives us 10x plus 50 and less than or equal to,"},{"Start":"08:09.640 ","End":"08:16.480","Text":"multiply out here, I\u0027ve got 21x minus 60."},{"Start":"08:16.480 ","End":"08:21.095","Text":"Now I have to split it up into two inequalities."},{"Start":"08:21.095 ","End":"08:27.960","Text":"I have that 90 is less than 10x plus 50"},{"Start":"08:27.960 ","End":"08:33.690","Text":"and 10x plus 50"},{"Start":"08:33.690 ","End":"08:40.660","Text":"is less than or equal to 21x minus 60."},{"Start":"08:40.660 ","End":"08:46.725","Text":"From here, if I bring the xs to the left and the numbers to the right,"},{"Start":"08:46.725 ","End":"08:53.742","Text":"I have minus 10x is less than minus 40,"},{"Start":"08:53.742 ","End":"08:56.215","Text":"it\u0027s 50 minus 90."},{"Start":"08:56.215 ","End":"08:58.220","Text":"Let\u0027s go on with this one,"},{"Start":"08:58.220 ","End":"09:00.680","Text":"won\u0027t even write it,"},{"Start":"09:00.680 ","End":"09:03.710","Text":"divide by minus 10."},{"Start":"09:03.710 ","End":"09:07.500","Text":"We invert the inequality."},{"Start":"09:07.500 ","End":"09:12.645","Text":"If I divide minus 40 over minus 10 then the answer is plus 4."},{"Start":"09:12.645 ","End":"09:14.475","Text":"As for the other one,"},{"Start":"09:14.475 ","End":"09:22.630","Text":"I have 10 minus 21 is minus 11x less than or equal to."},{"Start":"09:22.630 ","End":"09:24.560","Text":"If I take 50 over,"},{"Start":"09:24.560 ","End":"09:28.070","Text":"I\u0027ve got minus 110."},{"Start":"09:28.070 ","End":"09:33.230","Text":"Once again, I have to divide by a negative number minus 11,"},{"Start":"09:33.230 ","End":"09:38.510","Text":"so I reverse the inequality and then we get that x is"},{"Start":"09:38.510 ","End":"09:44.860","Text":"bigger or equal to 110 over 11 is just 10."},{"Start":"09:44.860 ","End":"09:47.720","Text":"Now just like in the previous one,"},{"Start":"09:47.720 ","End":"09:50.375","Text":"and I\u0027ll scroll up a bit so we can see that,"},{"Start":"09:50.375 ","End":"09:52.880","Text":"we had two inequalities going in"},{"Start":"09:52.880 ","End":"09:56.960","Text":"the same direction each time something is bigger or equal to."},{"Start":"09:56.960 ","End":"10:03.095","Text":"In this case, bigger than 6 automatically implied bigger than minus 17."},{"Start":"10:03.095 ","End":"10:09.320","Text":"Similarly here, if x is bigger or equal to 10 then for sure it\u0027s got to be bigger than 4."},{"Start":"10:09.320 ","End":"10:15.585","Text":"This one becomes redundant and our answer is just x bigger or equal to 10."},{"Start":"10:15.585 ","End":"10:20.475","Text":"I don\u0027t think we need a diagram just for this."},{"Start":"10:20.475 ","End":"10:24.765","Text":"Something is bigger or equal to 10 obviously it\u0027s bigger than 4."},{"Start":"10:24.765 ","End":"10:29.215","Text":"We come to the last one."},{"Start":"10:29.215 ","End":"10:35.254","Text":"Again, I\u0027ll multiply out to get rid of fractions first."},{"Start":"10:35.254 ","End":"10:40.070","Text":"The common denominator I\u0027m going to multiply will be 4 times 3,"},{"Start":"10:40.070 ","End":"10:42.335","Text":"so I multiply everything by 12."},{"Start":"10:42.335 ","End":"10:44.360","Text":"Unlike in the previous one, it\u0027s positive,"},{"Start":"10:44.360 ","End":"10:48.475","Text":"so we don\u0027t have to reverse anything."},{"Start":"10:48.475 ","End":"10:51.650","Text":"Here we have to multiply it by 12."},{"Start":"10:51.650 ","End":"10:54.050","Text":"Here 4 into 12 goes 3 times,"},{"Start":"10:54.050 ","End":"10:57.300","Text":"3 into 12 goes 4 times."},{"Start":"10:57.500 ","End":"11:04.390","Text":"We get minus 12 less than or equal to, let\u0027s see,"},{"Start":"11:04.670 ","End":"11:15.815","Text":"3 times this is 6x minus 18 and less than 4x plus 8."},{"Start":"11:15.815 ","End":"11:20.170","Text":"At this point, we split it up into two inequalities,"},{"Start":"11:20.170 ","End":"11:26.100","Text":"minus 12 less than or equal to 6x minus 18 and"},{"Start":"11:26.100 ","End":"11:34.110","Text":"6x minus 18 less than 4x plus 8."},{"Start":"11:34.110 ","End":"11:38.080","Text":"This one, xs on the left numbers on the right,"},{"Start":"11:38.080 ","End":"11:43.880","Text":"gives minus 6x is less than or equal to,"},{"Start":"11:43.880 ","End":"11:49.305","Text":"let\u0027s see, minus 18 plus 12 is minus 6."},{"Start":"11:49.305 ","End":"11:52.530","Text":"Divide by minus 6 it\u0027s negative."},{"Start":"11:52.530 ","End":"11:54.545","Text":"The first thing I do just to remind myself,"},{"Start":"11:54.545 ","End":"11:56.555","Text":"negative is right the direction."},{"Start":"11:56.555 ","End":"11:58.355","Text":"Now dividing by minus 6,"},{"Start":"11:58.355 ","End":"12:05.340","Text":"I get x is bigger or equal to 1 because minus 6 over minus 6 is 1."},{"Start":"12:05.340 ","End":"12:09.975","Text":"For the other one, x is on the left so that 2x,"},{"Start":"12:09.975 ","End":"12:16.470","Text":"6 minus 4 less than 18 plus 8 is 26."},{"Start":"12:16.470 ","End":"12:17.850","Text":"2 is positive,"},{"Start":"12:17.850 ","End":"12:22.290","Text":"so if I divide by it I get x less than 13."},{"Start":"12:22.290 ","End":"12:25.610","Text":"This time the inequalities are in opposite direction,"},{"Start":"12:25.610 ","End":"12:29.870","Text":"so something\u0027s bigger or equal to 1 and less than 13."},{"Start":"12:29.870 ","End":"12:34.236","Text":"That just gives us a double inequality,"},{"Start":"12:34.236 ","End":"12:37.640","Text":"that x is between 1 including 1,"},{"Start":"12:37.640 ","End":"12:40.100","Text":"and less than 13, not including."},{"Start":"12:40.100 ","End":"12:43.980","Text":"That\u0027s the answer to e. We are done."}],"Thumbnail":null,"ID":5058},{"Watched":false,"Name":"Exercise 6","Duration":"12m 22s","ChapterTopicVideoID":5057,"CourseChapterTopicPlaylistID":3875,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.650","Text":"In this exercise,"},{"Start":"00:01.650 ","End":"00:04.454","Text":"we have 5 inequalities."},{"Start":"00:04.454 ","End":"00:07.570","Text":"But this time we have or,"},{"Start":"00:07.570 ","End":"00:10.095","Text":"previously we had and,"},{"Start":"00:10.095 ","End":"00:12.810","Text":"and when we had an and,"},{"Start":"00:12.810 ","End":"00:15.675","Text":"that was equivalent to an intersection."},{"Start":"00:15.675 ","End":"00:18.390","Text":"When we have or,"},{"Start":"00:18.390 ","End":"00:26.550","Text":"that is mathematically the concept of a union of 2 intervals,"},{"Start":"00:26.550 ","End":"00:32.070","Text":"and it\u0027s written mathematically as U. I\u0027ll just remind you that we only have and."},{"Start":"00:32.070 ","End":"00:36.675","Text":"It\u0027s this way and it\u0027s called intersection."},{"Start":"00:36.675 ","End":"00:38.910","Text":"Intersection is when we"},{"Start":"00:38.910 ","End":"00:48.350","Text":"take the common pot and union is when we take both bits together and combine."},{"Start":"00:48.350 ","End":"00:50.510","Text":"Well, we\u0027ll see this in the exercises."},{"Start":"00:50.510 ","End":"00:54.980","Text":"Let\u0027s begin with the first 1."},{"Start":"00:54.980 ","End":"00:58.400","Text":"We have this or this. We solve each 1 separately first."},{"Start":"00:58.400 ","End":"01:00.290","Text":"Here we bring the 6 over,"},{"Start":"01:00.290 ","End":"01:05.280","Text":"so we get 3x bigger than 18+6 is 24,"},{"Start":"01:05.280 ","End":"01:06.900","Text":"and then divide by 3,"},{"Start":"01:06.900 ","End":"01:08.160","Text":"which is positive,"},{"Start":"01:08.160 ","End":"01:11.565","Text":"so we get x bigger than 8."},{"Start":"01:11.565 ","End":"01:15.240","Text":"In this 1, we bring the 4 over,"},{"Start":"01:15.240 ","End":"01:21.300","Text":"so we get 5x is less than or equal to 10,"},{"Start":"01:21.300 ","End":"01:23.175","Text":"and then 5 is positive,"},{"Start":"01:23.175 ","End":"01:29.890","Text":"so we get x less than or equal to 10/5."},{"Start":"01:29.890 ","End":"01:33.070","Text":"I have the word or in the middle."},{"Start":"01:33.440 ","End":"01:38.350","Text":"I\u0027ll do a sketch, it\u0027s not necessary."},{"Start":"01:38.540 ","End":"01:43.535","Text":"Here I\u0027ve sketched the number line and it increases to the right."},{"Start":"01:43.535 ","End":"01:46.265","Text":"The 2 interesting points are 2 and 8,"},{"Start":"01:46.265 ","End":"01:50.390","Text":"doesn\u0027t matter really where you put them as long as you put 2 to the left of 8."},{"Start":"01:50.390 ","End":"01:56.105","Text":"Let\u0027s say this place here is 2 and this here is 8."},{"Start":"01:56.105 ","End":"01:59.855","Text":"The first inequality, x bigger than 8."},{"Start":"01:59.855 ","End":"02:04.130","Text":"We start by drawing a hollow circle above the 8,"},{"Start":"02:04.130 ","End":"02:07.100","Text":"then we want a line going to the right,"},{"Start":"02:07.100 ","End":"02:12.520","Text":"and I put an arrow on it means it goes on indefinitely or to infinity."},{"Start":"02:12.520 ","End":"02:18.080","Text":"On the other 1, I have an inequality that includes the 2,"},{"Start":"02:18.080 ","End":"02:19.670","Text":"so 2 would have"},{"Start":"02:19.670 ","End":"02:26.650","Text":"a solid circle and less than or equal to means I\u0027m going from here and to the left."},{"Start":"02:26.650 ","End":"02:31.140","Text":"The little arrows to say it goes on indefinitely."},{"Start":"02:32.500 ","End":"02:35.600","Text":"The union, the or,"},{"Start":"02:35.600 ","End":"02:41.570","Text":"just means combined whatever is in here or in here, or if you like,"},{"Start":"02:41.570 ","End":"02:46.645","Text":"you could think of it as maybe this thing would cast a shadow,"},{"Start":"02:46.645 ","End":"02:54.959","Text":"and where would the shadow of this thing fall on here and it would cover this bit,"},{"Start":"02:54.959 ","End":"02:57.675","Text":"including the 2,"},{"Start":"02:57.675 ","End":"03:04.980","Text":"and it would cover this bit not including the 8."},{"Start":"03:04.980 ","End":"03:07.740","Text":"I have chosen a color that\u0027s too thick."},{"Start":"03:07.740 ","End":"03:10.655","Text":"But basically it\u0027s the same thing as the purple."},{"Start":"03:10.655 ","End":"03:13.937","Text":"In general, when they don\u0027t overlap,"},{"Start":"03:13.937 ","End":"03:15.470","Text":"when we take the union,"},{"Start":"03:15.470 ","End":"03:18.600","Text":"we just leave it as this or this."},{"Start":"03:19.780 ","End":"03:22.265","Text":"That\u0027s how we leave the answer."},{"Start":"03:22.265 ","End":"03:26.710","Text":"x bigger than 8 or x less than or equal to 2."},{"Start":"03:26.710 ","End":"03:29.675","Text":"There is no short way of doing this."},{"Start":"03:29.675 ","End":"03:32.660","Text":"If they overlap as they will in future exercises,"},{"Start":"03:32.660 ","End":"03:33.890","Text":"we\u0027ll see what we do then,"},{"Start":"03:33.890 ","End":"03:36.415","Text":"but this 1 looks a bit trivial."},{"Start":"03:36.415 ","End":"03:40.065","Text":"Anyway, there we are for a."},{"Start":"03:40.065 ","End":"03:42.700","Text":"Onto part b."},{"Start":"03:42.700 ","End":"03:45.199","Text":"Let\u0027s solve each 1 separately."},{"Start":"03:45.199 ","End":"03:50.675","Text":"This 1 gives us 2x is bigger or equal to bringing the 8 over,"},{"Start":"03:50.675 ","End":"03:56.700","Text":"that\u0027s 12, so x is bigger or equal to 6."},{"Start":"03:57.200 ","End":"04:06.735","Text":"This 1 gives us, on the left we get 3x from 4x-x is less than -10-8 is -18,"},{"Start":"04:06.735 ","End":"04:09.120","Text":"divided by 3, it\u0027s positive."},{"Start":"04:09.120 ","End":"04:12.180","Text":"I should\u0027ve mentioned here, 2 is positive."},{"Start":"04:12.180 ","End":"04:15.650","Text":"It doesn\u0027t hurt to mention at each time because people still"},{"Start":"04:15.650 ","End":"04:19.865","Text":"forget you divide by a negative number and you forget to change the sign."},{"Start":"04:19.865 ","End":"04:26.580","Text":"x is less than dividing by 3 is -6."},{"Start":"04:28.160 ","End":"04:30.695","Text":"Put an or in."},{"Start":"04:30.695 ","End":"04:36.455","Text":"Once again, they don\u0027t overlap because on the number line,"},{"Start":"04:36.455 ","End":"04:39.620","Text":"we will get something like this diagram."},{"Start":"04:39.620 ","End":"04:41.420","Text":"I went ahead and drew it,"},{"Start":"04:41.420 ","End":"04:44.780","Text":"just to make sure that the -6 is to the left of the 6,"},{"Start":"04:44.780 ","End":"04:47.990","Text":"and because here we have greater or equal to this 1 is"},{"Start":"04:47.990 ","End":"04:51.440","Text":"a solid circle and x less than -6 is a hollow circle."},{"Start":"04:51.440 ","End":"04:54.590","Text":"But once again, we see that they don\u0027t overlap."},{"Start":"04:54.590 ","End":"04:58.050","Text":"All the x\u0027s here,"},{"Start":"04:59.510 ","End":"05:03.590","Text":"not including the -6 and all the x\u0027s,"},{"Start":"05:03.590 ","End":"05:09.315","Text":"here are all part of this or this, the combined."},{"Start":"05:09.315 ","End":"05:11.510","Text":"There\u0027s nothing to abbreviate."},{"Start":"05:11.510 ","End":"05:13.310","Text":"The answer is just this or this."},{"Start":"05:13.310 ","End":"05:14.615","Text":"I\u0027ll just write it again."},{"Start":"05:14.615 ","End":"05:16.610","Text":"x bigger or equal to 6,"},{"Start":"05:16.610 ","End":"05:21.470","Text":"or x less than -6 and that\u0027s the answer."},{"Start":"05:21.470 ","End":"05:23.210","Text":"Let\u0027s go to another 1."},{"Start":"05:23.210 ","End":"05:27.030","Text":"Maybe this time there will be an overlap and it\u0027ll be more interesting."},{"Start":"05:29.350 ","End":"05:36.765","Text":"This inequality would give us 7x bigger than,"},{"Start":"05:36.765 ","End":"05:38.985","Text":"30-9, 21,"},{"Start":"05:38.985 ","End":"05:41.010","Text":"dividing by positive 7,"},{"Start":"05:41.010 ","End":"05:49.320","Text":"so x is bigger than 3 and here on the left we have -2x on the right."},{"Start":"05:49.320 ","End":"05:53.990","Text":"10/-2 which is negative,"},{"Start":"05:53.990 ","End":"05:57.260","Text":"we immediately reverse the inequality."},{"Start":"05:57.260 ","End":"06:00.910","Text":"x is less than 10 over -2,"},{"Start":"06:00.910 ","End":"06:05.020","Text":"and this is -5."},{"Start":"06:05.020 ","End":"06:09.710","Text":"Now notice that both inequalities going in the same direction,"},{"Start":"06:09.710 ","End":"06:11.525","Text":"x bigger than and x bigger than,"},{"Start":"06:11.525 ","End":"06:14.570","Text":"so there will be an overlap."},{"Start":"06:14.570 ","End":"06:16.790","Text":"Here\u0027s the number line,"},{"Start":"06:16.790 ","End":"06:18.510","Text":"-5 is on the left,"},{"Start":"06:18.510 ","End":"06:19.640","Text":"3 is on the right,"},{"Start":"06:19.640 ","End":"06:22.505","Text":"so this is -5."},{"Start":"06:22.505 ","End":"06:26.020","Text":"This is 3 just anywhere as long as they\u0027re in the right order."},{"Start":"06:26.020 ","End":"06:31.214","Text":"x bigger than -5 means"},{"Start":"06:31.214 ","End":"06:37.824","Text":"the -5 is not included and we want to go to the right,"},{"Start":"06:37.824 ","End":"06:40.760","Text":"so something like this."},{"Start":"06:40.760 ","End":"06:43.970","Text":"That\u0027s the x bigger than -5,"},{"Start":"06:43.970 ","End":"06:47.700","Text":"and the other 1 x bigger than 3,"},{"Start":"06:47.700 ","End":"06:53.075","Text":"we need again a hollow circle for the 3 because it\u0027s not included."},{"Start":"06:53.075 ","End":"06:56.116","Text":"Again, we want to go to the right,"},{"Start":"06:56.116 ","End":"06:59.510","Text":"so something like this."},{"Start":"06:59.510 ","End":"07:01.940","Text":"This would be the x bigger than 3."},{"Start":"07:01.940 ","End":"07:07.295","Text":"Now if we take the combined range that they cover,"},{"Start":"07:07.295 ","End":"07:10.220","Text":"the combined interval, or I like to think of it"},{"Start":"07:10.220 ","End":"07:14.375","Text":"sometimes as the sun shining from above, what\u0027s the shadow?"},{"Start":"07:14.375 ","End":"07:18.985","Text":"Then it\u0027s obviously everything from -5 onwards."},{"Start":"07:18.985 ","End":"07:22.230","Text":"I\u0027ll sketch it in this color."},{"Start":"07:22.230 ","End":"07:28.790","Text":"From -5, not including and everything to the right."},{"Start":"07:28.790 ","End":"07:31.420","Text":"Basically we just get this 1,"},{"Start":"07:31.420 ","End":"07:36.320","Text":"and so the answer is that this or this is just"},{"Start":"07:36.320 ","End":"07:42.695","Text":"the same as x bigger than -5."},{"Start":"07:42.695 ","End":"07:46.130","Text":"Because this doesn\u0027t have any possibilities,"},{"Start":"07:46.130 ","End":"07:49.685","Text":"this is already included in this."},{"Start":"07:49.685 ","End":"07:54.635","Text":"If I tell you that something\u0027s bigger than -5 or bigger than 3,"},{"Start":"07:54.635 ","End":"07:59.510","Text":"either way it\u0027s bigger than -5 and this covers both possibilities."},{"Start":"07:59.510 ","End":"08:03.135","Text":"This is included as redundant, doesn\u0027t add anything."},{"Start":"08:03.135 ","End":"08:05.745","Text":"This is the answer."},{"Start":"08:05.745 ","End":"08:08.450","Text":"When 1 is included in the other,"},{"Start":"08:08.450 ","End":"08:09.890","Text":"in the case of an or,"},{"Start":"08:09.890 ","End":"08:12.005","Text":"we take the larger 1,"},{"Start":"08:12.005 ","End":"08:14.150","Text":"just like as in the case of and,"},{"Start":"08:14.150 ","End":"08:18.180","Text":"do we took the smaller 1 we wanted to combine."},{"Start":"08:18.610 ","End":"08:23.460","Text":"Onto part d,"},{"Start":"08:23.810 ","End":"08:27.185","Text":"we\u0027ll solve each 1 separately first."},{"Start":"08:27.185 ","End":"08:35.155","Text":"This 1 is 3x is bigger than -18,"},{"Start":"08:35.155 ","End":"08:37.430","Text":"and 3 is positive,"},{"Start":"08:37.430 ","End":"08:38.790","Text":"so we can divide by it."},{"Start":"08:38.790 ","End":"08:41.840","Text":"x bigger than -6,"},{"Start":"08:41.840 ","End":"08:47.180","Text":"or x-9x"},{"Start":"08:47.180 ","End":"08:53.617","Text":"is -8x,"},{"Start":"08:53.617 ","End":"08:57.675","Text":"-10 goes over as +10,"},{"Start":"08:57.675 ","End":"08:59.885","Text":"so it\u0027s 24 here."},{"Start":"08:59.885 ","End":"09:02.810","Text":"This time I\u0027m dividing by -8,"},{"Start":"09:02.810 ","End":"09:04.130","Text":"which is negative,"},{"Start":"09:04.130 ","End":"09:07.690","Text":"so I\u0027m going to get a reverse inequality."},{"Start":"09:07.690 ","End":"09:10.155","Text":"x is going to be bigger than,"},{"Start":"09:10.155 ","End":"09:15.250","Text":"24 over -8 is -3."},{"Start":"09:15.950 ","End":"09:20.380","Text":"Now, let\u0027s see what happens here."},{"Start":"09:20.830 ","End":"09:25.310","Text":"I drew the number line and I already put in the -6 and -3."},{"Start":"09:25.310 ","End":"09:29.165","Text":"Let\u0027s see what x bigger than -6 looks like."},{"Start":"09:29.165 ","End":"09:31.850","Text":"We start from the -6,"},{"Start":"09:31.850 ","End":"09:36.575","Text":"not including and we go to the right, like so."},{"Start":"09:36.575 ","End":"09:40.820","Text":"The other 1 x bigger than -3,"},{"Start":"09:42.050 ","End":"09:46.825","Text":"so we don\u0027t include the -3,"},{"Start":"09:46.825 ","End":"09:52.680","Text":"but we\u0027re also going to the right, like this."},{"Start":"09:53.440 ","End":"09:56.600","Text":"This is exactly the same as the previous 1,"},{"Start":"09:56.600 ","End":"09:58.550","Text":"just with different numbers."},{"Start":"09:58.550 ","End":"10:01.745","Text":"1 is included in the other,"},{"Start":"10:01.745 ","End":"10:04.080","Text":"in this case also,"},{"Start":"10:04.080 ","End":"10:06.575","Text":"and so we just take the larger 1,"},{"Start":"10:06.575 ","End":"10:08.600","Text":"which is this 1."},{"Start":"10:08.600 ","End":"10:15.940","Text":"The answer here is x bigger than -6."},{"Start":"10:16.460 ","End":"10:20.060","Text":"Onto part e, the last part,"},{"Start":"10:20.060 ","End":"10:23.270","Text":"this 1 exhibit something different."},{"Start":"10:23.270 ","End":"10:25.490","Text":"We\u0027ll see it\u0027s a spoiler,"},{"Start":"10:25.490 ","End":"10:28.385","Text":"but they are going to overlap,"},{"Start":"10:28.385 ","End":"10:30.620","Text":"but neither will be included in the other."},{"Start":"10:30.620 ","End":"10:31.895","Text":"Let\u0027s see how we\u0027ll handle that."},{"Start":"10:31.895 ","End":"10:35.100","Text":"Let\u0027s just start solving."},{"Start":"10:35.100 ","End":"10:37.840","Text":"We get, bringing the x\u0027s over here,"},{"Start":"10:37.840 ","End":"10:43.185","Text":"that 5x is bigger than 20,"},{"Start":"10:43.185 ","End":"10:46.560","Text":"x bigger than 4."},{"Start":"10:46.560 ","End":"10:49.320","Text":"For this 1, we get,"},{"Start":"10:49.320 ","End":"10:57.990","Text":"5x-2x is 3x less than this plus this is 30. x is"},{"Start":"10:57.990 ","End":"11:06.920","Text":"less than 10 or I went ahead and did the whole diagram."},{"Start":"11:06.920 ","End":"11:09.950","Text":"I put 4 and 10 here and here."},{"Start":"11:09.950 ","End":"11:13.380","Text":"Bigger than 4 is the top 1."},{"Start":"11:13.380 ","End":"11:16.560","Text":"Not including 4, but anything larger."},{"Start":"11:16.560 ","End":"11:17.970","Text":"The x less than 10,"},{"Start":"11:17.970 ","End":"11:20.650","Text":"doesn\u0027t include 10, but everything smaller."},{"Start":"11:20.650 ","End":"11:27.140","Text":"Now I want the combined interval,"},{"Start":"11:27.140 ","End":"11:29.855","Text":"which is this interval together with this interval,"},{"Start":"11:29.855 ","End":"11:32.045","Text":"or the way I think of it as the shadow,"},{"Start":"11:32.045 ","End":"11:35.390","Text":"this is like a roof another roof and the sun shining from above."},{"Start":"11:35.390 ","End":"11:37.820","Text":"What does it cover? Well, if you think about it,"},{"Start":"11:37.820 ","End":"11:39.470","Text":"it covers the whole line."},{"Start":"11:39.470 ","End":"11:41.795","Text":"Everything is covered."},{"Start":"11:41.795 ","End":"11:44.360","Text":"We get the whole number line."},{"Start":"11:44.360 ","End":"11:47.390","Text":"If you paste this together with this or you look at the shadow or"},{"Start":"11:47.390 ","End":"11:51.290","Text":"whatever analogy you want to use between them."},{"Start":"11:51.290 ","End":"11:55.570","Text":"Every number has to be either less than 10 or bigger than 4."},{"Start":"11:55.570 ","End":"11:58.045","Text":"It can\u0027t be otherwise."},{"Start":"11:58.045 ","End":"12:01.130","Text":"Together they cover everything."},{"Start":"12:01.130 ","End":"12:03.965","Text":"Sometimes it\u0027s both less than 10 and bigger than 4,"},{"Start":"12:03.965 ","End":"12:07.045","Text":"but at least 1 or the other has to hold because we can see it."},{"Start":"12:07.045 ","End":"12:08.750","Text":"They cover everything."},{"Start":"12:08.750 ","End":"12:12.290","Text":"We write all x for the answer."},{"Start":"12:12.290 ","End":"12:15.020","Text":"This or this means all x,"},{"Start":"12:15.020 ","End":"12:17.105","Text":"the whole number line."},{"Start":"12:17.105 ","End":"12:20.570","Text":"That\u0027s the answer to this inequality,"},{"Start":"12:20.570 ","End":"12:23.020","Text":"and we are done."}],"Thumbnail":null,"ID":5059}],"ID":3875},{"Name":"Polynomial Inequalities","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Quadratic Inequalities","Duration":"16m 51s","ChapterTopicVideoID":5069,"CourseChapterTopicPlaylistID":3876,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.640","Text":"Continuing with Inequalities,"},{"Start":"00:02.640 ","End":"00:07.410","Text":"the next topic will be Polynomial Inequalities."},{"Start":"00:07.410 ","End":"00:09.495","Text":"We\u0027ve already done linear,"},{"Start":"00:09.495 ","End":"00:15.240","Text":"so let\u0027s next do Quadratic degree 2,"},{"Start":"00:15.240 ","End":"00:20.500","Text":"and after that, we\u0027ll go to more general higher degree polynomials."},{"Start":"00:20.500 ","End":"00:23.435","Text":"I\u0027ll start with an example."},{"Start":"00:23.435 ","End":"00:26.735","Text":"The example will be"},{"Start":"00:26.735 ","End":"00:35.525","Text":"x^2 plus 5 is less than 6x."},{"Start":"00:35.525 ","End":"00:38.960","Text":"The first thing to do is to bring everything on"},{"Start":"00:38.960 ","End":"00:43.400","Text":"the left-hand side and arrange it in decreasing order."},{"Start":"00:43.400 ","End":"00:45.440","Text":"In other words, just like we normally see,"},{"Start":"00:45.440 ","End":"00:53.215","Text":"a quadratic x^2 minus 6x plus 5 less than 0,"},{"Start":"00:53.215 ","End":"00:59.210","Text":"and then we solve the equation equal 0."},{"Start":"00:59.210 ","End":"01:01.670","Text":"In other words, we find the roots of the polynomial."},{"Start":"01:01.670 ","End":"01:03.719","Text":"So I\u0027ll do that at the side,"},{"Start":"01:03.719 ","End":"01:11.940","Text":"the roots of the polynomial are 1 and 5."},{"Start":"01:13.170 ","End":"01:15.505","Text":"Then we draw a sketch."},{"Start":"01:15.505 ","End":"01:18.010","Text":"Let\u0027s start with the number line,"},{"Start":"01:18.010 ","End":"01:23.970","Text":"the roots are 1 and 5,"},{"Start":"01:23.970 ","End":"01:27.910","Text":"so I\u0027ve put those on the number line,"},{"Start":"01:27.910 ","End":"01:38.610","Text":"and then I like to draw some dotted lines just faint through these points."},{"Start":"01:38.610 ","End":"01:40.755","Text":"This is optional,"},{"Start":"01:40.755 ","End":"01:46.700","Text":"but the idea is to think of the number line as divided into 3 bits."},{"Start":"01:46.700 ","End":"01:50.340","Text":"Less than 1, between 1 and 5 and greater than 5,"},{"Start":"01:50.340 ","End":"01:52.770","Text":"and then there\u0027s 1 and 5 themselves,"},{"Start":"01:52.770 ","End":"01:57.970","Text":"and the idea is to draw some approximate sketch just to"},{"Start":"01:57.970 ","End":"02:04.875","Text":"represent where the expression is positive, negative, or 0."},{"Start":"02:04.875 ","End":"02:08.115","Text":"Now, this expression x squared minus 6x plus 5."},{"Start":"02:08.115 ","End":"02:12.280","Text":"I know it\u0027s 0 that 105 and I represent this by"},{"Start":"02:12.280 ","End":"02:18.890","Text":"drawing a dot here and here,"},{"Start":"02:18.890 ","End":"02:26.920","Text":"and then I\u0027d like to know in each of these 3 intervals whether it\u0027s positive or negative."},{"Start":"02:26.920 ","End":"02:31.525","Text":"I know it has to be always 1 over the other"},{"Start":"02:31.525 ","End":"02:41.720","Text":"because a polynomial can\u0027t cross from positive to negative without going through 0."},{"Start":"02:41.720 ","End":"02:44.165","Text":"These are the only places we are at 0."},{"Start":"02:44.165 ","End":"02:46.595","Text":"So the easiest way to check,"},{"Start":"02:46.595 ","End":"02:47.870","Text":"let\u0027s take the middle interval,"},{"Start":"02:47.870 ","End":"02:49.280","Text":"for example, between 1 and 5."},{"Start":"02:49.280 ","End":"02:51.950","Text":"How do I know if it\u0027s positive or negative?"},{"Start":"02:51.950 ","End":"02:55.130","Text":"Because it doesn\u0027t change sign in the middle,"},{"Start":"02:55.130 ","End":"02:57.335","Text":"all I have to do is take a sample point."},{"Start":"02:57.335 ","End":"03:01.580","Text":"So I\u0027ll choose, for example,"},{"Start":"03:01.580 ","End":"03:04.760","Text":"3 as a sample point."},{"Start":"03:04.760 ","End":"03:08.840","Text":"Any number in this interval will do plug-in 3 and"},{"Start":"03:08.840 ","End":"03:12.410","Text":"I only care about whether the answer is positive or negative."},{"Start":"03:12.410 ","End":"03:20.185","Text":"Let\u0027s see 3^2 is 9 minus 18 plus 5."},{"Start":"03:20.185 ","End":"03:21.995","Text":"It comes out negative."},{"Start":"03:21.995 ","End":"03:25.490","Text":"I\u0027m going to use what is called the snake method,"},{"Start":"03:25.490 ","End":"03:27.140","Text":"and you\u0027ll soon see why,"},{"Start":"03:27.140 ","End":"03:30.590","Text":"and I indicate the fact that it\u0027s negative."},{"Start":"03:30.590 ","End":"03:34.960","Text":"Well, first of all, I choose a point below the axis."},{"Start":"03:34.960 ","End":"03:40.055","Text":"Below the 3 here to show negativity."},{"Start":"03:40.055 ","End":"03:42.950","Text":"Positive is above, negative is below,"},{"Start":"03:42.950 ","End":"03:45.400","Text":"and then other way it\u0027s not accurate,"},{"Start":"03:45.400 ","End":"03:48.680","Text":"we just show that it\u0027s negative everywhere in"},{"Start":"03:48.680 ","End":"03:53.675","Text":"this interval by just drawing some kind of line."},{"Start":"03:53.675 ","End":"03:57.740","Text":"Doesn\u0027t have to be accurate or anything."},{"Start":"03:57.740 ","End":"03:59.180","Text":"We\u0027re not drawing the graph."},{"Start":"03:59.180 ","End":"04:05.405","Text":"We\u0027re just showing that it\u0027s negative between 1 and 5 and let 0 here and here,"},{"Start":"04:05.405 ","End":"04:12.630","Text":"and then we want to continue the sketch to say what happens when x is bigger than 5."},{"Start":"04:12.760 ","End":"04:15.740","Text":"So once again, choose a sample point."},{"Start":"04:15.740 ","End":"04:18.500","Text":"Let\u0027s take 6 as a sample point."},{"Start":"04:18.500 ","End":"04:21.400","Text":"Substitute 6 for x,"},{"Start":"04:21.400 ","End":"04:26.010","Text":"we\u0027ve got x^2 is 36 minus 36 plus 5."},{"Start":"04:26.010 ","End":"04:32.145","Text":"Obviously positive, so I indicate that by drawing a dot above the 6,"},{"Start":"04:32.145 ","End":"04:38.260","Text":"above the x-axis is positive and just continue the snake."},{"Start":"04:38.260 ","End":"04:40.080","Text":"Let\u0027s go through here,"},{"Start":"04:40.080 ","End":"04:42.665","Text":"we don\u0027t know how exactly it continues,"},{"Start":"04:42.665 ","End":"04:44.135","Text":"goes on indefinitely,"},{"Start":"04:44.135 ","End":"04:47.915","Text":"but the important thing is it\u0027s stays positive in"},{"Start":"04:47.915 ","End":"04:52.710","Text":"all of this interval up to wherever infinity,"},{"Start":"04:52.710 ","End":"04:58.290","Text":"and then we have the last remaining interval, what\u0027s below 1."},{"Start":"04:58.290 ","End":"05:01.230","Text":"Again, choose a sample point,"},{"Start":"05:01.230 ","End":"05:05.940","Text":"say 0 will be in this interval,"},{"Start":"05:05.940 ","End":"05:09.090","Text":"and then I substitute 0 in"},{"Start":"05:09.090 ","End":"05:15.560","Text":"the polynomial expression and I see 0 minus 0 plus 5 positive,"},{"Start":"05:15.560 ","End":"05:21.400","Text":"so I put a point above the 0 to indicate positive,"},{"Start":"05:21.400 ","End":"05:29.670","Text":"and then it\u0027s positive in all of the interval from 1 downwards not including the 1,"},{"Start":"05:29.670 ","End":"05:33.690","Text":"so I just continue it somehow."},{"Start":"05:33.690 ","End":"05:38.390","Text":"And this is the snake,"},{"Start":"05:38.390 ","End":"05:46.715","Text":"and what it shows us is where the expression is positive,"},{"Start":"05:46.715 ","End":"05:48.680","Text":"which is above the axis,"},{"Start":"05:48.680 ","End":"05:52.050","Text":"and where it\u0027s negative below the axis."},{"Start":"05:52.070 ","End":"05:55.820","Text":"Now, that I have the sketch,"},{"Start":"05:55.820 ","End":"06:00.770","Text":"only now do we actually look at what the inequality is,"},{"Start":"06:00.770 ","End":"06:03.575","Text":"and I noticed that it\u0027s a less than,"},{"Start":"06:03.575 ","End":"06:06.200","Text":"but this snake sketch,"},{"Start":"06:06.200 ","End":"06:09.380","Text":"we would do regardless of whether it\u0027s bigger than,"},{"Start":"06:09.380 ","End":"06:10.580","Text":"bigger than or equal to,"},{"Start":"06:10.580 ","End":"06:13.180","Text":"less than or equal to whatever,"},{"Start":"06:13.180 ","End":"06:16.745","Text":"and now that I see that it\u0027s less than,"},{"Start":"06:16.745 ","End":"06:20.720","Text":"less than means below 0."},{"Start":"06:20.720 ","End":"06:25.580","Text":"So I look at which places the snake is below the 0,"},{"Start":"06:25.580 ","End":"06:27.500","Text":"meaning below the axis,"},{"Start":"06:27.500 ","End":"06:30.400","Text":"and I can see that it\u0027s this part here."},{"Start":"06:30.400 ","End":"06:33.995","Text":"There\u0027s lots of variations on this sketching technique."},{"Start":"06:33.995 ","End":"06:37.730","Text":"1 way to indicate this part is by highlighting it,"},{"Start":"06:37.730 ","End":"06:42.710","Text":"so I could just go over this bit and highlight it,"},{"Start":"06:42.710 ","End":"06:46.340","Text":"and after I\u0027ve highlighted it,"},{"Start":"06:46.340 ","End":"06:55.880","Text":"I can see that the highlighting corresponds to the x is between 1 and 5."},{"Start":"06:55.880 ","End":"06:59.930","Text":"A variation of highlighting is shading."},{"Start":"06:59.930 ","End":"07:02.510","Text":"Shading, I might do something like"},{"Start":"07:02.510 ","End":"07:08.465","Text":"this to indicate that this is the part I\u0027m interested in,"},{"Start":"07:08.465 ","End":"07:10.370","Text":"whatever it is,"},{"Start":"07:10.370 ","End":"07:15.715","Text":"either way, I can see that the part that I need,"},{"Start":"07:15.715 ","End":"07:20.594","Text":"the bit between 1 and 5, not inclusive,"},{"Start":"07:20.594 ","End":"07:25.335","Text":"so it\u0027s between 1 and 5,"},{"Start":"07:25.335 ","End":"07:31.730","Text":"not including, and this would be the answer."},{"Start":"07:31.730 ","End":"07:37.550","Text":"We would say that write it as a double inequality that x"},{"Start":"07:37.550 ","End":"07:44.885","Text":"is less than 5 and greater than 1."},{"Start":"07:44.885 ","End":"07:49.754","Text":"The answer is 1 less than x,"},{"Start":"07:49.754 ","End":"07:56.620","Text":"less than 5 and that\u0027s the answer."},{"Start":"07:56.620 ","End":"08:01.340","Text":"Like to show you is still another variation on the sketch."},{"Start":"08:01.340 ","End":"08:05.215","Text":"If you don\u0027t like snakes or something,"},{"Start":"08:05.215 ","End":"08:10.700","Text":"what you can do is after we\u0027ve done the sample points,"},{"Start":"08:10.700 ","End":"08:15.470","Text":"we just write down for each sample point whether it was positive or negative."},{"Start":"08:15.470 ","End":"08:20.780","Text":"So over 0, I would put a big plus sign"},{"Start":"08:20.780 ","End":"08:26.420","Text":"because it\u0027s positive and over the 3 turns out to be negative."},{"Start":"08:26.420 ","End":"08:27.905","Text":"I put a minus,"},{"Start":"08:27.905 ","End":"08:29.120","Text":"and over the 6,"},{"Start":"08:29.120 ","End":"08:31.609","Text":"I would just write plus,"},{"Start":"08:31.609 ","End":"08:39.335","Text":"or I could write in words positive, negative, positive, whatever."},{"Start":"08:39.335 ","End":"08:45.225","Text":"Now, we know where the expression is positive 0,"},{"Start":"08:45.225 ","End":"08:48.060","Text":"negative 0, positive,"},{"Start":"08:48.060 ","End":"08:49.610","Text":"and at that point again,"},{"Start":"08:49.610 ","End":"08:51.530","Text":"we look at what we are interested in."},{"Start":"08:51.530 ","End":"08:55.805","Text":"We\u0027re interested in less than 0, which means negative."},{"Start":"08:55.805 ","End":"08:58.625","Text":"The negative is from here to here,"},{"Start":"08:58.625 ","End":"09:01.730","Text":"and I\u0027m not including the endpoints,"},{"Start":"09:01.730 ","End":"09:03.979","Text":"and then I would write it in words."},{"Start":"09:03.979 ","End":"09:06.830","Text":"The 1 less than x,"},{"Start":"09:06.830 ","End":"09:09.160","Text":"less than 5,"},{"Start":"09:09.160 ","End":"09:15.765","Text":"and that\u0027s just an alternative to those who don\u0027t like this snake sketch."},{"Start":"09:15.765 ","End":"09:20.650","Text":"Even this is optional if you can see the negative applies from"},{"Start":"09:20.650 ","End":"09:25.180","Text":"here to here as endless variations for the sketch is what I\u0027m saying,"},{"Start":"09:25.180 ","End":"09:28.030","Text":"and in the example exercises,"},{"Start":"09:28.030 ","End":"09:30.010","Text":"sometimes I\u0027ll do it this way,"},{"Start":"09:30.010 ","End":"09:32.020","Text":"sometimes this way, sometimes shading,"},{"Start":"09:32.020 ","End":"09:34.990","Text":"sometimes highlighting, there is no big difference,"},{"Start":"09:34.990 ","End":"09:39.445","Text":"it\u0027s all in the graphics and you\u0027ll choose whichever is suitable for you."},{"Start":"09:39.445 ","End":"09:43.330","Text":"Okay. Let\u0027s do another exercise."},{"Start":"09:43.330 ","End":"09:52.420","Text":"This time, I\u0027ll take almost the same as here x^2 plus 5,"},{"Start":"09:52.420 ","End":"09:57.440","Text":"but this time greater or equal to 6x."},{"Start":"09:57.440 ","End":"10:01.165","Text":"Let\u0027s see how that would change everything."},{"Start":"10:01.165 ","End":"10:05.635","Text":"Will I still get the same quadratic expression,"},{"Start":"10:05.635 ","End":"10:08.470","Text":"x^2 minus 6x plus 5?"},{"Start":"10:08.470 ","End":"10:11.500","Text":"But the only thing that would change would be instead"},{"Start":"10:11.500 ","End":"10:15.310","Text":"of less than I get greater than or equal to."},{"Start":"10:15.310 ","End":"10:19.570","Text":"Just highlight that to emphasize this time it bigger than or"},{"Start":"10:19.570 ","End":"10:23.770","Text":"equal to 0 and I can reuse 1 of these 2,"},{"Start":"10:23.770 ","End":"10:28.345","Text":"I\u0027ll say I\u0027ll use this format and I\u0027ll copy it over here."},{"Start":"10:28.345 ","End":"10:37.090","Text":"The only difference is that because we are now collecting positives and 0,"},{"Start":"10:37.090 ","End":"10:39.505","Text":"is that we don\u0027t take this bit."},{"Start":"10:39.505 ","End":"10:42.190","Text":"We take where it\u0027s positive and where it\u0027s 0."},{"Start":"10:42.190 ","End":"10:45.400","Text":"In other words, we take 1,"},{"Start":"10:45.400 ","End":"10:47.230","Text":"we take 5,"},{"Start":"10:47.230 ","End":"10:53.995","Text":"that\u0027s the zeros, and here\u0027s the positives and here positives where the pluses are."},{"Start":"10:53.995 ","End":"10:57.055","Text":"This is the new shape."},{"Start":"10:57.055 ","End":"10:58.960","Text":"Instead of writing it like this,"},{"Start":"10:58.960 ","End":"11:01.690","Text":"this time it\u0027s 2 separate intervals."},{"Start":"11:01.690 ","End":"11:03.145","Text":"We needed an O,"},{"Start":"11:03.145 ","End":"11:09.610","Text":"we need to say that this part\u0027s x less than 1,"},{"Start":"11:09.610 ","End":"11:14.875","Text":"this part\u0027s x bigger than 5 and together we get x"},{"Start":"11:14.875 ","End":"11:21.610","Text":"less than 1 or x greater than 5."},{"Start":"11:21.610 ","End":"11:23.660","Text":"I\u0027ll highlight it."},{"Start":"11:24.540 ","End":"11:26.965","Text":"That\u0027s another 1 done."},{"Start":"11:26.965 ","End":"11:34.120","Text":"Basically it goes to show you that it\u0027s the polynomial that determines the diagram,"},{"Start":"11:34.120 ","End":"11:35.920","Text":"whether it\u0027s like this or like this,"},{"Start":"11:35.920 ","End":"11:39.535","Text":"and only when you come to the actual shading or choosing,"},{"Start":"11:39.535 ","End":"11:44.720","Text":"do you look at the actual direction or the type of inequality."},{"Start":"11:47.610 ","End":"11:52.400","Text":"Let\u0027s go on to do another quadratic."},{"Start":"11:52.440 ","End":"11:55.720","Text":"Here is the next example,"},{"Start":"11:55.720 ","End":"11:59.080","Text":"I have it written out already and already"},{"Start":"11:59.080 ","End":"12:03.310","Text":"everything\u0027s on the left-hand side so that\u0027s 1 less step."},{"Start":"12:03.310 ","End":"12:08.275","Text":"Remember the next step is to find the roots of this polynomial."},{"Start":"12:08.275 ","End":"12:13.450","Text":"If I do this at the side and I do the computation for you,"},{"Start":"12:13.450 ","End":"12:16.450","Text":"it was 1 of those where we get a 0 under the root sign."},{"Start":"12:16.450 ","End":"12:21.400","Text":"We get the same answer twice x=3 or x=3."},{"Start":"12:21.400 ","End":"12:25.465","Text":"There\u0027s only 1 root or it\u0027s a double root."},{"Start":"12:25.465 ","End":"12:30.820","Text":"The root is x=3,"},{"Start":"12:30.820 ","End":"12:33.730","Text":"although you could say it\u0027s 3 or 3."},{"Start":"12:33.730 ","End":"12:36.925","Text":"Actually, if you factorize this,"},{"Start":"12:36.925 ","End":"12:40.940","Text":"you would get x minus 3^2,"},{"Start":"12:40.950 ","End":"12:45.790","Text":"and it could be done this way."},{"Start":"12:45.790 ","End":"12:48.520","Text":"If you saw it was a perfect square you wouldn\u0027t need to"},{"Start":"12:48.520 ","End":"12:51.130","Text":"solve and then you\u0027d know just x=3."},{"Start":"12:51.130 ","End":"12:55.720","Text":"Anyway, let\u0027s do the simplified sketch 1 of these."},{"Start":"12:55.720 ","End":"12:58.900","Text":"Here\u0027s the number line and there\u0027s only 1 point to"},{"Start":"12:58.900 ","End":"13:03.655","Text":"mark and that is the point where x is 3."},{"Start":"13:03.655 ","End":"13:06.895","Text":"This time we only have 2 intervals,"},{"Start":"13:06.895 ","End":"13:09.415","Text":"this left part and the right part."},{"Start":"13:09.415 ","End":"13:11.410","Text":"Let\u0027s take sample points."},{"Start":"13:11.410 ","End":"13:15.970","Text":"Well, I\u0027ll take 0 because it\u0027s always easy to calculate and here I\u0027ll"},{"Start":"13:15.970 ","End":"13:21.085","Text":"take 4 and let\u0027s see what the values are."},{"Start":"13:21.085 ","End":"13:26.140","Text":"If I plug in 0 into this polynomial,"},{"Start":"13:26.140 ","End":"13:29.050","Text":"I get 9, which is positive."},{"Start":"13:29.050 ","End":"13:33.745","Text":"So I\u0027ll write the plus here."},{"Start":"13:33.745 ","End":"13:38.030","Text":"I should have really put the dotted lines here."},{"Start":"13:39.060 ","End":"13:42.573","Text":"Next, I\u0027ll try the other point 4,"},{"Start":"13:42.573 ","End":"13:44.245","Text":"should plug 4 in."},{"Start":"13:44.245 ","End":"13:47.720","Text":"I checked it, to get something positive."},{"Start":"13:47.720 ","End":"13:50.325","Text":"Actually, you can see it from here,"},{"Start":"13:50.325 ","End":"13:53.520","Text":"because if I put 4 in 4 minus 3^2,"},{"Start":"13:53.520 ","End":"13:55.830","Text":"it\u0027s going to be positive."},{"Start":"13:55.830 ","End":"13:59.070","Text":"This is always going to be positive except when x is 3,"},{"Start":"13:59.070 ","End":"14:01.860","Text":"in which case it\u0027s 0, which is this point here."},{"Start":"14:01.860 ","End":"14:05.810","Text":"We have only positives or 0."},{"Start":"14:05.810 ","End":"14:09.760","Text":"Now at this point, I look at what I need,"},{"Start":"14:09.760 ","End":"14:15.475","Text":"and I need positive and that\u0027s what bigger than 0 means, positive."},{"Start":"14:15.475 ","End":"14:18.520","Text":"I want everything, almost everything."},{"Start":"14:18.520 ","End":"14:20.140","Text":"I want this and I want this,"},{"Start":"14:20.140 ","End":"14:22.540","Text":"I just don\u0027t want the 0."},{"Start":"14:22.540 ","End":"14:25.570","Text":"If I sketch it, there\u0027s a hole here."},{"Start":"14:25.570 ","End":"14:30.310","Text":"I don\u0027t want the value 3 because that\u0027s where it\u0027s 0,"},{"Start":"14:30.310 ","End":"14:36.025","Text":"but everywhere here it\u0027s positive and everywhere here it\u0027s positive."},{"Start":"14:36.025 ","End":"14:40.360","Text":"Basically I\u0027ve got the whole number line but with the hole in it."},{"Start":"14:40.360 ","End":"14:42.318","Text":"There\u0027s 2 ways I can write that."},{"Start":"14:42.318 ","End":"14:44.485","Text":"1 way of writing it, well,"},{"Start":"14:44.485 ","End":"14:46.510","Text":"here it\u0027s x less than 3,"},{"Start":"14:46.510 ","End":"14:48.460","Text":"and here x bigger than 3."},{"Start":"14:48.460 ","End":"14:52.915","Text":"I could write x is less than 3,"},{"Start":"14:52.915 ","End":"14:56.455","Text":"or x is bigger than 3."},{"Start":"14:56.455 ","End":"14:58.840","Text":"This is a perfectly good answer,"},{"Start":"14:58.840 ","End":"15:00.850","Text":"but it\u0027s possible to simplify this."},{"Start":"15:00.850 ","End":"15:06.790","Text":"You could actually just say that x is not equal to 3."},{"Start":"15:06.790 ","End":"15:10.615","Text":"I mean, if it\u0027s bigger than or smaller than and it\u0027s just different from."},{"Start":"15:10.615 ","End":"15:14.185","Text":"This would be a nicer way to write the answer."},{"Start":"15:14.185 ","End":"15:19.165","Text":"Actually, we didn\u0027t really need a sketch here because if you see something squared,"},{"Start":"15:19.165 ","End":"15:22.720","Text":"you know that it\u0027s always positive except when"},{"Start":"15:22.720 ","End":"15:27.565","Text":"the brackets is 0 or x is 3 and then it\u0027s 0."},{"Start":"15:27.565 ","End":"15:31.430","Text":"Just to give a variation,"},{"Start":"15:31.800 ","End":"15:36.355","Text":"suppose I had bigger or equal to 0 here."},{"Start":"15:36.355 ","End":"15:39.160","Text":"I\u0027ll take some variations of the problem let\u0027s say,"},{"Start":"15:39.160 ","End":"15:43.180","Text":"3a had the same thing but with bigger or equal to 0."},{"Start":"15:43.180 ","End":"15:50.185","Text":"Then I would be collecting positives and zeros or be taking everything."},{"Start":"15:50.185 ","End":"15:56.320","Text":"Then the answer would be all x because this wouldn\u0027t be a hole anymore."},{"Start":"15:56.320 ","End":"15:57.880","Text":"Let me ask something else."},{"Start":"15:57.880 ","End":"16:02.980","Text":"Suppose I took less than 0 that\u0027s 3b,"},{"Start":"16:02.980 ","End":"16:05.320","Text":"same question but with less than 0."},{"Start":"16:05.320 ","End":"16:07.900","Text":"Then there is no place where it\u0027s negative."},{"Start":"16:07.900 ","End":"16:09.220","Text":"Less than 0 would be negative."},{"Start":"16:09.220 ","End":"16:11.350","Text":"I\u0027m looking for negative I can\u0027t find any,"},{"Start":"16:11.350 ","End":"16:13.675","Text":"then it would be no x."},{"Start":"16:13.675 ","End":"16:16.585","Text":"Let\u0027s take another variation."},{"Start":"16:16.585 ","End":"16:17.890","Text":"This is really 4 in 1."},{"Start":"16:17.890 ","End":"16:19.780","Text":"Let\u0027s take question 3c."},{"Start":"16:19.780 ","End":"16:23.020","Text":"Same question but with less than or equal to 0,"},{"Start":"16:23.020 ","End":"16:24.790","Text":"less than 3 equal to 0,"},{"Start":"16:24.790 ","End":"16:26.679","Text":"I\u0027m looking for negatives and zeros."},{"Start":"16:26.679 ","End":"16:28.150","Text":"There\u0027s no negatives."},{"Start":"16:28.150 ","End":"16:29.800","Text":"I can only take the 0,"},{"Start":"16:29.800 ","End":"16:32.890","Text":"so the answer would be just x=3."},{"Start":"16:32.890 ","End":"16:40.135","Text":"1 single-point would make this polynomial less than or equal to 0 when x is precisely 3."},{"Start":"16:40.135 ","End":"16:44.710","Text":"That\u0027s covered a lot of variations and I think that\u0027s"},{"Start":"16:44.710 ","End":"16:50.210","Text":"about it for quadratics and we\u0027ll move on to the next topic."}],"Thumbnail":null,"ID":5060},{"Watched":false,"Name":"Quadratic Inequalities with AND OR","Duration":"10m 17s","ChapterTopicVideoID":5068,"CourseChapterTopicPlaylistID":3876,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.150","Text":"Continuing with inequalities, I\u0027d"},{"Start":"00:03.150 ","End":"00:07.065","Text":"like to show you how to combine 2 recently learned concepts."},{"Start":"00:07.065 ","End":"00:12.990","Text":"We\u0027ve learned how to solve inequalities for quadratics, including linear."},{"Start":"00:12.990 ","End":"00:18.495","Text":"I\u0027ll throw a linear in with quadratics degree 1 or 2 inequalities."},{"Start":"00:18.495 ","End":"00:23.310","Text":"We also learned about how to combine basic inequalities with AND, OR."},{"Start":"00:23.310 ","End":"00:26.865","Text":"I\u0027d like to do a hybrid here."},{"Start":"00:26.865 ","End":"00:29.730","Text":"I\u0027ll show you what I mean with an example."},{"Start":"00:29.730 ","End":"00:35.565","Text":"The first example is x^2 minus"},{"Start":"00:35.565 ","End":"00:42.985","Text":"8x plus 7 is less than or equal to 0,"},{"Start":"00:42.985 ","End":"00:50.415","Text":"and 3x minus 2 is less than 10."},{"Start":"00:50.415 ","End":"00:52.975","Text":"A quadratic and a linear."},{"Start":"00:52.975 ","End":"00:55.820","Text":"We solve each one separately first,"},{"Start":"00:55.820 ","End":"00:58.400","Text":"let\u0027s start with the quadratic."},{"Start":"00:58.400 ","End":"01:01.560","Text":"The roots of the quadratic,"},{"Start":"01:01.600 ","End":"01:05.150","Text":"I\u0027ll write them without showing you how I got to it."},{"Start":"01:05.150 ","End":"01:08.090","Text":"The roots are 1 and 7."},{"Start":"01:08.090 ","End":"01:12.260","Text":"I know you can solve quadratic equations."},{"Start":"01:12.260 ","End":"01:14.975","Text":"That gives us a little sketch."},{"Start":"01:14.975 ","End":"01:16.985","Text":"When I take the number line,"},{"Start":"01:16.985 ","End":"01:21.830","Text":"I put in the points 1 and 7,"},{"Start":"01:21.830 ","End":"01:26.220","Text":"I choose sample points for the intervals,"},{"Start":"01:26.220 ","End":"01:28.035","Text":"let\u0027s say, 2,"},{"Start":"01:28.035 ","End":"01:31.590","Text":"0, and 8."},{"Start":"01:31.590 ","End":"01:37.485","Text":"I\u0027m doing this just quickly because we\u0027ve already done this thing."},{"Start":"01:37.485 ","End":"01:43.700","Text":"I substitute and if I plug in 0 into the quadratic,"},{"Start":"01:43.700 ","End":"01:46.895","Text":"I get 7 which is positive."},{"Start":"01:46.895 ","End":"01:50.330","Text":"When I plug in 2, I get something negative,"},{"Start":"01:50.330 ","End":"01:51.770","Text":"when I plug in 8,"},{"Start":"01:51.770 ","End":"01:54.235","Text":"I get something positive."},{"Start":"01:54.235 ","End":"01:56.700","Text":"I\u0027ll skip the calculations."},{"Start":"01:56.700 ","End":"02:00.930","Text":"We know that at 1 and at 7, it\u0027s 0."},{"Start":"02:00.930 ","End":"02:06.500","Text":"Then we just connect the dots in some way"},{"Start":"02:06.500 ","End":"02:12.710","Text":"just to show us where the expression is positive,"},{"Start":"02:12.710 ","End":"02:15.890","Text":"this bit and this bit where it\u0027s negative which is this bit,"},{"Start":"02:15.890 ","End":"02:18.515","Text":"and we\u0027re at 0 here and here."},{"Start":"02:18.515 ","End":"02:22.670","Text":"Then we see that we\u0027re looking for is less than or equal to 0."},{"Start":"02:22.670 ","End":"02:25.850","Text":"In other words, we want negatives and zeros,"},{"Start":"02:25.850 ","End":"02:28.970","Text":"so it\u0027s this bit including this and this."},{"Start":"02:28.970 ","End":"02:33.380","Text":"I would sketch this by putting a solid dot here,"},{"Start":"02:33.380 ","End":"02:35.360","Text":"a solid dot here,"},{"Start":"02:35.360 ","End":"02:38.525","Text":"and the line between them."},{"Start":"02:38.525 ","End":"02:45.395","Text":"That\u0027s the solution for this one which happens to be,"},{"Start":"02:45.395 ","End":"02:47.425","Text":"I\u0027ll just write it here,"},{"Start":"02:47.425 ","End":"02:50.240","Text":"1 less than or equal to x,"},{"Start":"02:50.240 ","End":"02:52.010","Text":"less than or equal to 7."},{"Start":"02:52.010 ","End":"02:54.605","Text":"That\u0027s the solution to this bit."},{"Start":"02:54.605 ","End":"02:58.114","Text":"This bit, we don\u0027t need a sketch for, it\u0027s linear."},{"Start":"02:58.114 ","End":"02:59.945","Text":"We can do it in our heads."},{"Start":"02:59.945 ","End":"03:02.615","Text":"3x is less than 12."},{"Start":"03:02.615 ","End":"03:05.240","Text":"Then dividing by 3 which is positive,"},{"Start":"03:05.240 ","End":"03:07.625","Text":"we get x less than 4."},{"Start":"03:07.625 ","End":"03:09.820","Text":"I\u0027ll write that over here,"},{"Start":"03:09.820 ","End":"03:11.880","Text":"x less than 4."},{"Start":"03:11.880 ","End":"03:16.150","Text":"We want the AND of these 2."},{"Start":"03:16.150 ","End":"03:18.725","Text":"We\u0027ve sketched already this bit,"},{"Start":"03:18.725 ","End":"03:21.005","Text":"x less than 4."},{"Start":"03:21.005 ","End":"03:24.425","Text":"I\u0027ll put 4 on the map, so to speak,"},{"Start":"03:24.425 ","End":"03:27.530","Text":"and then we want less than 4,"},{"Start":"03:27.530 ","End":"03:31.710","Text":"so 4 is not included,"},{"Start":"03:31.710 ","End":"03:32.940","Text":"so a hallow dot,"},{"Start":"03:32.940 ","End":"03:35.899","Text":"and less than means go to the left."},{"Start":"03:35.899 ","End":"03:40.730","Text":"Something like this and indefinitely down to minus infinity."},{"Start":"03:40.730 ","End":"03:42.950","Text":"Because at the end,"},{"Start":"03:42.950 ","End":"03:45.475","Text":"what I need is the overlap."},{"Start":"03:45.475 ","End":"03:48.735","Text":"I can just shade it like this."},{"Start":"03:48.735 ","End":"03:54.270","Text":"We see what we get is between 1 and 4,"},{"Start":"03:54.270 ","End":"03:59.910","Text":"but including the 1 and not including the 4,"},{"Start":"03:59.910 ","End":"04:03.015","Text":"so this would be our answer."},{"Start":"04:03.015 ","End":"04:05.300","Text":"If I write it down here,"},{"Start":"04:05.300 ","End":"04:12.975","Text":"this comes out to be 1 less than or equal to x less than 4,"},{"Start":"04:12.975 ","End":"04:14.610","Text":"and that\u0027s the answer,"},{"Start":"04:14.610 ","End":"04:17.460","Text":"and that\u0027s an example of AND."},{"Start":"04:17.460 ","End":"04:20.625","Text":"Now, we\u0027ll take an example with OR,"},{"Start":"04:20.625 ","End":"04:26.100","Text":"so that will be number 2."},{"Start":"04:26.100 ","End":"04:32.385","Text":"Let\u0027s take the first inequality is x^2"},{"Start":"04:32.385 ","End":"04:38.685","Text":"minus 4x minus 5 is less than 0."},{"Start":"04:38.685 ","End":"04:48.705","Text":"The second one which I combine with OR is x^2 minus 4,"},{"Start":"04:48.705 ","End":"04:51.480","Text":"this time, bigger than 0."},{"Start":"04:51.480 ","End":"04:55.640","Text":"Now, as before, we\u0027re going to solve each one"},{"Start":"04:55.640 ","End":"04:59.705","Text":"separately and then see how to use the OR to combine them."},{"Start":"04:59.705 ","End":"05:04.850","Text":"The roots, and I\u0027m not going to show you the work,"},{"Start":"05:04.850 ","End":"05:06.170","Text":"I\u0027ll just present the roots,"},{"Start":"05:06.170 ","End":"05:09.590","Text":"which means we solve the equation, this equals 0."},{"Start":"05:09.590 ","End":"05:15.380","Text":"We get minus 1 and 5."},{"Start":"05:15.380 ","End":"05:17.495","Text":"I\u0027ll do this one also."},{"Start":"05:17.495 ","End":"05:22.230","Text":"I\u0027ll tell you that the roots are plus or minus 2."},{"Start":"05:22.230 ","End":"05:26.280","Text":"I mean, you can see, if x^2 minus 4 is 0, then x^2 is 4,"},{"Start":"05:26.280 ","End":"05:28.170","Text":"so x^2 plus or minus 2,"},{"Start":"05:28.170 ","End":"05:32.895","Text":"so it\u0027s minus 2 or 2."},{"Start":"05:32.895 ","End":"05:37.875","Text":"Now, let\u0027s do a little sketch for each,"},{"Start":"05:37.875 ","End":"05:40.640","Text":"working on them both at the same time."},{"Start":"05:40.640 ","End":"05:47.945","Text":"In this one, the important points are minus 1 and 5."},{"Start":"05:47.945 ","End":"05:52.400","Text":"I\u0027m going to skip some of the stages."},{"Start":"05:52.400 ","End":"05:54.620","Text":"Normally, we would take a sample point here,"},{"Start":"05:54.620 ","End":"05:55.955","Text":"here, and here."},{"Start":"05:55.955 ","End":"06:01.705","Text":"Here, I might take 0 and then plug it in and see that I get a negative."},{"Start":"06:01.705 ","End":"06:04.200","Text":"If you plug in here, let\u0027s say,"},{"Start":"06:04.200 ","End":"06:05.940","Text":"6 minus 2,"},{"Start":"06:05.940 ","End":"06:08.075","Text":"you see that these 2 both come up positive."},{"Start":"06:08.075 ","End":"06:09.229","Text":"In short, here,"},{"Start":"06:09.229 ","End":"06:10.760","Text":"it\u0027s 0, here,"},{"Start":"06:10.760 ","End":"06:12.109","Text":"it\u0027s 0, here,"},{"Start":"06:12.109 ","End":"06:14.480","Text":"it\u0027s negative, here,"},{"Start":"06:14.480 ","End":"06:17.270","Text":"it\u0027s positive, here, it\u0027s positive."},{"Start":"06:17.270 ","End":"06:23.200","Text":"We draw what we call the snake which goes through these points,"},{"Start":"06:23.200 ","End":"06:26.695","Text":"which just goes to show where it\u0027s positive 0 and negative,"},{"Start":"06:26.695 ","End":"06:29.990","Text":"negative being below the axis and so on."},{"Start":"06:30.320 ","End":"06:39.250","Text":"In this case, the roots would be minus 2 and 2."},{"Start":"06:39.950 ","End":"06:44.890","Text":"Once again, I\u0027m going to just do this briefly."},{"Start":"06:44.890 ","End":"06:47.290","Text":"If we put in sample point of 0,"},{"Start":"06:47.290 ","End":"06:52.380","Text":"we get that it\u0027s minus 4 which is negative."},{"Start":"06:52.380 ","End":"06:53.830","Text":"If you take some points here and here,"},{"Start":"06:53.830 ","End":"06:54.840","Text":"we\u0027ll see it\u0027s positive."},{"Start":"06:54.840 ","End":"06:57.310","Text":"So we get negative, positive,"},{"Start":"06:57.310 ","End":"07:00.910","Text":"positive, and these 2 points at 0."},{"Start":"07:00.910 ","End":"07:04.880","Text":"We also get some sketch like this,"},{"Start":"07:04.880 ","End":"07:07.820","Text":"which just illustrates where it\u0027s positive,"},{"Start":"07:07.820 ","End":"07:09.650","Text":"negative, or 0."},{"Start":"07:09.650 ","End":"07:11.299","Text":"Now, in each case,"},{"Start":"07:11.299 ","End":"07:13.865","Text":"we look at what we want."},{"Start":"07:13.865 ","End":"07:17.720","Text":"The first one, we\u0027re looking for negatives only,"},{"Start":"07:17.720 ","End":"07:20.570","Text":"that\u0027s what less than 0 means."},{"Start":"07:20.570 ","End":"07:24.755","Text":"This time, I\u0027ll use the highlighting method rather than the shading."},{"Start":"07:24.755 ","End":"07:29.345","Text":"Less than 0, so it\u0027s this part here,"},{"Start":"07:29.345 ","End":"07:33.395","Text":"not including these two because we\u0027re strictly less than."},{"Start":"07:33.395 ","End":"07:40.380","Text":"For greater than, it\u0027s this bit here and this bit here."},{"Start":"07:40.600 ","End":"07:43.400","Text":"We look at the interval,"},{"Start":"07:43.400 ","End":"07:48.020","Text":"the x part for what this highlighted stuff is."},{"Start":"07:48.020 ","End":"07:53.215","Text":"It means between minus 1 and 5,"},{"Start":"07:53.215 ","End":"07:56.420","Text":"not inclusive, something like this."},{"Start":"07:56.420 ","End":"08:04.930","Text":"This is the first inequality which I could write as minus 1 less than x less than 5."},{"Start":"08:04.930 ","End":"08:08.150","Text":"For the other one, there\u0027s two branches,"},{"Start":"08:08.150 ","End":"08:15.275","Text":"it would be from 2 and to the right indefinitely."},{"Start":"08:15.275 ","End":"08:18.260","Text":"In other words, x bigger than 2."},{"Start":"08:18.260 ","End":"08:25.340","Text":"The other bit would be from minus 2 downwards,"},{"Start":"08:25.340 ","End":"08:27.695","Text":"not including the minus 2."},{"Start":"08:27.695 ","End":"08:33.470","Text":"The way I would write this would be x less"},{"Start":"08:33.470 ","End":"08:39.195","Text":"than minus 2 or x bigger than 2."},{"Start":"08:39.195 ","End":"08:43.820","Text":"Now, I want to put these two bits,"},{"Start":"08:43.820 ","End":"08:49.630","Text":"these two solutions for the separate inequalities into one picture."},{"Start":"08:49.630 ","End":"08:51.770","Text":"In this common picture,"},{"Start":"08:51.770 ","End":"08:55.084","Text":"I\u0027ll mark all the points. Let\u0027s see."},{"Start":"08:55.084 ","End":"08:58.130","Text":"We need a minus 2,"},{"Start":"08:58.130 ","End":"09:00.335","Text":"we need a minus 1,"},{"Start":"09:00.335 ","End":"09:02.495","Text":"we need 2,"},{"Start":"09:02.495 ","End":"09:05.165","Text":"and we need 5."},{"Start":"09:05.165 ","End":"09:07.625","Text":"It doesn\u0027t have to be to scale or anything,"},{"Start":"09:07.625 ","End":"09:09.990","Text":"just as long as they\u0027re in the right order."},{"Start":"09:09.990 ","End":"09:14.690","Text":"This one, I copied to here,"},{"Start":"09:14.690 ","End":"09:18.560","Text":"and this one, I drew over here,"},{"Start":"09:18.560 ","End":"09:20.440","Text":"that\u0027s this bit and this bit."},{"Start":"09:20.440 ","End":"09:23.165","Text":"Then when we take the all of them,"},{"Start":"09:23.165 ","End":"09:26.390","Text":"then we superimpose them or I like"},{"Start":"09:26.390 ","End":"09:29.390","Text":"to think of it like this is the ground and these are roofs,"},{"Start":"09:29.390 ","End":"09:31.040","Text":"and the sun is shining down,"},{"Start":"09:31.040 ","End":"09:32.630","Text":"and which part is in the shade,"},{"Start":"09:32.630 ","End":"09:35.930","Text":"or whatever image works for you."},{"Start":"09:35.930 ","End":"09:39.377","Text":"What we get is what I\u0027ve shaded here,"},{"Start":"09:39.377 ","End":"09:42.020","Text":"so there\u0027s two branches."},{"Start":"09:42.020 ","End":"09:47.970","Text":"This is written as x less than minus 2,"},{"Start":"09:47.970 ","End":"09:49.590","Text":"which is this bit,"},{"Start":"09:49.590 ","End":"09:55.905","Text":"or x bigger than minus 1."},{"Start":"09:55.905 ","End":"09:57.950","Text":"That\u0027s the answer here,"},{"Start":"09:57.950 ","End":"10:02.465","Text":"and I forgot to highlight in question 1 there."},{"Start":"10:02.465 ","End":"10:09.690","Text":"That\u0027s it. In the next clip,"},{"Start":"10:09.690 ","End":"10:16.380","Text":"we\u0027ll move on to polynomials of degree higher than 2."}],"Thumbnail":null,"ID":5061},{"Watched":false,"Name":"Polynomial Inequalities","Duration":"17m 40s","ChapterTopicVideoID":5067,"CourseChapterTopicPlaylistID":3876,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.370","Text":"Continuing it with inequalities,"},{"Start":"00:02.370 ","End":"00:04.860","Text":"we\u0027re gonna discuss polynomials."},{"Start":"00:04.860 ","End":"00:09.975","Text":"We\u0027ve already done quadratics and linear."},{"Start":"00:09.975 ","End":"00:17.020","Text":"We\u0027re going to focus on polynomials with degree three or more, cubic and higher."},{"Start":"00:17.210 ","End":"00:23.115","Text":"Let me say now that there is no general method that will work for all polynomials."},{"Start":"00:23.115 ","End":"00:27.885","Text":"I\u0027m going to assume that we can find the roots."},{"Start":"00:27.885 ","End":"00:31.545","Text":"If we can\u0027t find the roots it won\u0027t be any good."},{"Start":"00:31.545 ","End":"00:33.315","Text":"When we do find the roots,"},{"Start":"00:33.315 ","End":"00:38.855","Text":"typically will be using the what we call the snake method."},{"Start":"00:38.855 ","End":"00:42.440","Text":"Let\u0027s start straightaway with an example."},{"Start":"00:42.440 ","End":"00:43.550","Text":"Let me"},{"Start":"00:43.550 ","End":"00:45.150","Text":"take"},{"Start":"00:51.890 ","End":"00:56.790","Text":"(x-1)(x+1)(x-3)"},{"Start":"00:56.790 ","End":"00:59.585","Text":"is bigger than 0."},{"Start":"00:59.585 ","End":"01:01.010","Text":"Now this is a cubic."},{"Start":"01:01.010 ","End":"01:02.720","Text":"If we multiply it out,"},{"Start":"01:02.720 ","End":"01:05.110","Text":"we\u0027re going to get x^3 something."},{"Start":"01:05.110 ","End":"01:07.385","Text":"We\u0027re starting off easy."},{"Start":"01:07.385 ","End":"01:09.785","Text":"This is already been factorized."},{"Start":"01:09.785 ","End":"01:13.680","Text":"We can immediately say what the roots are."},{"Start":"01:14.030 ","End":"01:23.190","Text":"You can see that the roots are 1, -1 and 3."},{"Start":"01:23.190 ","End":"01:26.600","Text":"If this polynomial is going to be 0,"},{"Start":"01:26.600 ","End":"01:29.810","Text":"then one of the factors has got to be 0."},{"Start":"01:29.810 ","End":"01:32.000","Text":"If x-1 is 0, then x is 1."},{"Start":"01:32.000 ","End":"01:36.635","Text":"If this is 0, x is -1 or 3, and so on."},{"Start":"01:36.635 ","End":"01:40.850","Text":"Now we draw a little sketch."},{"Start":"01:40.850 ","End":"01:43.040","Text":"Here\u0027s a number line,"},{"Start":"01:43.040 ","End":"01:46.610","Text":"and let\u0027s put these routes on the number line,"},{"Start":"01:46.610 ","End":"01:49.190","Text":"but must get them in the right order."},{"Start":"01:49.190 ","End":"01:52.970","Text":"Scale doesn\u0027t matter, but the order matters -1,"},{"Start":"01:52.970 ","End":"01:56.485","Text":"1, and 3."},{"Start":"01:56.485 ","End":"02:00.965","Text":"Let\u0027s choose some sample points for each of the intervals."},{"Start":"02:00.965 ","End":"02:03.560","Text":"Let\u0027s take -2."},{"Start":"02:03.560 ","End":"02:05.660","Text":"Let\u0027s take here 0,"},{"Start":"02:05.660 ","End":"02:09.660","Text":"let\u0027s take 2 and let\u0027s take 4."},{"Start":"02:10.060 ","End":"02:14.360","Text":"Remember that all we care about is positive, negative or 0."},{"Start":"02:14.360 ","End":"02:18.200","Text":"If I put in -2, this is negative,"},{"Start":"02:18.200 ","End":"02:19.835","Text":"this is negative,"},{"Start":"02:19.835 ","End":"02:21.380","Text":"infact they\u0027re are all negative,"},{"Start":"02:21.380 ","End":"02:23.510","Text":"3 negatives is a negative."},{"Start":"02:23.510 ","End":"02:27.769","Text":"I just put some value here below the axis."},{"Start":"02:27.769 ","End":"02:29.330","Text":"Of course that these points,"},{"Start":"02:29.330 ","End":"02:30.695","Text":"we might as well mark them."},{"Start":"02:30.695 ","End":"02:35.575","Text":"That\u0027s where the polynomial is 0."},{"Start":"02:35.575 ","End":"02:39.540","Text":"At 0, we\u0027re going to get negative,"},{"Start":"02:39.540 ","End":"02:41.370","Text":"positive, negative."},{"Start":"02:41.370 ","End":"02:43.125","Text":"That\u0027s positive."},{"Start":"02:43.125 ","End":"02:45.720","Text":"At 2, will get 2-1 is positive,"},{"Start":"02:45.720 ","End":"02:47.880","Text":"2+1 is positive and negative."},{"Start":"02:47.880 ","End":"02:49.875","Text":"It\u0027s negative."},{"Start":"02:49.875 ","End":"02:55.085","Text":"At 4, all these expressions are positive, so it\u0027s positive."},{"Start":"02:55.085 ","End":"02:59.160","Text":"Then we draw what we call the snake."},{"Start":"02:59.630 ","End":"03:02.150","Text":"Doesn\u0027t really matter. As I said,"},{"Start":"03:02.150 ","End":"03:04.010","Text":"it\u0027s not a graph,"},{"Start":"03:04.010 ","End":"03:12.059","Text":"it\u0027s just a schematic representation of where we\u0027re above or below, around the axis."},{"Start":"03:12.070 ","End":"03:18.515","Text":"At this point we look at what type of inequality we have."},{"Start":"03:18.515 ","End":"03:23.660","Text":"What we have is a greater than 0, which means positive."},{"Start":"03:23.660 ","End":"03:25.595","Text":"We\u0027re collecting positives,"},{"Start":"03:25.595 ","End":"03:28.400","Text":"and need to do some shading or highlighting,"},{"Start":"03:28.400 ","End":"03:29.900","Text":"let\u0027s go for highlighting this time."},{"Start":"03:29.900 ","End":"03:32.835","Text":"The positive is above the axis,"},{"Start":"03:32.835 ","End":"03:39.690","Text":"is this, not including these points and this."},{"Start":"03:40.400 ","End":"03:46.880","Text":"I just need the axis that\u0027s the bit on the axis corresponding to each of these."},{"Start":"03:46.880 ","End":"03:48.530","Text":"I have from here to here,"},{"Start":"03:48.530 ","End":"03:55.440","Text":"I\u0027ll draw it over here where I have some room from -1 up to 1,"},{"Start":"03:55.440 ","End":"04:03.150","Text":"not including and from 3 onwards indefinitely."},{"Start":"04:03.150 ","End":"04:07.745","Text":"If I put that as an expression,"},{"Start":"04:07.745 ","End":"04:13.560","Text":"it\u0027s -1 less than x less than 1,"},{"Start":"04:13.560 ","End":"04:20.195","Text":"or x between -1 and 1 or x bigger than 3,"},{"Start":"04:20.195 ","End":"04:23.675","Text":"or 3, less than x, whichever way."},{"Start":"04:23.675 ","End":"04:27.060","Text":"This is our answer."},{"Start":"04:28.070 ","End":"04:30.890","Text":"I want to point something out."},{"Start":"04:30.890 ","End":"04:34.760","Text":"Let\u0027s suppose I put the dotted lines here,"},{"Start":"04:34.760 ","End":"04:40.630","Text":"like I should have done that go vertically through the roots."},{"Start":"04:40.630 ","End":"04:45.760","Text":"Suppose we had just written the plus, minus, plus, minus."},{"Start":"04:45.760 ","End":"04:51.560","Text":"Then we would have negative here, positive here,"},{"Start":"04:51.560 ","End":"04:54.035","Text":"negative in this region,"},{"Start":"04:54.035 ","End":"04:59.975","Text":"and positive in this region with 0 at the roots."},{"Start":"04:59.975 ","End":"05:02.965","Text":"Notice that they alternate."},{"Start":"05:02.965 ","End":"05:06.260","Text":"In most questions in fact, they do."},{"Start":"05:06.260 ","End":"05:08.515","Text":"But this is not always the case."},{"Start":"05:08.515 ","End":"05:12.800","Text":"Don\u0027t make the mistake of assuming that if you\u0027ve checked that something\u0027s negative here,"},{"Start":"05:12.800 ","End":"05:15.665","Text":"then it\u0027s going to alternate positive, negative, positive."},{"Start":"05:15.665 ","End":"05:21.095","Text":"The next example is going to be one where there isn\u0027t an alternation like this."},{"Start":"05:21.095 ","End":"05:25.505","Text":"Let\u0027s get on to that. That will be number 2."},{"Start":"05:25.505 ","End":"05:30.300","Text":"Number 2 will be"},{"Start":"05:31.850 ","End":"05:43.720","Text":"x^4+2x^3 - 3x^2 is less than or equal to 0."},{"Start":"05:44.070 ","End":"05:49.165","Text":"Now, the good thing is that we have everything on one side,"},{"Start":"05:49.165 ","End":"05:53.920","Text":"so we don\u0027t have to do that step to get 0 on the right,"},{"Start":"05:53.920 ","End":"05:56.859","Text":"but we don\u0027t have it factored,"},{"Start":"05:56.859 ","End":"05:59.480","Text":"so it\u0027s a little bit more work."},{"Start":"05:59.570 ","End":"06:01.680","Text":"It\u0027s a degree 4,"},{"Start":"06:01.680 ","End":"06:04.920","Text":"but look, there\u0027s no degree 1 and 2 here."},{"Start":"06:04.920 ","End":"06:09.330","Text":"It looked like I can take x^2 outside the brackets. Let\u0027s do that."},{"Start":"06:09.330 ","End":"06:19.999","Text":"We get x^2 times x^2+2x-3 is less than or equal to 0."},{"Start":"06:19.999 ","End":"06:22.855","Text":"Now to find the roots of this,"},{"Start":"06:22.855 ","End":"06:24.490","Text":"I have a product."},{"Start":"06:24.490 ","End":"06:29.349","Text":"Either x^2 is 0, or this is 0."},{"Start":"06:29.349 ","End":"06:37.750","Text":"Now if this is 0, that gives me just one root which is 0."},{"Start":"06:37.750 ","End":"06:39.970","Text":"Actually, it\u0027s debatable."},{"Start":"06:39.970 ","End":"06:42.590","Text":"It\u0027s a double root at 0."},{"Start":"06:42.690 ","End":"06:45.925","Text":"I\u0027m not really going to get into that."},{"Start":"06:45.925 ","End":"06:47.020","Text":"I\u0027ll write it twice."},{"Start":"06:47.020 ","End":"06:49.330","Text":"It\u0027s a double root, but really as far as I\u0027m concerned,"},{"Start":"06:49.330 ","End":"06:53.380","Text":"it\u0027s just one root 0 that occurs twice."},{"Start":"06:53.380 ","End":"06:56.000","Text":"From here, I won\u0027t do all the work."},{"Start":"06:56.000 ","End":"06:57.710","Text":"But if you solve the quadratic,"},{"Start":"06:57.710 ","End":"07:04.215","Text":"then you get -3 and 1 from this bit."},{"Start":"07:04.215 ","End":"07:07.780","Text":"Anyway, I\u0027ll just combine these two."},{"Start":"07:07.850 ","End":"07:10.480","Text":"I only need it once."},{"Start":"07:10.480 ","End":"07:12.290","Text":"These are the roots."},{"Start":"07:12.290 ","End":"07:15.175","Text":"Then we draw a number line."},{"Start":"07:15.175 ","End":"07:19.220","Text":"Let\u0027s put the roots on the number line but in the correct order."},{"Start":"07:19.220 ","End":"07:22.145","Text":"First I have the -3,"},{"Start":"07:22.145 ","End":"07:26.060","Text":"then the 0, and then the 1."},{"Start":"07:26.060 ","End":"07:28.160","Text":"Let\u0027s say scale doesn\u0027t matter,"},{"Start":"07:28.160 ","End":"07:30.960","Text":"but you got to get them in the right order."},{"Start":"07:31.060 ","End":"07:34.810","Text":"I\u0027ll put some dotted lines."},{"Start":"07:34.810 ","End":"07:38.950","Text":"It sometimes helps to see things."},{"Start":"07:39.020 ","End":"07:43.200","Text":"Now got to choose which method we want."},{"Start":"07:43.660 ","End":"07:46.130","Text":"I don\u0027t think I\u0027ll go for the snake this time."},{"Start":"07:46.130 ","End":"07:52.455","Text":"I\u0027ll go for the plus-minus system just for a change."},{"Start":"07:52.455 ","End":"07:54.705","Text":"Pick some sample points,"},{"Start":"07:54.705 ","End":"07:59.755","Text":"let\u0027s say -4, let\u0027s say -1."},{"Start":"07:59.755 ","End":"08:01.670","Text":"Here I can\u0027t choose a whole number,"},{"Start":"08:01.670 ","End":"08:02.779","Text":"then I take a fraction,"},{"Start":"08:02.779 ","End":"08:06.470","Text":"so I\u0027ll take a 1/2 or 0.5, whatever you prefer."},{"Start":"08:06.470 ","End":"08:08.720","Text":"Here I\u0027ll take 2."},{"Start":"08:08.720 ","End":"08:13.655","Text":"Now let\u0027s plug these into here."},{"Start":"08:13.655 ","End":"08:19.700","Text":"It occurs to me it would be simpler if I factorize this,"},{"Start":"08:19.700 ","End":"08:20.720","Text":"since I have the roots,"},{"Start":"08:20.720 ","End":"08:25.615","Text":"I know this is going to be x+3,"},{"Start":"08:25.615 ","End":"08:28.800","Text":"x-1 on the left here."},{"Start":"08:28.800 ","End":"08:31.665","Text":"If I put in -4,"},{"Start":"08:31.665 ","End":"08:34.590","Text":"well -4^2 is going to be positive."},{"Start":"08:34.590 ","End":"08:37.605","Text":"-4+3 is negative and this is negative."},{"Start":"08:37.605 ","End":"08:44.000","Text":"Positive, negative, negative, makes that positive."},{"Start":"08:44.000 ","End":"08:46.230","Text":"Then if I put in -1,"},{"Start":"08:46.230 ","End":"08:49.019","Text":"the first one is always gonna be positive."},{"Start":"08:49.240 ","End":"08:55.640","Text":"I\u0027ve got -1+3 is positive and this one is negative."},{"Start":"08:55.640 ","End":"09:01.370","Text":"I\u0027ve got negative in this interval between 0 and 1,"},{"Start":"09:01.370 ","End":"09:08.675","Text":"this sample 0.5, positive 1/2+3 is positive, 1/2-1, negative."},{"Start":"09:08.675 ","End":"09:13.160","Text":"It\u0027s negative again, no alternation."},{"Start":"09:13.160 ","End":"09:15.440","Text":"Usually it is, but not always,"},{"Start":"09:15.440 ","End":"09:17.630","Text":"so you can\u0027t assume it."},{"Start":"09:18.750 ","End":"09:22.195","Text":"Finally, the two, positive,"},{"Start":"09:22.195 ","End":"09:26.450","Text":"positive, positive, so positive."},{"Start":"09:26.450 ","End":"09:28.940","Text":"These points, it\u0027s 0,"},{"Start":"09:28.940 ","End":"09:32.390","Text":"which I denote by just putting a dot there."},{"Start":"09:32.390 ","End":"09:35.750","Text":"I know everywhere where it\u0027s positive, negative or 0."},{"Start":"09:35.750 ","End":"09:39.420","Text":"Now let\u0027s look what we need, what we\u0027re collecting."},{"Start":"09:39.590 ","End":"09:46.865","Text":"We want the less than or equal to 0 means we want zeros and negatives, but not positives."},{"Start":"09:46.865 ","End":"09:49.625","Text":"I\u0027ll highlight those that are good."},{"Start":"09:49.625 ","End":"09:52.040","Text":"0, the negative, the 0,"},{"Start":"09:52.040 ","End":"09:55.055","Text":"negative, and 0, these are the ones I\u0027m collecting."},{"Start":"09:55.055 ","End":"09:57.245","Text":"If you look at it altogether,"},{"Start":"09:57.245 ","End":"10:02.225","Text":"it just gives me everything from -3 to 1 inclusive."},{"Start":"10:02.225 ","End":"10:06.540","Text":"I\u0027ve got these points and I\u0027ve got the intervals in between."},{"Start":"10:06.610 ","End":"10:15.360","Text":"Altogether what I\u0027ve got is from here up to here, inclusive."},{"Start":"10:15.360 ","End":"10:18.660","Text":"I would write the answer as,"},{"Start":"10:18.660 ","End":"10:23.260","Text":"-3 less than or equal to x,"},{"Start":"10:23.260 ","End":"10:26.570","Text":"less than or equal to 1."},{"Start":"10:26.990 ","End":"10:30.000","Text":"And that is the answer."},{"Start":"10:30.000 ","End":"10:33.705","Text":"Now I\u0027d like to go back to the beginning"},{"Start":"10:33.705 ","End":"10:42.600","Text":"and then I\u0027m going to slightly complicate these same questions by adding,"},{"Start":"10:42.600 ","End":"10:47.865","Text":"just like with quadratics an AND/OR inequality."},{"Start":"10:47.865 ","End":"10:49.530","Text":"In the case of the first one,"},{"Start":"10:49.530 ","End":"10:52.260","Text":"I\u0027m going to add this"},{"Start":"10:52.260 ","End":"11:00.779","Text":"and x times x minus 5 is less than or equal to 0."},{"Start":"11:00.779 ","End":"11:05.040","Text":"Once again, given it factorized and 0 on the right,"},{"Start":"11:05.040 ","End":"11:07.570","Text":"just to make things easier."},{"Start":"11:07.790 ","End":"11:13.425","Text":"For this inequality, we can see immediately that the roots are"},{"Start":"11:13.425 ","End":"11:19.125","Text":"0 and 5 and let\u0027s draw the sketch here."},{"Start":"11:19.125 ","End":"11:20.460","Text":"Put a number line there,"},{"Start":"11:20.460 ","End":"11:24.690","Text":"so we have 0 and we have 5."},{"Start":"11:24.690 ","End":"11:31.515","Text":"Some dotted lines and sample points."},{"Start":"11:31.515 ","End":"11:33.120","Text":"I minus 1,1,"},{"Start":"11:33.120 ","End":"11:42.165","Text":"and 6, plug in minus 1 and we get both a negative, so it\u0027s positive."},{"Start":"11:42.165 ","End":"11:47.835","Text":"Plug in 1 and we get 1 times 1 minus 5 positive times negative is negative."},{"Start":"11:47.835 ","End":"11:51.840","Text":"Plug in 6, 6 times 6 minus 5, both are positive."},{"Start":"11:51.840 ","End":"11:57.510","Text":"Here, it\u0027s 0."},{"Start":"11:57.510 ","End":"12:00.907","Text":"I won\u0027t bother with the snake but let\u0027s just see what we\u0027re looking for."},{"Start":"12:00.907 ","End":"12:03.150","Text":"We\u0027re looking for less than or equal to 0,"},{"Start":"12:03.150 ","End":"12:07.270","Text":"in other words, negatives and 0. This is good."},{"Start":"12:07.970 ","End":"12:14.955","Text":"That gives us from 0 up to 5, inclusive."},{"Start":"12:14.955 ","End":"12:17.070","Text":"This would be our answer."},{"Start":"12:17.070 ","End":"12:21.330","Text":"In other words, 0 less than or equal to x,"},{"Start":"12:21.330 ","End":"12:24.160","Text":"less than or equal to 5."},{"Start":"12:24.410 ","End":"12:29.190","Text":"All we have to do now is take the end."},{"Start":"12:29.190 ","End":"12:34.875","Text":"So I\u0027ll just write down that I\u0027m combining this one with this one with the end."},{"Start":"12:34.875 ","End":"12:44.550","Text":"We\u0027ll do a third diagram just to end these 2 so I need to mark all the points."},{"Start":"12:44.550 ","End":"12:45.690","Text":"Let\u0027s see what do we have?"},{"Start":"12:45.690 ","End":"12:47.400","Text":"We have minus 1,"},{"Start":"12:47.400 ","End":"12:52.755","Text":"we have 0, we have 1,"},{"Start":"12:52.755 ","End":"12:56.710","Text":"we have 3,"},{"Start":"12:56.720 ","End":"13:00.450","Text":"we have 5,"},{"Start":"13:00.450 ","End":"13:09.435","Text":"and this one, I\u0027ll draw here from minus 1 to 1,"},{"Start":"13:09.435 ","End":"13:14.500","Text":"not inclusive, and from 3 onwards."},{"Start":"13:14.540 ","End":"13:20.055","Text":"This one from 0 to 5 inclusive so that would be from"},{"Start":"13:20.055 ","End":"13:29.530","Text":"0 solid dot to 5 inclusive."},{"Start":"13:29.990 ","End":"13:36.090","Text":"The end gives us the overlap so this bit and this bit."},{"Start":"13:36.090 ","End":"13:37.830","Text":"If I draw it,"},{"Start":"13:37.830 ","End":"13:43.035","Text":"what I get is from underneath then"},{"Start":"13:43.035 ","End":"13:49.980","Text":"from 0 solid dot up to 1 hollow dot."},{"Start":"13:49.980 ","End":"13:55.665","Text":"We\u0027ve also got the other bits from 3 to 5,"},{"Start":"13:55.665 ","End":"13:58.710","Text":"like this and the answer,"},{"Start":"13:58.710 ","End":"13:59.820","Text":"I\u0027ll just write it over here."},{"Start":"13:59.820 ","End":"14:03.750","Text":"The end of these 2 is between 0 and 1, in other words,"},{"Start":"14:03.750 ","End":"14:11.070","Text":"0 is less than or equal to x,"},{"Start":"14:11.070 ","End":"14:13.755","Text":"less than 1,"},{"Start":"14:13.755 ","End":"14:19.230","Text":"or 3 is less than x,"},{"Start":"14:19.230 ","End":"14:22.635","Text":"less than or equal to 5."},{"Start":"14:22.635 ","End":"14:30.285","Text":"This bit is the combined answer for this part of the end and this part of the end here."},{"Start":"14:30.285 ","End":"14:32.220","Text":"Okay, the second one,"},{"Start":"14:32.220 ","End":"14:35.955","Text":"I\u0027ll complicate by adding an O."},{"Start":"14:35.955 ","End":"14:40.180","Text":"Let\u0027s scroll down a bit first."},{"Start":"14:40.760 ","End":"14:43.860","Text":"I will write O."},{"Start":"14:43.860 ","End":"14:45.825","Text":"Better move this out of the way."},{"Start":"14:45.825 ","End":"14:55.150","Text":"Let\u0027s take something easier linear 2x minus 1 is bigger than 0."},{"Start":"14:55.580 ","End":"14:58.035","Text":"We solve this one."},{"Start":"14:58.035 ","End":"15:04.470","Text":"There\u0027s no need when it\u0027s a linear to use a snake diagram or one of these diagrams,"},{"Start":"15:04.470 ","End":"15:10.740","Text":"we can just add 1 to both sides 2x is bigger than 1,"},{"Start":"15:10.740 ","End":"15:12.700","Text":"and then divide by 2,"},{"Start":"15:12.700 ","End":"15:18.125","Text":"noting that 2 is positive so we can divide by positive and we get x"},{"Start":"15:18.125 ","End":"15:25.360","Text":"is bigger than 1/2 and x is bigger than 1/2."},{"Start":"15:25.360 ","End":"15:29.205","Text":"Now, I can make a sketch."},{"Start":"15:29.205 ","End":"15:31.440","Text":"Only one point of interest,"},{"Start":"15:31.440 ","End":"15:33.705","Text":"and that\u0027s the 1/2."},{"Start":"15:33.705 ","End":"15:38.730","Text":"The inequality looks like a hollow dot and everything to the right of"},{"Start":"15:38.730 ","End":"15:44.650","Text":"1/2 so that\u0027s this one."},{"Start":"15:44.650 ","End":"15:46.745","Text":"I\u0027ll just highlight this."},{"Start":"15:46.745 ","End":"15:51.650","Text":"Now what we want to do is to combine this one and"},{"Start":"15:51.650 ","End":"15:57.450","Text":"this one using the OR operator."},{"Start":"15:57.450 ","End":"16:01.335","Text":"Let\u0027s put both of these on the same sketch."},{"Start":"16:01.335 ","End":"16:03.570","Text":"In the combined diagram,"},{"Start":"16:03.570 ","End":"16:04.890","Text":"I need all these points."},{"Start":"16:04.890 ","End":"16:07.605","Text":"I need minus 3."},{"Start":"16:07.605 ","End":"16:11.760","Text":"I need 0, I need 1/2 from here,"},{"Start":"16:11.760 ","End":"16:13.890","Text":"I need 1 from here,"},{"Start":"16:13.890 ","End":"16:16.125","Text":"and that\u0027s about it."},{"Start":"16:16.125 ","End":"16:18.945","Text":"Let\u0027s take this inequality,"},{"Start":"16:18.945 ","End":"16:22.380","Text":"x between minus 3 and 1 inclusive."},{"Start":"16:22.380 ","End":"16:27.570","Text":"We have the minus 3 with a solid dot and"},{"Start":"16:27.570 ","End":"16:36.270","Text":"the 1 with a solid dot and join it up."},{"Start":"16:36.270 ","End":"16:39.690","Text":"We don\u0027t need the 0 I just realized."},{"Start":"16:39.690 ","End":"16:41.565","Text":"Then the second one,"},{"Start":"16:41.565 ","End":"16:50.595","Text":"x bigger than 1/2 so it\u0027s 1/2 not included and anything to the right of that."},{"Start":"16:50.595 ","End":"16:53.415","Text":"Now we need to owe them,"},{"Start":"16:53.415 ","End":"16:59.925","Text":"which means that I take anything that\u0027s covered by at least one of them."},{"Start":"16:59.925 ","End":"17:02.430","Text":"I like to imagine that this is the ground and"},{"Start":"17:02.430 ","End":"17:05.010","Text":"these are 2 roofs and the sun is shining and where does"},{"Start":"17:05.010 ","End":"17:10.439","Text":"the shade go or whatever picture you want to squish them down together."},{"Start":"17:10.439 ","End":"17:14.910","Text":"What we get basically is from here"},{"Start":"17:14.910 ","End":"17:21.225","Text":"all the way to the right on indefinitely,"},{"Start":"17:21.225 ","End":"17:26.340","Text":"which is what I would call x bigger or equal"},{"Start":"17:26.340 ","End":"17:34.620","Text":"to minus 3 and that would be the answer for the combined."},{"Start":"17:34.620 ","End":"17:37.155","Text":"Just need to highlight this."},{"Start":"17:37.155 ","End":"17:40.210","Text":"I think that\u0027s enough for now."}],"Thumbnail":null,"ID":5062},{"Watched":false,"Name":"Exercise 1","Duration":"16m 45s","ChapterTopicVideoID":5060,"CourseChapterTopicPlaylistID":3876,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.690","Text":"In this exercise, we have a bunch of inequalities,"},{"Start":"00:03.690 ","End":"00:05.100","Text":"4 to be precise."},{"Start":"00:05.100 ","End":"00:08.740","Text":"Each of them is a quadratic inequality."},{"Start":"00:08.840 ","End":"00:11.760","Text":"Let\u0027s start with the first 1."},{"Start":"00:11.760 ","End":"00:15.885","Text":"We have that x^2 is less than a 144."},{"Start":"00:15.885 ","End":"00:27.030","Text":"The usual way is to bring everything to 1 side and then get x^2 minus a 144 less than 0."},{"Start":"00:27.200 ","End":"00:29.220","Text":"When we have a quadratic,"},{"Start":"00:29.220 ","End":"00:33.295","Text":"we try to factorize it or at least to find its roots."},{"Start":"00:33.295 ","End":"00:36.650","Text":"In this case, because there\u0027s a missing x term,"},{"Start":"00:36.650 ","End":"00:40.290","Text":"we can do it easier."},{"Start":"00:40.290 ","End":"00:48.815","Text":"Actually, we could use the difference of squares formula because 144 is 12^2."},{"Start":"00:48.815 ","End":"00:52.055","Text":"We could write this as x minus 12,"},{"Start":"00:52.055 ","End":"00:54.030","Text":"x plus 12,"},{"Start":"00:54.030 ","End":"00:56.405","Text":"and in case you\u0027re not sure,"},{"Start":"00:56.405 ","End":"01:05.370","Text":"I\u0027m just using the formula that a^2 minus b ^2 is a minus b times a plus b,"},{"Start":"01:05.370 ","End":"01:07.950","Text":"the difference of squares formula."},{"Start":"01:09.700 ","End":"01:12.980","Text":"Once we have it factorized,"},{"Start":"01:12.980 ","End":"01:15.800","Text":"we know that the roots,"},{"Start":"01:15.800 ","End":"01:19.160","Text":"also 12 and minus 12,"},{"Start":"01:19.160 ","End":"01:23.990","Text":"the roots are the places where this thing is 0."},{"Start":"01:23.990 ","End":"01:27.035","Text":"At this point we draw a sketch."},{"Start":"01:27.035 ","End":"01:29.720","Text":"That\u0027s the best way to go about it."},{"Start":"01:29.720 ","End":"01:32.270","Text":"Let\u0027s start with a number line,"},{"Start":"01:32.270 ","End":"01:35.405","Text":"and let\u0027s put in the 2 roots."},{"Start":"01:35.405 ","End":"01:40.175","Text":"Let\u0027s say the points are here and here."},{"Start":"01:40.175 ","End":"01:42.590","Text":"The left 1 has to be minus 12."},{"Start":"01:42.590 ","End":"01:45.240","Text":"The right 1 is 12."},{"Start":"01:46.100 ","End":"01:50.120","Text":"These 2 points divide the line into 3 intervals up"},{"Start":"01:50.120 ","End":"01:53.930","Text":"to here between the 2 and to the right of this 1."},{"Start":"01:53.930 ","End":"01:58.175","Text":"What we want to do is find out in each of the intervals,"},{"Start":"01:58.175 ","End":"02:03.020","Text":"whether the polynomial is positive or negative,"},{"Start":"02:03.020 ","End":"02:06.425","Text":"it won\u0027t change in the middle is 0 here and here,"},{"Start":"02:06.425 ","End":"02:09.665","Text":"because these are the roots, but here it will be always plus or always minus."},{"Start":"02:09.665 ","End":"02:12.710","Text":"What we do is we choose a sample point."},{"Start":"02:12.710 ","End":"02:15.380","Text":"Sometimes we do this in our heads."},{"Start":"02:15.380 ","End":"02:17.075","Text":"Sometimes I\u0027ll write it down."},{"Start":"02:17.075 ","End":"02:19.820","Text":"Whenever I can choose 0, I like to choose 0."},{"Start":"02:19.820 ","End":"02:21.935","Text":"It\u0027s always easy to substitute."},{"Start":"02:21.935 ","End":"02:25.860","Text":"Here I\u0027ll choose 13."},{"Start":"02:25.860 ","End":"02:28.500","Text":"I often choose the next whole number along."},{"Start":"02:28.500 ","End":"02:31.870","Text":"Here I\u0027ll choose minus 13."},{"Start":"02:31.870 ","End":"02:37.130","Text":"Then I substitute this in this expression or in the original,"},{"Start":"02:37.130 ","End":"02:40.895","Text":"but it\u0027s easier usually to substitute it in the factorized expression."},{"Start":"02:40.895 ","End":"02:43.190","Text":"Because all I really want to know is not the answer,"},{"Start":"02:43.190 ","End":"02:45.260","Text":"but just whether it\u0027s positive or negative."},{"Start":"02:45.260 ","End":"02:47.210","Text":"I\u0027ve put in 0."},{"Start":"02:47.210 ","End":"02:52.775","Text":"Here, I have a negative times a positive,"},{"Start":"02:52.775 ","End":"02:55.010","Text":"which is a negative."},{"Start":"02:55.010 ","End":"02:57.960","Text":"I draw some point here."},{"Start":"02:58.570 ","End":"03:02.450","Text":"Sorry, it\u0027s going to be below negative,"},{"Start":"03:02.450 ","End":"03:04.820","Text":"so I draw some point below."},{"Start":"03:04.820 ","End":"03:08.930","Text":"Let\u0027s try 13.13 minus 12 is positive,"},{"Start":"03:08.930 ","End":"03:11.000","Text":"13 plus 12 is positive."},{"Start":"03:11.000 ","End":"03:12.260","Text":"Altogether positive."},{"Start":"03:12.260 ","End":"03:16.360","Text":"I draw some point above the axis."},{"Start":"03:16.360 ","End":"03:21.615","Text":"For minus 13, this is going to be negative and this is going to be negative."},{"Start":"03:21.615 ","End":"03:23.120","Text":"I\u0027m also going to get a positive."},{"Start":"03:23.120 ","End":"03:24.425","Text":"I\u0027m thinking above."},{"Start":"03:24.425 ","End":"03:30.825","Text":"Sometimes I\u0027ll actually just write something like plus,"},{"Start":"03:30.825 ","End":"03:34.270","Text":"minus, plus, and"},{"Start":"03:34.270 ","End":"03:37.670","Text":"sometimes we actually draw the points and maybe even do a little sketch."},{"Start":"03:37.670 ","End":"03:46.200","Text":"Remember it goes through these points which are the roots."},{"Start":"03:46.900 ","End":"03:49.505","Text":"Let\u0027s see what interests us."},{"Start":"03:49.505 ","End":"03:52.385","Text":"What interests us is the negative values."},{"Start":"03:52.385 ","End":"03:55.585","Text":"I can see that the line,"},{"Start":"03:55.585 ","End":"03:59.390","Text":"sometimes called the snake is negative between here and here,"},{"Start":"03:59.390 ","End":"04:01.355","Text":"sometimes its shading,"},{"Start":"04:01.355 ","End":"04:03.680","Text":"makes it more obvious."},{"Start":"04:03.680 ","End":"04:12.155","Text":"But I think we can see that the negative portion is between 12 and minus 12,"},{"Start":"04:12.155 ","End":"04:16.200","Text":"not including the 12 and minus 12."},{"Start":"04:16.200 ","End":"04:18.515","Text":"If we write this algebraically,"},{"Start":"04:18.515 ","End":"04:22.250","Text":"it\u0027s minus 12 less than x,"},{"Start":"04:22.250 ","End":"04:23.915","Text":"less than 12,"},{"Start":"04:23.915 ","End":"04:27.635","Text":"and that is the answer."},{"Start":"04:27.635 ","End":"04:32.050","Text":"I\u0027ll highlight it, strange color."},{"Start":"04:32.050 ","End":"04:34.155","Text":"That\u0027s better, anyway."},{"Start":"04:34.155 ","End":"04:36.120","Text":"Onto Part b."},{"Start":"04:36.120 ","End":"04:39.620","Text":"Part b, we also have a quadratic,"},{"Start":"04:39.620 ","End":"04:41.990","Text":"it\u0027s also missing quadratic,"},{"Start":"04:41.990 ","End":"04:45.780","Text":"it\u0027s missing the constant part."},{"Start":"04:45.860 ","End":"04:50.135","Text":"What we can do actually is take out a factor."},{"Start":"04:50.135 ","End":"04:53.945","Text":"Then in general if you see that everything\u0027s divisible by 3,"},{"Start":"04:53.945 ","End":"04:56.140","Text":"let\u0027s take 3 out."},{"Start":"04:56.140 ","End":"05:00.900","Text":"Let\u0027s even take minus 3 out because I\u0027d like to have positive x^2."},{"Start":"05:00.900 ","End":"05:02.460","Text":"If we take minus 3,"},{"Start":"05:02.460 ","End":"05:04.185","Text":"we\u0027re left with x^2,"},{"Start":"05:04.185 ","End":"05:07.250","Text":"and minus 3, if we divide it into 12,"},{"Start":"05:07.250 ","End":"05:09.490","Text":"comes out minus 4,"},{"Start":"05:09.490 ","End":"05:15.515","Text":"and that\u0027s x, of course,"},{"Start":"05:15.515 ","End":"05:19.070","Text":"is bigger than 0."},{"Start":"05:19.070 ","End":"05:21.400","Text":"I\u0027d like to get rid of this minus 3,"},{"Start":"05:21.400 ","End":"05:24.770","Text":"so let\u0027s divide both sides by minus 3."},{"Start":"05:24.770 ","End":"05:27.485","Text":"But be careful this is a negative number."},{"Start":"05:27.485 ","End":"05:31.415","Text":"If it\u0027s a negative number and you multiply or divide by a negative,"},{"Start":"05:31.415 ","End":"05:35.035","Text":"then you got to change the sign of the inequality."},{"Start":"05:35.035 ","End":"05:44.050","Text":"Dividing by 3 gives us that x^2 minus 4x is less than 0 over minus 3 is 0."},{"Start":"05:44.800 ","End":"05:49.010","Text":"Now this is easy to factorize because,"},{"Start":"05:49.010 ","End":"05:55.060","Text":"see it\u0027s x times x minus 4 is less than 0."},{"Start":"05:55.060 ","End":"06:02.070","Text":"The roots when x is 0 or when x minus 4 is 0, which is 4."},{"Start":"06:02.070 ","End":"06:03.910","Text":"These are the 2 roots,"},{"Start":"06:03.910 ","End":"06:07.310","Text":"and let\u0027s draw them on the number line."},{"Start":"06:07.310 ","End":"06:10.270","Text":"Here\u0027s a number line."},{"Start":"06:10.270 ","End":"06:15.370","Text":"Here are the points 0 and 4."},{"Start":"06:15.920 ","End":"06:21.950","Text":"Let\u0027s see if we can take some sample points."},{"Start":"06:21.950 ","End":"06:24.560","Text":"Sometimes I actually write them down."},{"Start":"06:24.560 ","End":"06:27.260","Text":"Sometimes we just do everything in our heads."},{"Start":"06:27.260 ","End":"06:31.280","Text":"This time I\u0027ll write down some sample points."},{"Start":"06:31.280 ","End":"06:33.380","Text":"Let\u0027s say we take 2 here,"},{"Start":"06:33.380 ","End":"06:35.660","Text":"let\u0027s say we take 5 here,"},{"Start":"06:35.660 ","End":"06:39.060","Text":"let\u0027s say we take minus 1 here."},{"Start":"06:39.290 ","End":"06:43.815","Text":"If I put minus 1 here,"},{"Start":"06:43.815 ","End":"06:47.265","Text":"I substitute it in this or this, it\u0027s easier in this."},{"Start":"06:47.265 ","End":"06:49.020","Text":"Minus 1 is negative,"},{"Start":"06:49.020 ","End":"06:51.200","Text":"minus 1 minus 4 is also negative,"},{"Start":"06:51.200 ","End":"06:55.010","Text":"negative times negative is positive."},{"Start":"06:55.010 ","End":"06:59.525","Text":"I just put any point above the lines is not going to be an accurate sketch."},{"Start":"06:59.525 ","End":"07:05.845","Text":"Sometimes I just write plus here to say that it\u0027s positive."},{"Start":"07:05.845 ","End":"07:09.060","Text":"You choose your own variation that suits you."},{"Start":"07:09.060 ","End":"07:12.000","Text":"For 2, we see that 2 is positive,"},{"Start":"07:12.000 ","End":"07:13.520","Text":"but 2 minus 4 is negative,"},{"Start":"07:13.520 ","End":"07:16.340","Text":"positive times negative is negative."},{"Start":"07:16.340 ","End":"07:18.635","Text":"So it\u0027s a point below the axis."},{"Start":"07:18.635 ","End":"07:20.825","Text":"I\u0027m doing both at the same time."},{"Start":"07:20.825 ","End":"07:24.485","Text":"I write minus, you do 1 over the other, whichever."},{"Start":"07:24.485 ","End":"07:28.410","Text":"Then and x is 5, 5 is positive,"},{"Start":"07:28.410 ","End":"07:30.284","Text":"5 minus 4 is positive,"},{"Start":"07:30.284 ","End":"07:34.000","Text":"so it\u0027s positive, so I draw a line again above the axis."},{"Start":"07:34.000 ","End":"07:35.640","Text":"This is an either or."},{"Start":"07:35.640 ","End":"07:37.395","Text":"In fact let\u0027s just stick with 1,"},{"Start":"07:37.395 ","End":"07:41.000","Text":"analyze the extra stuff."},{"Start":"07:41.000 ","End":"07:42.725","Text":"Both methods are good."},{"Start":"07:42.725 ","End":"07:47.660","Text":"Then we can draw a quick sketch."},{"Start":"07:47.660 ","End":"07:52.669","Text":"At the roots, it actually crosses the axis, that\u0027s where it\u0027s 0."},{"Start":"07:52.669 ","End":"07:54.815","Text":"We\u0027ve got something like this."},{"Start":"07:54.815 ","End":"07:59.485","Text":"What we want is the negative."},{"Start":"07:59.485 ","End":"08:03.030","Text":"The negative begins below the axis,"},{"Start":"08:03.030 ","End":"08:05.205","Text":"so it\u0027s this part here,"},{"Start":"08:05.205 ","End":"08:13.270","Text":"which means between 0 and 4 not including."},{"Start":"08:13.310 ","End":"08:22.520","Text":"The answer is 0 less than x, less than 4."},{"Start":"08:22.520 ","End":"08:26.920","Text":"We are done with Part B."},{"Start":"08:26.920 ","End":"08:32.440","Text":"Let\u0027s move on to Part C. What do we have here?"},{"Start":"08:32.440 ","End":"08:36.650","Text":"We have a quadratic bigger than 0."},{"Start":"08:38.610 ","End":"08:43.060","Text":"Well, first of all, this is a square and this is a square with a plus."},{"Start":"08:43.060 ","End":"08:47.020","Text":"You think maybe one of those special binomial products."},{"Start":"08:47.020 ","End":"08:49.615","Text":"I\u0027ll remind you what I\u0027m talking about."},{"Start":"08:49.615 ","End":"08:52.735","Text":"There\u0027s 2 of them. There is one with a minus, one with the plus."},{"Start":"08:52.735 ","End":"08:55.045","Text":"But the one with the minus says"},{"Start":"08:55.045 ","End":"09:05.035","Text":"that a^2-2ab+b^2 is (a-b)^2."},{"Start":"09:05.035 ","End":"09:07.750","Text":"You more commonly see it in this direction,"},{"Start":"09:07.750 ","End":"09:11.570","Text":"but we\u0027re going to use it in this direction."},{"Start":"09:12.360 ","End":"09:15.280","Text":"We see the a^2 and the b^2."},{"Start":"09:15.280 ","End":"09:20.800","Text":"We know a has to be x and b has to be 5,"},{"Start":"09:20.800 ","End":"09:22.855","Text":"and this has to be something squared."},{"Start":"09:22.855 ","End":"09:25.660","Text":"We know that it has to be a minus and a squared."},{"Start":"09:25.660 ","End":"09:30.160","Text":"But you do have to check since we didn\u0027t validate the middle term."},{"Start":"09:30.160 ","End":"09:32.665","Text":"We never checked that this is 2ab,"},{"Start":"09:32.665 ","End":"09:34.735","Text":"that\u0027s one thing you have to do."},{"Start":"09:34.735 ","End":"09:36.520","Text":"Twice a times b,"},{"Start":"09:36.520 ","End":"09:39.625","Text":"twice x times 5 really is 10x."},{"Start":"09:39.625 ","End":"09:42.205","Text":"This is correct."},{"Start":"09:42.205 ","End":"09:45.680","Text":"We have bigger than 0."},{"Start":"09:47.190 ","End":"09:54.070","Text":"In this case, there is only 1 root,"},{"Start":"09:54.070 ","End":"09:56.800","Text":"because this is x minus 5 times x minus 5."},{"Start":"09:56.800 ","End":"09:59.335","Text":"Only x equals 5 is a root."},{"Start":"09:59.335 ","End":"10:04.540","Text":"If you do it with the sketch method,"},{"Start":"10:04.540 ","End":"10:06.744","Text":"then we draw a number line."},{"Start":"10:06.744 ","End":"10:10.378","Text":"But there\u0027s only one interesting point, 0.5,"},{"Start":"10:10.378 ","End":"10:12.730","Text":"5 is the root,"},{"Start":"10:12.730 ","End":"10:14.815","Text":"the thing that makes it 0."},{"Start":"10:14.815 ","End":"10:20.365","Text":"We\u0027ll just put some number 5 here."},{"Start":"10:20.365 ","End":"10:23.170","Text":"Now, at this point,"},{"Start":"10:23.170 ","End":"10:26.050","Text":"the expression is 0."},{"Start":"10:26.050 ","End":"10:28.225","Text":"Let\u0027s check what it is on the right."},{"Start":"10:28.225 ","End":"10:33.850","Text":"Let\u0027s take, for example, 6 and 4."},{"Start":"10:33.850 ","End":"10:35.815","Text":"If I put 6 in here,"},{"Start":"10:35.815 ","End":"10:39.640","Text":"6 minus 5 is 1, squared is positive."},{"Start":"10:39.640 ","End":"10:41.560","Text":"If I put 4 here,"},{"Start":"10:41.560 ","End":"10:43.780","Text":"4 minus 5 is minus 1,"},{"Start":"10:43.780 ","End":"10:46.225","Text":"the squared is still positive."},{"Start":"10:46.225 ","End":"10:51.260","Text":"It\u0027s positive here and positive here."},{"Start":"10:51.470 ","End":"10:58.860","Text":"Basically, we could say straight away that it\u0027s positive everywhere except 5."},{"Start":"10:58.860 ","End":"11:01.275","Text":"We can also draw the sketch."},{"Start":"11:01.275 ","End":"11:03.510","Text":"Instead of just writing plus and plus,"},{"Start":"11:03.510 ","End":"11:08.880","Text":"we could put a point above and a point above the axis."},{"Start":"11:08.880 ","End":"11:11.670","Text":"Actually, we always know that at the root is 0."},{"Start":"11:11.670 ","End":"11:15.999","Text":"The picture looks something like this."},{"Start":"11:15.999 ","End":"11:22.300","Text":"The place where it\u0027s positive is everywhere except for here, when it\u0027s 0."},{"Start":"11:22.300 ","End":"11:25.060","Text":"It\u0027s momentarily 0 but positive."},{"Start":"11:25.060 ","End":"11:32.695","Text":"What we would actually say is x can be anything except 5."},{"Start":"11:32.695 ","End":"11:34.240","Text":"This is actually the answer."},{"Start":"11:34.240 ","End":"11:37.430","Text":"X not equal to 5."},{"Start":"11:38.490 ","End":"11:43.150","Text":"But it could\u0027ve been a short way of doing it. I\u0027ll mention it."},{"Start":"11:43.150 ","End":"11:44.830","Text":"Once you see this,"},{"Start":"11:44.830 ","End":"11:47.920","Text":"if something squared is bigger than 0,"},{"Start":"11:47.920 ","End":"11:52.240","Text":"everything squared is bigger than 0 except for 0."},{"Start":"11:52.240 ","End":"11:57.595","Text":"We could have said straight away that x minus 5 is not equal to 0."},{"Start":"11:57.595 ","End":"12:00.640","Text":"Then we would have gotten this straightaway."},{"Start":"12:00.640 ","End":"12:04.900","Text":"Because the only time that x minus 5 is 0 is when x is 5."},{"Start":"12:04.900 ","End":"12:08.470","Text":"And if this is not 0,"},{"Start":"12:08.470 ","End":"12:11.080","Text":"then the x is not 5."},{"Start":"12:11.080 ","End":"12:17.170","Text":"By the way, it\u0027s not a mistake to write x is less than 5,"},{"Start":"12:17.170 ","End":"12:19.210","Text":"or x is bigger than 5."},{"Start":"12:19.210 ","End":"12:21.700","Text":"It\u0027s not a mistake. It would be okay to write that."},{"Start":"12:21.700 ","End":"12:25.060","Text":"It\u0027s just that this is more compact or more elegant."},{"Start":"12:25.060 ","End":"12:28.160","Text":"But you could actually write it this way."},{"Start":"12:28.350 ","End":"12:34.240","Text":"That does Part C. Now let\u0027s get onto Part D,"},{"Start":"12:34.240 ","End":"12:36.355","Text":"which is the last part."},{"Start":"12:36.355 ","End":"12:38.170","Text":"We have this inequality."},{"Start":"12:38.170 ","End":"12:39.940","Text":"Let\u0027s write it in standard form,"},{"Start":"12:39.940 ","End":"12:45.400","Text":"x^2 minus 12x plus 32 bigger than 0."},{"Start":"12:45.400 ","End":"12:49.100","Text":"Put the whole quadratic on the left in the right order."},{"Start":"12:50.340 ","End":"12:55.030","Text":"Let\u0027s see. We want to try and factorize it."},{"Start":"12:55.030 ","End":"13:03.100","Text":"We can either factorize it as a trinomial if you know how to do that."},{"Start":"13:03.100 ","End":"13:08.440","Text":"Or we can solve the equation and get 2 roots."},{"Start":"13:08.440 ","End":"13:10.555","Text":"Either way, you know what,"},{"Start":"13:10.555 ","End":"13:14.425","Text":"I\u0027ll do it with the quadratic formula."},{"Start":"13:14.425 ","End":"13:16.660","Text":"Sometimes I just give you the answer to save time,"},{"Start":"13:16.660 ","End":"13:19.570","Text":"but let\u0027s just time do it and say,"},{"Start":"13:19.570 ","End":"13:25.840","Text":"we have the roots for the equation where this is 0."},{"Start":"13:25.840 ","End":"13:31.480","Text":"We have x is 12 plus or minus the square root of b squared is a"},{"Start":"13:31.480 ","End":"13:36.385","Text":"144 minus 4 times"},{"Start":"13:36.385 ","End":"13:43.120","Text":"a times c is minus 32 over 2."},{"Start":"13:43.120 ","End":"13:48.490","Text":"Now, 4 times 32 is 128."},{"Start":"13:48.490 ","End":"13:54.055","Text":"What we get is 12 plus or minus 128 from"},{"Start":"13:54.055 ","End":"14:01.255","Text":"44 is 36 over 2."},{"Start":"14:01.255 ","End":"14:04.270","Text":"Square root of 36 is just 6."},{"Start":"14:04.270 ","End":"14:07.930","Text":"What we get is 12 plus 6 over 2,"},{"Start":"14:07.930 ","End":"14:15.580","Text":"which comes out to be please excuse me,"},{"Start":"14:15.580 ","End":"14:18.040","Text":"I did the subtraction wrong in my head."},{"Start":"14:18.040 ","End":"14:24.265","Text":"It is not 36, it is only 16 and the square root is 4, sorry about that."},{"Start":"14:24.265 ","End":"14:30.670","Text":"12 plus 4 over 2 is 16 over 2 is 8,"},{"Start":"14:30.670 ","End":"14:33.955","Text":"12 minus 4 is 8 over 2 is 4."},{"Start":"14:33.955 ","End":"14:36.940","Text":"These would be 2 roots,"},{"Start":"14:36.940 ","End":"14:41.166","Text":"and then we would factorize it as x minus 8,"},{"Start":"14:41.166 ","End":"14:45.190","Text":"x minus 4, bigger than 0."},{"Start":"14:45.190 ","End":"14:47.875","Text":"I\u0027ll just write again."},{"Start":"14:47.875 ","End":"14:51.610","Text":"The roots are 8 and 4."},{"Start":"14:51.610 ","End":"14:59.980","Text":"If I make a sketch like this and add the 2 numbers,"},{"Start":"14:59.980 ","End":"15:01.600","Text":"the left one has to be 4,"},{"Start":"15:01.600 ","End":"15:04.480","Text":"the right one has to be 8."},{"Start":"15:04.480 ","End":"15:07.615","Text":"Let me not bother with the sketch this time."},{"Start":"15:07.615 ","End":"15:10.764","Text":"We\u0027ll just do everything in our heads."},{"Start":"15:10.764 ","End":"15:13.750","Text":"I\u0027ll just write the sign."},{"Start":"15:13.750 ","End":"15:17.590","Text":"If x, let\u0027s say is here I\u0027ll choose the point 0."},{"Start":"15:17.590 ","End":"15:20.395","Text":"0 minus 8, 0 minus 4 are both negative."},{"Start":"15:20.395 ","End":"15:22.435","Text":"Here it\u0027s positive."},{"Start":"15:22.435 ","End":"15:24.950","Text":"Between 4 and 8, I\u0027ll choose, let\u0027s say 6,"},{"Start":"15:24.950 ","End":"15:27.160","Text":"6 minus 8 is negative,"},{"Start":"15:27.160 ","End":"15:29.005","Text":"but 6 minus 4 is positive,"},{"Start":"15:29.005 ","End":"15:32.125","Text":"negative times positive is negative."},{"Start":"15:32.125 ","End":"15:34.390","Text":"If I take say, 10,"},{"Start":"15:34.390 ","End":"15:35.725","Text":"10 minus 8,"},{"Start":"15:35.725 ","End":"15:37.780","Text":"10 minus 4, they\u0027re both positive."},{"Start":"15:37.780 ","End":"15:39.400","Text":"That\u0027s a positive."},{"Start":"15:39.400 ","End":"15:43.090","Text":"What I want is positive."},{"Start":"15:43.090 ","End":"15:46.780","Text":"This part is the bigger than 0, it\u0027s the plus."},{"Start":"15:46.780 ","End":"15:53.140","Text":"I see that the plus occurs in this part here."},{"Start":"15:53.140 ","End":"15:56.770","Text":"We\u0027ll just go over it this part here,"},{"Start":"15:56.770 ","End":"16:02.860","Text":"up to but not including the 4 and from the 8 onwards."},{"Start":"16:02.860 ","End":"16:13.370","Text":"What we say is x less than 4 or x bigger than 8."},{"Start":"16:13.410 ","End":"16:17.890","Text":"If you did a sketch for those who liked them,"},{"Start":"16:17.890 ","End":"16:19.420","Text":"I\u0027ll just throw one in quickly."},{"Start":"16:19.420 ","End":"16:21.880","Text":"A positive would mean we throw something above,"},{"Start":"16:21.880 ","End":"16:26.860","Text":"something below, here\u0027s 0."},{"Start":"16:26.860 ","End":"16:30.640","Text":"It goes through here, something like this."},{"Start":"16:30.640 ","End":"16:32.710","Text":"Then we say we want the positive,"},{"Start":"16:32.710 ","End":"16:35.095","Text":"so it\u0027s here and it\u0027s here."},{"Start":"16:35.095 ","End":"16:39.010","Text":"We see it\u0027s to the right of 8 and to the left of 4."},{"Start":"16:39.010 ","End":"16:41.920","Text":"Anyway, this is the answer."},{"Start":"16:41.920 ","End":"16:45.170","Text":"That\u0027s the last part. So we\u0027re done."}],"Thumbnail":null,"ID":5063},{"Watched":false,"Name":"Exercise 2","Duration":"13m 46s","ChapterTopicVideoID":5061,"CourseChapterTopicPlaylistID":3876,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.840","Text":"In this exercise, we have four different inequalities to solve."},{"Start":"00:03.840 ","End":"00:05.295","Text":"They\u0027re all quadratic."},{"Start":"00:05.295 ","End":"00:07.260","Text":"Some are bigger or equal to 0,"},{"Start":"00:07.260 ","End":"00:09.030","Text":"some less than 0."},{"Start":"00:09.030 ","End":"00:11.850","Text":"Let\u0027s do them one at a time."},{"Start":"00:11.850 ","End":"00:13.815","Text":"We\u0027ll start with a,"},{"Start":"00:13.815 ","End":"00:20.245","Text":"and you happen to notice that they\u0027re all divisible by 2,"},{"Start":"00:20.245 ","End":"00:25.000","Text":"then makes things simpler if we just get rid of that 2."},{"Start":"00:25.370 ","End":"00:28.720","Text":"Let\u0027s just divide everything by 2."},{"Start":"00:28.720 ","End":"00:30.485","Text":"Since 2 is positive,"},{"Start":"00:30.485 ","End":"00:33.620","Text":"we don\u0027t have to change the direction of the inequality."},{"Start":"00:33.620 ","End":"00:35.960","Text":"Also 0 over 2 is 0,"},{"Start":"00:35.960 ","End":"00:39.370","Text":"so really we\u0027re just dividing the left-hand side by 2,"},{"Start":"00:39.370 ","End":"00:46.930","Text":"so x^2 plus x plus 12 is bigger or equal to 0."},{"Start":"00:46.930 ","End":"00:51.140","Text":"The next thing to do is to factorize this."},{"Start":"00:51.140 ","End":"00:54.440","Text":"You either find the roots with the quadratic formula or you"},{"Start":"00:54.440 ","End":"00:58.145","Text":"use the trinomial decomposition method."},{"Start":"00:58.145 ","End":"01:01.740","Text":"Either way, I won\u0027t waste time with that,"},{"Start":"01:01.740 ","End":"01:07.745","Text":"you will get that it doesn\u0027t decompose."},{"Start":"01:07.745 ","End":"01:09.560","Text":"The trinomial method won\u0027t work,"},{"Start":"01:09.560 ","End":"01:11.090","Text":"and if you try the formula,"},{"Start":"01:11.090 ","End":"01:14.135","Text":"you will get under the square root sign,"},{"Start":"01:14.135 ","End":"01:17.549","Text":"the b^2 minus 4ac will be negative."},{"Start":"01:17.549 ","End":"01:20.520","Text":"This one has no solutions,"},{"Start":"01:20.520 ","End":"01:22.520","Text":"and if it has no roots,"},{"Start":"01:22.520 ","End":"01:26.735","Text":"that means that it\u0027s always positive or always negative."},{"Start":"01:26.735 ","End":"01:30.189","Text":"I\u0027ll just write down that it has no roots,"},{"Start":"01:30.189 ","End":"01:34.969","Text":"no decomposition, and then if we let, for example,"},{"Start":"01:34.969 ","End":"01:40.940","Text":"x equals 0, then what we get is something which is positive,"},{"Start":"01:40.940 ","End":"01:42.530","Text":"we get specifically 12,"},{"Start":"01:42.530 ","End":"01:44.734","Text":"but that\u0027s not important that it\u0027s positive."},{"Start":"01:44.734 ","End":"01:48.380","Text":"If it\u0027s positive at one point and it never crosses the 0,"},{"Start":"01:48.380 ","End":"01:50.780","Text":"it will be always positive."},{"Start":"01:50.780 ","End":"01:57.220","Text":"This thing is, it\u0027s bigger than 0 for all x,"},{"Start":"02:01.070 ","End":"02:04.235","Text":"and then particularly if it\u0027s bigger than 0,"},{"Start":"02:04.235 ","End":"02:06.755","Text":"then it\u0027s also bigger or equal to 0."},{"Start":"02:06.755 ","End":"02:12.485","Text":"Basically our answer is that this is true for all x,"},{"Start":"02:12.485 ","End":"02:15.200","Text":"of course you can always try and plug in values."},{"Start":"02:15.200 ","End":"02:17.885","Text":"Whatever you put in here you\u0027ll get something positive,"},{"Start":"02:17.885 ","End":"02:22.775","Text":"strictly positive, but in particular, bigger or equal to 0."},{"Start":"02:22.775 ","End":"02:25.070","Text":"Now let\u0027s go for the next one."},{"Start":"02:25.070 ","End":"02:29.250","Text":"The next one is nothing to factorize and everything\u0027s on the left."},{"Start":"02:30.910 ","End":"02:34.835","Text":"You try and decompose this one."},{"Start":"02:34.835 ","End":"02:36.800","Text":"Again it doesn\u0027t work,"},{"Start":"02:36.800 ","End":"02:39.830","Text":"so we try the formula method."},{"Start":"02:39.830 ","End":"02:45.545","Text":"I\u0027ve tried this, it doesn\u0027t work either because one of the first things you would want to"},{"Start":"02:45.545 ","End":"02:50.440","Text":"check would be the b^2 minus 4ac before you even apply the whole formula,"},{"Start":"02:50.440 ","End":"02:52.025","Text":"that\u0027s part of the formula."},{"Start":"02:52.025 ","End":"02:57.080","Text":"This is equal to 1^2 minus 4 times"},{"Start":"02:57.080 ","End":"03:04.545","Text":"1 times 24,1 minus 96,"},{"Start":"03:04.545 ","End":"03:08.730","Text":"obviously negative, so there\u0027s no square root,"},{"Start":"03:08.730 ","End":"03:11.815","Text":"so this one also has no roots."},{"Start":"03:11.815 ","End":"03:16.880","Text":"If it has no roots and it\u0027s always positive or always negative,"},{"Start":"03:16.880 ","End":"03:21.055","Text":"in this case it\u0027s always positive,"},{"Start":"03:21.055 ","End":"03:26.975","Text":"just plug in any value like x is equal 0 is the easiest, 24 is positive."},{"Start":"03:26.975 ","End":"03:28.850","Text":"This thing is always positive."},{"Start":"03:28.850 ","End":"03:30.079","Text":"If it\u0027s always positive,"},{"Start":"03:30.079 ","End":"03:32.890","Text":"it\u0027s always bigger or equal to 0."},{"Start":"03:32.890 ","End":"03:37.450","Text":"The answer again is all x."},{"Start":"03:37.510 ","End":"03:41.665","Text":"Every x satisfies this inequality."},{"Start":"03:41.665 ","End":"03:45.060","Text":"Even more it satisfied that it\u0027s strictly bigger than 0,"},{"Start":"03:45.060 ","End":"03:47.840","Text":"but all we require is bigger or equal to."},{"Start":"03:47.840 ","End":"03:51.860","Text":"That\u0027s two down, two more to go."},{"Start":"03:51.860 ","End":"03:56.790","Text":"We have still c and d. Let\u0027s see."},{"Start":"03:58.070 ","End":"04:01.680","Text":"My personal recommendation, I don\u0027t know,"},{"Start":"04:01.680 ","End":"04:02.930","Text":"maybe just because I\u0027m used to it,"},{"Start":"04:02.930 ","End":"04:05.630","Text":"I like to have the a as being positive."},{"Start":"04:05.630 ","End":"04:08.075","Text":"Somehow one makes less than mistakes,"},{"Start":"04:08.075 ","End":"04:10.940","Text":"even if you have to convert it to a positive."},{"Start":"04:10.940 ","End":"04:15.975","Text":"I\u0027m going to multiply or divide both sides by minus 1,"},{"Start":"04:15.975 ","End":"04:18.365","Text":"makes no difference to multiply or divide."},{"Start":"04:18.365 ","End":"04:20.450","Text":"But because this is negative,"},{"Start":"04:20.450 ","End":"04:22.160","Text":"I immediately remind myself,"},{"Start":"04:22.160 ","End":"04:27.095","Text":"I like to write this first so I don\u0027t forget and copy it."},{"Start":"04:27.095 ","End":"04:31.310","Text":"Dividing minus 1 meaning to reverse every older signs."},{"Start":"04:31.310 ","End":"04:34.040","Text":"Usually you don\u0027t make a mistake with that."},{"Start":"04:34.040 ","End":"04:42.645","Text":"X^2 minus 13x minus 30 is bigger than 0."},{"Start":"04:42.645 ","End":"04:47.840","Text":"At this point you might want to factorize"},{"Start":"04:47.840 ","End":"04:53.210","Text":"it either by finding the roots with the formula, or using trinomial."},{"Start":"04:53.210 ","End":"04:59.885","Text":"Why don\u0027t we just practice the trinomial method?"},{"Start":"04:59.885 ","End":"05:02.550","Text":"Because we won\u0027t be doing it every time, just from time to time,"},{"Start":"05:02.550 ","End":"05:07.475","Text":"and mostly I\u0027ll be using the formula to find the roots as to the two ways of factorizing."},{"Start":"05:07.475 ","End":"05:13.610","Text":"What we do is, we look for two numbers whose product"},{"Start":"05:13.610 ","End":"05:20.360","Text":"something times something has got to equal ac,"},{"Start":"05:20.360 ","End":"05:25.550","Text":"and ac is minus 30 and the same two numbers,"},{"Start":"05:25.550 ","End":"05:30.645","Text":"when added, I\u0027ve got to give b,"},{"Start":"05:30.645 ","End":"05:33.610","Text":"and that\u0027s minus 13."},{"Start":"05:33.610 ","End":"05:35.420","Text":"Same two numbers here."},{"Start":"05:35.420 ","End":"05:38.825","Text":"You have something and something and then the same two somethings."},{"Start":"05:38.825 ","End":"05:42.830","Text":"What we do is when the product is negative,"},{"Start":"05:42.830 ","End":"05:45.050","Text":"you know that they are of opposite signs,"},{"Start":"05:45.050 ","End":"05:47.795","Text":"so in a way, if they are of opposite signs,"},{"Start":"05:47.795 ","End":"05:51.830","Text":"the addition is a bit like a subtraction because they\u0027re pulling in opposite directions."},{"Start":"05:51.830 ","End":"05:57.400","Text":"I like to think what two numbers have a difference of 13 and the product of 30,"},{"Start":"05:57.400 ","End":"06:00.570","Text":"and later I\u0027ll take care of the signs."},{"Start":"06:01.180 ","End":"06:05.885","Text":"It\u0027s easier to start with the product because there\u0027s not that many combinations."},{"Start":"06:05.885 ","End":"06:07.640","Text":"Could be 6 times 5,"},{"Start":"06:07.640 ","End":"06:09.500","Text":"they don\u0027t have a difference of 13."},{"Start":"06:09.500 ","End":"06:12.240","Text":"You finally hit on 15 times 2, yes,"},{"Start":"06:12.240 ","End":"06:14.030","Text":"if they do have a difference of 13,"},{"Start":"06:14.030 ","End":"06:15.260","Text":"so it\u0027s 15 and 2."},{"Start":"06:15.260 ","End":"06:17.015","Text":"Now what are the signs?"},{"Start":"06:17.015 ","End":"06:21.890","Text":"Because this is a minus, the minus is going to be bigger than the plus,"},{"Start":"06:21.890 ","End":"06:23.330","Text":"so we\u0027re going to have say,"},{"Start":"06:23.330 ","End":"06:26.535","Text":"a minus 15 and a plus 2."},{"Start":"06:26.535 ","End":"06:30.270","Text":"Then if you multiply the minus 15 times 2,"},{"Start":"06:30.270 ","End":"06:31.620","Text":"we do get minus 30."},{"Start":"06:31.620 ","End":"06:33.030","Text":"These are our two numbers,"},{"Start":"06:33.030 ","End":"06:35.175","Text":"now what do we do with them?"},{"Start":"06:35.175 ","End":"06:37.620","Text":"I\u0027ll just mention this in general,"},{"Start":"06:37.620 ","End":"06:42.300","Text":"is the ac and this is the b from the quadratic."},{"Start":"06:42.300 ","End":"06:45.870","Text":"Because this plus this equals the b, the minus 13,"},{"Start":"06:45.870 ","End":"06:49.905","Text":"we split the minus 13 up into two bits."},{"Start":"06:49.905 ","End":"06:53.240","Text":"Write this as x^2 and instead of minus 13,"},{"Start":"06:53.240 ","End":"06:57.020","Text":"we look at here and say minus 15 plus 2."},{"Start":"06:57.020 ","End":"07:01.120","Text":"Minus 15x plus 2x,"},{"Start":"07:01.120 ","End":"07:05.625","Text":"then minus 30 is bigger than 0."},{"Start":"07:05.625 ","End":"07:08.825","Text":"Once you\u0027ve done it this way,"},{"Start":"07:08.825 ","End":"07:13.190","Text":"breaking it up, you\u0027re guaranteed that it will factorize."},{"Start":"07:13.190 ","End":"07:15.755","Text":"I take the x out of the first pair."},{"Start":"07:15.755 ","End":"07:17.685","Text":"We do it in groups, in pairs."},{"Start":"07:17.685 ","End":"07:26.965","Text":"X times x minus 15 plus 2(x-15), bigger than 0."},{"Start":"07:26.965 ","End":"07:30.230","Text":"Then, because we have the same bracket here,"},{"Start":"07:30.230 ","End":"07:31.880","Text":"and we have the x plus 2,"},{"Start":"07:31.880 ","End":"07:34.420","Text":"we can take the x plus 2,"},{"Start":"07:34.420 ","End":"07:36.960","Text":"x minus 15 outside the brackets,"},{"Start":"07:36.960 ","End":"07:38.670","Text":"I\u0027ll write it in second place."},{"Start":"07:38.670 ","End":"07:42.735","Text":"What\u0027s left is x plus 2, bigger than 0."},{"Start":"07:42.735 ","End":"07:46.935","Text":"Now we have factorization,"},{"Start":"07:46.935 ","End":"07:51.215","Text":"and then we know which numbers to put on the number line if we do the sketch."},{"Start":"07:51.215 ","End":"07:52.880","Text":"If x plus 2 is 0,"},{"Start":"07:52.880 ","End":"07:55.800","Text":"then x is minus 2."},{"Start":"07:55.800 ","End":"07:58.370","Text":"If x minus 15 is 0, then x is 15."},{"Start":"07:58.370 ","End":"08:00.505","Text":"These are the two roots."},{"Start":"08:00.505 ","End":"08:06.150","Text":"Let\u0027s do it with a quick sketch, a number line."},{"Start":"08:06.150 ","End":"08:09.010","Text":"Then the two points,"},{"Start":"08:09.050 ","End":"08:13.395","Text":"minus 2 and 15."},{"Start":"08:13.395 ","End":"08:15.530","Text":"Nothing has to be to any kind of scale,"},{"Start":"08:15.530 ","End":"08:17.300","Text":"just to get them in the right order."},{"Start":"08:17.300 ","End":"08:20.190","Text":"15 is bigger than minus 2."},{"Start":"08:21.190 ","End":"08:25.700","Text":"We\u0027ll just mentally sample the intervals."},{"Start":"08:25.700 ","End":"08:31.799","Text":"If I put in x equals 0 from this interval,"},{"Start":"08:31.799 ","End":"08:33.600","Text":"0 plus 2 is positive,"},{"Start":"08:33.600 ","End":"08:35.550","Text":"0 minus 15 is negative,"},{"Start":"08:35.550 ","End":"08:37.895","Text":"so we get negative here."},{"Start":"08:37.895 ","End":"08:41.030","Text":"If we\u0027re bigger than 15, I don\u0027t know, say 20,"},{"Start":"08:41.030 ","End":"08:45.535","Text":"this is positive, this is positive, product is positive."},{"Start":"08:45.535 ","End":"08:49.560","Text":"If 4x equals, I don\u0027t know, minus 3 even."},{"Start":"08:49.560 ","End":"08:51.330","Text":"Minus 3 plus 2 is negative,"},{"Start":"08:51.330 ","End":"08:52.920","Text":"this one is even more negative."},{"Start":"08:52.920 ","End":"08:55.130","Text":"Negative times negative is positive."},{"Start":"08:55.130 ","End":"08:57.230","Text":"Then we look, what are we interested in?"},{"Start":"08:57.230 ","End":"08:59.780","Text":"We\u0027re interested in bigger than 0,"},{"Start":"08:59.780 ","End":"09:01.909","Text":"bigger than 0 to me means positive."},{"Start":"09:01.909 ","End":"09:03.410","Text":"This is the one we want,"},{"Start":"09:03.410 ","End":"09:05.880","Text":"and we want this one here."},{"Start":"09:06.340 ","End":"09:10.445","Text":"What we can do is without any sketches,"},{"Start":"09:10.445 ","End":"09:14.715","Text":"positive is up to but not including the two of these points."},{"Start":"09:14.715 ","End":"09:19.505","Text":"The polynomial is 0,"},{"Start":"09:19.505 ","End":"09:22.030","Text":"so strictly positive here and here."},{"Start":"09:22.030 ","End":"09:23.390","Text":"What does this give?"},{"Start":"09:23.390 ","End":"09:28.820","Text":"Pictorially, it\u0027s from the minus 2 and leftwards,"},{"Start":"09:28.820 ","End":"09:30.620","Text":"so downwards or whatever."},{"Start":"09:30.620 ","End":"09:34.775","Text":"Then from 15 and in this direction."},{"Start":"09:34.775 ","End":"09:37.430","Text":"What we say is,"},{"Start":"09:37.430 ","End":"09:40.880","Text":"x less than minus 2,"},{"Start":"09:40.880 ","End":"09:45.890","Text":"or x bigger than 15,"},{"Start":"09:45.890 ","End":"09:50.704","Text":"and this is our answer."},{"Start":"09:50.704 ","End":"09:53.765","Text":"Onto the last one,"},{"Start":"09:53.765 ","End":"09:58.695","Text":"d. Let\u0027s see what we have here."},{"Start":"09:58.695 ","End":"10:03.530","Text":"We have a quadratic, and an inequality."},{"Start":"10:03.530 ","End":"10:07.735","Text":"Like I said, my personal thing is I like the a to be positive."},{"Start":"10:07.735 ","End":"10:11.380","Text":"You don\u0027t have to, but I\u0027m going to do it that way."},{"Start":"10:11.380 ","End":"10:15.865","Text":"I\u0027m going to multiply or divide by minus 1 on both sides."},{"Start":"10:15.865 ","End":"10:18.580","Text":"That means that I reverse because it\u0027s negative,"},{"Start":"10:18.580 ","End":"10:21.505","Text":"I reverse the inequality,"},{"Start":"10:21.505 ","End":"10:24.280","Text":"make it bigger than 0 and it\u0027s reversed all the signs."},{"Start":"10:24.280 ","End":"10:29.380","Text":"Plus 2x^2 minus 7x minus 4 bigger than 0."},{"Start":"10:29.380 ","End":"10:33.740","Text":"I\u0027m used to it this way, I find we\u0027ll make less mistakes if the a is positive."},{"Start":"10:33.950 ","End":"10:43.250","Text":"Once again, we could either solve the formula or try a trinomial decomposition method."},{"Start":"10:43.250 ","End":"10:46.610","Text":"This time I won\u0027t waste time with doing that,"},{"Start":"10:46.610 ","End":"10:48.785","Text":"I\u0027ll give you straight away the answer."},{"Start":"10:48.785 ","End":"10:53.570","Text":"2x plus 1, x minus 4."},{"Start":"10:53.570 ","End":"10:57.695","Text":"Certainly you can always multiply it out 2x times x is 2x^2."},{"Start":"10:57.695 ","End":"11:00.425","Text":"We have here a plus x and the minus 8x,"},{"Start":"11:00.425 ","End":"11:02.975","Text":"which is that, and the 1 times minus 4."},{"Start":"11:02.975 ","End":"11:05.125","Text":"We see that it works out."},{"Start":"11:05.125 ","End":"11:10.655","Text":"But [inaudible] is bigger than 0."},{"Start":"11:10.655 ","End":"11:21.330","Text":"Then the roots of this polynomial quadratic would be when this is 0."},{"Start":"11:21.330 ","End":"11:24.305","Text":"If this is 0, x is minus 1/2."},{"Start":"11:24.305 ","End":"11:27.260","Text":"If this is 0, x is 4."},{"Start":"11:27.260 ","End":"11:29.975","Text":"These are the two roots."},{"Start":"11:29.975 ","End":"11:37.130","Text":"I\u0027ll draw a number line and I\u0027ll plot the points here,"},{"Start":"11:37.130 ","End":"11:40.800","Text":"4 and minus a half."},{"Start":"11:42.860 ","End":"11:45.560","Text":"0 is in this interval here,"},{"Start":"11:45.560 ","End":"11:47.900","Text":"or like substituting 0, it\u0027s so easy."},{"Start":"11:47.900 ","End":"11:55.920","Text":"Then we get 1 times minus 4 plus minus is minus."},{"Start":"11:56.210 ","End":"11:59.500","Text":"Here if I take x is 5,"},{"Start":"11:59.500 ","End":"12:05.330","Text":"this is obviously positive and 5 minus 4 is also positive, productive positives."},{"Start":"12:05.330 ","End":"12:07.160","Text":"Here we\u0027ll get the product of negatives."},{"Start":"12:07.160 ","End":"12:10.205","Text":"If I take, minus 1 could do."},{"Start":"12:10.205 ","End":"12:13.830","Text":"Minus 1, that\u0027s minus 2 plus 1,"},{"Start":"12:13.830 ","End":"12:15.680","Text":"negative, this one is negative,"},{"Start":"12:15.680 ","End":"12:17.389","Text":"negative and negative is positive."},{"Start":"12:17.389 ","End":"12:18.935","Text":"What do we want?"},{"Start":"12:18.935 ","End":"12:21.290","Text":"We want bigger than 0, which is positive."},{"Start":"12:21.290 ","End":"12:23.690","Text":"We want this and we want this."},{"Start":"12:23.690 ","End":"12:26.375","Text":"Without the sketch even,"},{"Start":"12:26.375 ","End":"12:34.160","Text":"I will just write that either x is less than minus a half, not including,"},{"Start":"12:34.160 ","End":"12:37.880","Text":"because we want strictly positive here and here at 0,"},{"Start":"12:37.880 ","End":"12:41.720","Text":"and also, I\u0027ll put them all on this side here,"},{"Start":"12:41.720 ","End":"12:45.020","Text":"we have positive or x is bigger than 4."},{"Start":"12:45.020 ","End":"12:48.560","Text":"Let\u0027s take a bit of shortcuts from time to time because it\u0027s"},{"Start":"12:48.560 ","End":"12:55.420","Text":"tiresome to keep doing the full method."},{"Start":"12:55.750 ","End":"13:00.500","Text":"But you know what, I\u0027ll give you a quick reminder of what the snake method is."},{"Start":"13:00.500 ","End":"13:02.810","Text":"I copied this or most of it."},{"Start":"13:02.810 ","End":"13:06.110","Text":"Then we plug in values and when it\u0027s positive,"},{"Start":"13:06.110 ","End":"13:07.940","Text":"we put a point here,"},{"Start":"13:07.940 ","End":"13:09.980","Text":"somewhere above the axis and when it\u0027s negative,"},{"Start":"13:09.980 ","End":"13:12.005","Text":"we put a point below."},{"Start":"13:12.005 ","End":"13:13.580","Text":"Again it\u0027s positive,"},{"Start":"13:13.580 ","End":"13:15.125","Text":"we put a point above,"},{"Start":"13:15.125 ","End":"13:18.695","Text":"and we know that here at 0 and here at 0 so we draw."},{"Start":"13:18.695 ","End":"13:22.360","Text":"We call the snake because usually it depends several times,"},{"Start":"13:22.360 ","End":"13:23.870","Text":"if it\u0027s not a quadratic,"},{"Start":"13:23.870 ","End":"13:25.800","Text":"it\u0027s a cubic, for example."},{"Start":"13:25.800 ","End":"13:27.560","Text":"Then we say, what are we looking for?"},{"Start":"13:27.560 ","End":"13:28.910","Text":"We\u0027re looking for bigger than 0,"},{"Start":"13:28.910 ","End":"13:30.410","Text":"which means above the x-axis,"},{"Start":"13:30.410 ","End":"13:33.305","Text":"so we shade this and we shade this."},{"Start":"13:33.305 ","End":"13:36.530","Text":"Then we notice that this shading occurs when x"},{"Start":"13:36.530 ","End":"13:40.175","Text":"is less than 1/2 and this shading is when x is bigger than 4."},{"Start":"13:40.175 ","End":"13:42.980","Text":"Two very similar methods."},{"Start":"13:42.980 ","End":"13:46.710","Text":"Anyway, d is the final one, so we\u0027re done."}],"Thumbnail":null,"ID":5064},{"Watched":false,"Name":"Exercise 3","Duration":"12m 3s","ChapterTopicVideoID":5062,"CourseChapterTopicPlaylistID":3876,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.525","Text":"Here we have some inequalities to solve."},{"Start":"00:03.525 ","End":"00:06.120","Text":"They\u0027re pretty much all quadratic."},{"Start":"00:06.120 ","End":"00:10.650","Text":"The first one is the easiest because we have a 0 on"},{"Start":"00:10.650 ","End":"00:15.495","Text":"the right-hand side to compare with and we\u0027ve already got it factorized."},{"Start":"00:15.495 ","End":"00:18.660","Text":"What we have to do here is,"},{"Start":"00:18.660 ","End":"00:20.895","Text":"let\u0027s say, we use a sketch."},{"Start":"00:20.895 ","End":"00:24.525","Text":"We note the roots of this quadratic,"},{"Start":"00:24.525 ","End":"00:28.320","Text":"it\u0027s either x plus 2 is 0 gives us 1 root,"},{"Start":"00:28.320 ","End":"00:30.780","Text":"that means x is minus 2,"},{"Start":"00:30.780 ","End":"00:32.430","Text":"and if x plus 5 is 0,"},{"Start":"00:32.430 ","End":"00:33.975","Text":"the next is minus 5."},{"Start":"00:33.975 ","End":"00:38.775","Text":"These are the two roots of the quadratic on the left."},{"Start":"00:38.775 ","End":"00:40.050","Text":"Once we have the roots,"},{"Start":"00:40.050 ","End":"00:42.795","Text":"we can sketch them on the number line,"},{"Start":"00:42.795 ","End":"00:46.145","Text":"so here\u0027s the number line."},{"Start":"00:46.145 ","End":"00:48.970","Text":"Let\u0027s see which one is further to the left,"},{"Start":"00:48.970 ","End":"00:50.620","Text":"that would be the minus 5,"},{"Start":"00:50.620 ","End":"00:55.790","Text":"so this one is minus 5 say and this one is minus 2."},{"Start":"00:55.790 ","End":"00:59.439","Text":"Then we can sample each of the 3 intervals,"},{"Start":"00:59.439 ","End":"01:01.675","Text":"whether it\u0027s positive or negative."},{"Start":"01:01.675 ","End":"01:03.865","Text":"If we try, for example,"},{"Start":"01:03.865 ","End":"01:06.475","Text":"x is minus 6,"},{"Start":"01:06.475 ","End":"01:12.910","Text":"then we will get that this is negative and minus 6 plus 5 is negative,"},{"Start":"01:12.910 ","End":"01:16.030","Text":"so here we\u0027re positive."},{"Start":"01:16.030 ","End":"01:19.270","Text":"Let\u0027s go for the snake method,"},{"Start":"01:19.270 ","End":"01:20.740","Text":"we sketch a snake."},{"Start":"01:20.740 ","End":"01:25.275","Text":"Here we know that the value is 0 these are the roots."},{"Start":"01:25.275 ","End":"01:27.970","Text":"If x is between minus 5 and minus 2,"},{"Start":"01:27.970 ","End":"01:30.340","Text":"let\u0027s say minus 3,"},{"Start":"01:30.340 ","End":"01:32.575","Text":"this is going to be negative,"},{"Start":"01:32.575 ","End":"01:35.660","Text":"and this is going to be positive or put minus 3,"},{"Start":"01:35.660 ","End":"01:43.115","Text":"so it\u0027s going to be positive rather than negative times positive is negative."},{"Start":"01:43.115 ","End":"01:45.650","Text":"I\u0027ll put some value at minus 3,"},{"Start":"01:45.650 ","End":"01:46.910","Text":"I\u0027ll put a negative value."},{"Start":"01:46.910 ","End":"01:48.815","Text":"It really doesn\u0027t matter,"},{"Start":"01:48.815 ","End":"01:54.230","Text":"everything\u0027s roughly just to make sure you get it on the right side of the line."},{"Start":"01:54.230 ","End":"01:58.070","Text":"If x is, let\u0027s say 0, I like 0,"},{"Start":"01:58.070 ","End":"01:59.905","Text":"that would be somewhere here,"},{"Start":"01:59.905 ","End":"02:04.145","Text":"then this is positive and this is positive so the product is positive."},{"Start":"02:04.145 ","End":"02:12.635","Text":"At 0 we have something above the line and then we know that snake goes like this,"},{"Start":"02:12.635 ","End":"02:15.575","Text":"something at any rate."},{"Start":"02:15.575 ","End":"02:17.750","Text":"Now we should carefully look what we want."},{"Start":"02:17.750 ","End":"02:24.295","Text":"We want less than 0 below the axis, this bit here."},{"Start":"02:24.295 ","End":"02:28.745","Text":"We see that it happens between minus 5 and minus 2."},{"Start":"02:28.745 ","End":"02:34.490","Text":"Let\u0027s just write that minus 5 less than x,"},{"Start":"02:34.490 ","End":"02:36.530","Text":"less than minus 2,"},{"Start":"02:36.530 ","End":"02:39.935","Text":"and that\u0027s the answer."},{"Start":"02:39.935 ","End":"02:43.670","Text":"They started us off nice and easy."},{"Start":"02:43.670 ","End":"02:48.110","Text":"B gets a little bit more difficult."},{"Start":"02:48.110 ","End":"02:53.630","Text":"The reason it\u0027s more difficult is that there\u0027s not a 0 on the right-hand side."},{"Start":"02:53.630 ","End":"02:58.430","Text":"Actually turns out this factorization is no use at all unless it\u0027s 0."},{"Start":"02:58.430 ","End":"03:00.745","Text":"We\u0027re going to just have to open everything up,"},{"Start":"03:00.745 ","End":"03:03.260","Text":"like when you\u0027ve tidied up the room the"},{"Start":"03:03.260 ","End":"03:05.900","Text":"wrong way and make it a mess again and tidy it up again,"},{"Start":"03:05.900 ","End":"03:08.155","Text":"but this time with 0 on the right."},{"Start":"03:08.155 ","End":"03:12.225","Text":"Let\u0027s do the work, let\u0027s multiply out, x times x,"},{"Start":"03:12.225 ","End":"03:17.775","Text":"x^2 plus 2 x plus 4x plus 6x."},{"Start":"03:17.775 ","End":"03:19.650","Text":"Then we have plus a,"},{"Start":"03:19.650 ","End":"03:22.710","Text":"but bring the 35 over instead of plus 8,"},{"Start":"03:22.710 ","End":"03:29.180","Text":"subtract 35 we don\u0027t know what I said."},{"Start":"03:29.180 ","End":"03:33.880","Text":"Anyway, minus 27 is less than 0."},{"Start":"03:33.880 ","End":"03:36.930","Text":"We have two main methods to factor,"},{"Start":"03:36.930 ","End":"03:42.485","Text":"the trinomial method or the solving the equation roots method."},{"Start":"03:42.485 ","End":"03:46.100","Text":"I\u0027ll just do it for you and tell you that this is equal"},{"Start":"03:46.100 ","End":"03:54.300","Text":"to x plus 9x minus 3."},{"Start":"03:54.300 ","End":"03:59.250","Text":"Let\u0027s just check x times x is x^2 plus 9 x minus 3,"},{"Start":"03:59.250 ","End":"04:02.610","Text":"x plus 6x 9 times minus 3 is minus 27,"},{"Start":"04:02.610 ","End":"04:05.955","Text":"so that\u0027s right, less than 0."},{"Start":"04:05.955 ","End":"04:12.065","Text":"Now at this point it becomes similar to the first question, very similar."},{"Start":"04:12.065 ","End":"04:17.270","Text":"If I write the roots of the polynomial,"},{"Start":"04:17.270 ","End":"04:21.500","Text":"minus 9 and plus 3,"},{"Start":"04:21.500 ","End":"04:25.235","Text":"I\u0027m not going to draw the sketch again because it\u0027s so similar."},{"Start":"04:25.235 ","End":"04:26.780","Text":"Just like in this sketch,"},{"Start":"04:26.780 ","End":"04:28.580","Text":"it was between the 2 roots,"},{"Start":"04:28.580 ","End":"04:34.780","Text":"so leave the work to you and It\u0027s just copying what we did here,"},{"Start":"04:34.780 ","End":"04:42.770","Text":"and so we get minus 9 less than x, less than 3."},{"Start":"04:42.770 ","End":"04:47.170","Text":"Exactly the same picture but with different numbers in that\u0027s all."},{"Start":"04:47.550 ","End":"04:53.360","Text":"Now let\u0027s go on and see something more challenging."},{"Start":"04:53.970 ","End":"04:58.450","Text":"Here again, we have something factorized on the left there."},{"Start":"04:58.450 ","End":"05:05.539","Text":"Let\u0027s say the factoring is absolutely useless unless it\u0027s against 0, it\u0027s being compared."},{"Start":"05:05.630 ","End":"05:09.990","Text":"We just have to open the brackets."},{"Start":"05:09.990 ","End":"05:16.770","Text":"X^2 minus 3x minus 7x is minus 10x plus 21,"},{"Start":"05:16.770 ","End":"05:20.705","Text":"bigger or equal to 8x minus 56."},{"Start":"05:20.705 ","End":"05:23.615","Text":"Bring everything over to the left."},{"Start":"05:23.615 ","End":"05:29.700","Text":"x^2 minus 10x minus 8x minus 18x,"},{"Start":"05:29.700 ","End":"05:33.335","Text":"21 plus 56 is"},{"Start":"05:33.335 ","End":"05:40.750","Text":"77 bigger or equal to 0."},{"Start":"05:40.750 ","End":"05:49.575","Text":"Now we factorize this using any one of the techniques,"},{"Start":"05:49.575 ","End":"05:55.560","Text":"and what you will get is x minus 7,"},{"Start":"05:55.560 ","End":"06:00.880","Text":"x minus 11, bigger or equal to 0."},{"Start":"06:01.360 ","End":"06:05.255","Text":"The roots of the polynomial on the left,"},{"Start":"06:05.255 ","End":"06:12.360","Text":"7 and 11 with a plus draw a number line."},{"Start":"06:12.360 ","End":"06:14.955","Text":"I\u0027m going to put on these points,"},{"Start":"06:14.955 ","End":"06:22.125","Text":"the left most will be 7 and the right most 11 po10t to get them in the right order."},{"Start":"06:22.125 ","End":"06:28.130","Text":"Then I\u0027m going to check in each of the intervals if I\u0027m positive or negative,"},{"Start":"06:28.130 ","End":"06:32.035","Text":"and we know that here was 0."},{"Start":"06:32.035 ","End":"06:35.330","Text":"Here this thing is 0."},{"Start":"06:35.330 ","End":"06:39.400","Text":"I\u0027d like to substitute x equals 0."},{"Start":"06:39.400 ","End":"06:40.810","Text":"That\u0027s going to be somewhere here,"},{"Start":"06:40.810 ","End":"06:45.160","Text":"the 0,0 minus 7,0 minus 11."},{"Start":"06:45.160 ","End":"06:49.790","Text":"Both negative here where positive."},{"Start":"06:49.920 ","End":"06:55.495","Text":"Put a number here, I could just put a plus sign between 7 and 11,"},{"Start":"06:55.495 ","End":"06:59.380","Text":"let\u0027s go for 10.10 minus 7 is positive,"},{"Start":"06:59.380 ","End":"07:01.150","Text":"10 minus 11 is negative,"},{"Start":"07:01.150 ","End":"07:03.520","Text":"so here we are negative."},{"Start":"07:03.520 ","End":"07:13.190","Text":"Let\u0027s try something to the right of 11,12.12 minus 7 is positive 12 minus 11 is positive."},{"Start":"07:13.190 ","End":"07:16.145","Text":"Positive times positive is positive,"},{"Start":"07:16.145 ","End":"07:21.740","Text":"so If we sketch the graph using this is called"},{"Start":"07:21.740 ","End":"07:27.500","Text":"a snake methods because usually it wrinkles more here with just a quadratic,"},{"Start":"07:27.500 ","End":"07:29.810","Text":"so it goes down and up."},{"Start":"07:29.810 ","End":"07:31.565","Text":"What are we interested in?"},{"Start":"07:31.565 ","End":"07:33.545","Text":"Bigger or equal to 0."},{"Start":"07:33.545 ","End":"07:39.380","Text":"What we want is a maybe I\u0027ll shade it."},{"Start":"07:39.380 ","End":"07:44.570","Text":"But also emphasize that since we also include 0,"},{"Start":"07:44.570 ","End":"07:47.930","Text":"this thing is I\u0027ll make it larger,"},{"Start":"07:47.930 ","End":"07:50.675","Text":"maybe that will really emphasize it."},{"Start":"07:50.675 ","End":"07:52.925","Text":"These points are in,"},{"Start":"07:52.925 ","End":"07:57.120","Text":"and here\u0027s the bigger than 0 and here\u0027s the 0."},{"Start":"07:57.740 ","End":"08:01.180","Text":"What we end up getting,"},{"Start":"08:01.450 ","End":"08:05.030","Text":"write it over here. Is x."},{"Start":"08:05.030 ","End":"08:06.710","Text":"The only less than 7,"},{"Start":"08:06.710 ","End":"08:09.350","Text":"less than or equal to 7 because 7 is in,"},{"Start":"08:09.350 ","End":"08:12.514","Text":"because we have 0 \u0027s included."},{"Start":"08:12.514 ","End":"08:15.815","Text":"The other bit is x,"},{"Start":"08:15.815 ","End":"08:17.735","Text":"not only bigger than 11,"},{"Start":"08:17.735 ","End":"08:24.005","Text":"but bigger or equal to 11 because 11 is n. This would be our answer,"},{"Start":"08:24.005 ","End":"08:26.000","Text":"and I\u0027ll highlight it,"},{"Start":"08:26.000 ","End":"08:29.250","Text":"and then we can move on to the next."},{"Start":"08:29.330 ","End":"08:34.320","Text":"What have we got here?"},{"Start":"08:34.320 ","End":"08:38.450","Text":"Once again, we\u0027re going to have to do some work in expansion"},{"Start":"08:38.450 ","End":"08:42.050","Text":"and so on until we can get a quadratic."},{"Start":"08:42.050 ","End":"08:45.425","Text":"On the left-hand side, we have one of those special formulas,"},{"Start":"08:45.425 ","End":"08:49.425","Text":"a^2 minus 2 ab plus b^2. Remember that one?"},{"Start":"08:49.425 ","End":"08:53.190","Text":"X^2 minus twice x times 6, sorry,"},{"Start":"08:53.190 ","End":"08:58.240","Text":"minus twice x times 3 is minus 6x 3 squared is 9,"},{"Start":"08:58.240 ","End":"09:00.620","Text":"is bigger than,"},{"Start":"09:00.620 ","End":"09:01.700","Text":"multiply these 2,"},{"Start":"09:01.700 ","End":"09:07.450","Text":"I\u0027ve got x^2 minus x plus 6x is plus 5x,"},{"Start":"09:07.450 ","End":"09:11.235","Text":"minus one times 6 is minus 6,"},{"Start":"09:11.235 ","End":"09:15.735","Text":"and still minus 6x squared minus 3 x."},{"Start":"09:15.735 ","End":"09:19.365","Text":"Let\u0027s collect everything on the left,"},{"Start":"09:19.365 ","End":"09:21.950","Text":"and just while I\u0027m at it,"},{"Start":"09:21.950 ","End":"09:25.435","Text":"I just noticed that x^2 and x^2 cancel."},{"Start":"09:25.435 ","End":"09:29.980","Text":"On the left, we have x^2 Now let\u0027s see x terms."},{"Start":"09:29.980 ","End":"09:31.810","Text":"I have minus 6,"},{"Start":"09:31.810 ","End":"09:35.105","Text":"minus another 5 plus a 3,"},{"Start":"09:35.105 ","End":"09:41.655","Text":"minus 6 minus 11 plus 3 is minus 8x."},{"Start":"09:41.655 ","End":"09:52.800","Text":"Numbers 9 plus another 6 is plus 15 and strictly bigger than 0."},{"Start":"09:56.510 ","End":"10:00.690","Text":"Factorize this let\u0027s this time,"},{"Start":"10:00.690 ","End":"10:05.275","Text":"we\u0027ll use the method of the formula."},{"Start":"10:05.275 ","End":"10:08.075","Text":"Last time I remember doing the trinomial."},{"Start":"10:08.075 ","End":"10:11.960","Text":"If we take the equation equals 0,"},{"Start":"10:11.960 ","End":"10:17.225","Text":"then we get the roots are x equals minus b plus or minus."},{"Start":"10:17.225 ","End":"10:27.180","Text":"The square root of b^2 is 64 minus 4 times a times c. All over 2 a,"},{"Start":"10:27.180 ","End":"10:28.440","Text":"which is 2,"},{"Start":"10:28.440 ","End":"10:30.200","Text":"now under the square root sign,"},{"Start":"10:30.200 ","End":"10:32.900","Text":"we have 64 minus 60."},{"Start":"10:32.900 ","End":"10:37.085","Text":"We get the square root of 4, which is 2."},{"Start":"10:37.085 ","End":"10:42.950","Text":"We get 8 plus or minus 2 over 2."},{"Start":"10:42.950 ","End":"10:44.960","Text":"What does that give us?"},{"Start":"10:44.960 ","End":"10:47.255","Text":"If we take the plus,"},{"Start":"10:47.255 ","End":"10:54.215","Text":"we get 10 over 2 is 5,"},{"Start":"10:54.215 ","End":"10:56.000","Text":"and if we take the minus,"},{"Start":"10:56.000 ","End":"10:58.355","Text":"we get 6 over 2 is 3,"},{"Start":"10:58.355 ","End":"11:00.245","Text":"so 5 and 3 are the roots."},{"Start":"11:00.245 ","End":"11:05.270","Text":"Noting that the coefficient of x^2 is 1,"},{"Start":"11:05.270 ","End":"11:07.790","Text":"otherwise we can\u0027t just do it simply as that."},{"Start":"11:07.790 ","End":"11:10.985","Text":"We do x minus 1 root."},{"Start":"11:10.985 ","End":"11:17.130","Text":"X minus the other root is bigger than 0."},{"Start":"11:17.560 ","End":"11:21.650","Text":"Let me just say that we have a picture here staring at us."},{"Start":"11:21.650 ","End":"11:24.935","Text":"Here it\u0027s bigger or equal to and here it\u0027s bigger than."},{"Start":"11:24.935 ","End":"11:29.865","Text":"It\u0027s just the same as this except with a 3 and 5 instead,"},{"Start":"11:29.865 ","End":"11:32.255","Text":"and also we don\u0027t include the 0,"},{"Start":"11:32.255 ","End":"11:36.950","Text":"so we\u0027ll get that x is just like here."},{"Start":"11:36.950 ","End":"11:40.030","Text":"x is less than the smaller one."},{"Start":"11:40.030 ","End":"11:49.330","Text":"Smaller one is 3 or x is bigger than the large 1, which is 5."},{"Start":"11:49.330 ","End":"11:50.645","Text":"But this time, as I say,"},{"Start":"11:50.645 ","End":"11:55.355","Text":"not including because in this case we have a strict inequality,"},{"Start":"11:55.355 ","End":"11:59.070","Text":"so this is our answer."},{"Start":"12:00.380 ","End":"12:04.690","Text":"That\u0027s the final part of this question, so we\u0027re done."}],"Thumbnail":null,"ID":5065},{"Watched":false,"Name":"Exercise 4","Duration":"23m 15s","ChapterTopicVideoID":5063,"CourseChapterTopicPlaylistID":3876,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.288","Text":"In this exercise, we have some inequalities,"},{"Start":"00:04.288 ","End":"00:05.760","Text":"four to be precise."},{"Start":"00:05.760 ","End":"00:13.785","Text":"Each one of them is made up of a combination with and or or."},{"Start":"00:13.785 ","End":"00:17.970","Text":"We have just more work to do because there\u0027s"},{"Start":"00:17.970 ","End":"00:21.795","Text":"two inequalities to solve in each case and then we have to combine them."},{"Start":"00:21.795 ","End":"00:25.870","Text":"Let\u0027s get started on the first one."},{"Start":"00:26.090 ","End":"00:29.925","Text":"Here\u0027s a quadratic, this one is a linear."},{"Start":"00:29.925 ","End":"00:31.935","Text":"The quadratic one,"},{"Start":"00:31.935 ","End":"00:38.935","Text":"we solve by factorizing the polynomial, the quadratic."},{"Start":"00:38.935 ","End":"00:40.954","Text":"There are two methods,"},{"Start":"00:40.954 ","End":"00:45.510","Text":"but I\u0027m going to use the third method of just giving you the answer,"},{"Start":"00:45.590 ","End":"00:50.855","Text":"because the key here is not factorizing quadratic polynomials."},{"Start":"00:50.855 ","End":"00:54.065","Text":"This one turns out to be x minus 1,"},{"Start":"00:54.065 ","End":"00:58.945","Text":"x minus 6,"},{"Start":"00:58.945 ","End":"01:01.565","Text":"and it\u0027s still less than or equal to 0."},{"Start":"01:01.565 ","End":"01:05.140","Text":"Let\u0027s say there\u0027s the trinomial method where we break the middle piece up,"},{"Start":"01:05.140 ","End":"01:09.140","Text":"and then there\u0027s also the method of solving the equation and getting the two roots,"},{"Start":"01:09.140 ","End":"01:11.870","Text":"but not worth spending time on that."},{"Start":"01:11.870 ","End":"01:17.270","Text":"This is what you get and you can always check also by multiplying out to get this."},{"Start":"01:17.270 ","End":"01:21.850","Text":"Once we have this, then we do a quick sketch."},{"Start":"01:21.850 ","End":"01:24.810","Text":"A number line that goes this way,"},{"Start":"01:24.810 ","End":"01:28.065","Text":"we put in the two main points 1 and 6,"},{"Start":"01:28.065 ","End":"01:30.320","Text":"let\u0027s say here and here."},{"Start":"01:30.320 ","End":"01:32.750","Text":"Of course this is 1 and this is 6,"},{"Start":"01:32.750 ","End":"01:36.170","Text":"and then we can check in each of"},{"Start":"01:36.170 ","End":"01:41.270","Text":"the three intervals where it\u0027s positive and where it\u0027s negative."},{"Start":"01:41.270 ","End":"01:46.850","Text":"At these points the thing is zero because here at 1,"},{"Start":"01:46.850 ","End":"01:49.745","Text":"x minus 1 is 0 and here x minus 6 is 0."},{"Start":"01:49.745 ","End":"01:55.070","Text":"We want to see if it\u0027s above or below the axis in-between here I could take x equals 0,"},{"Start":"01:55.070 ","End":"02:00.425","Text":"if x is 0 and I have negative times negative is positive."},{"Start":"02:00.425 ","End":"02:03.215","Text":"So let\u0027s say I put a point here."},{"Start":"02:03.215 ","End":"02:06.290","Text":"It\u0027s all just schematic, nothing precise."},{"Start":"02:06.290 ","End":"02:08.875","Text":"If I take x here,"},{"Start":"02:08.875 ","End":"02:11.940","Text":"let\u0027s say 2 minus 1 is positive,"},{"Start":"02:11.940 ","End":"02:13.440","Text":"but 2 minus 6 is negative,"},{"Start":"02:13.440 ","End":"02:16.350","Text":"so here we get negative, so we\u0027re below."},{"Start":"02:16.350 ","End":"02:20.540","Text":"If I take x equals 7, they\u0027re both positive."},{"Start":"02:20.540 ","End":"02:23.825","Text":"X minus 1, x minus 6, so we\u0027re above."},{"Start":"02:23.825 ","End":"02:28.070","Text":"We join these with a curve,"},{"Start":"02:28.070 ","End":"02:29.965","Text":"what we call the snake,"},{"Start":"02:29.965 ","End":"02:32.720","Text":"and then we look, what do we want?"},{"Start":"02:32.720 ","End":"02:34.850","Text":"We want less than or equal to 0."},{"Start":"02:34.850 ","End":"02:37.630","Text":"Less than means below,"},{"Start":"02:37.630 ","End":"02:41.355","Text":"0 means we include these here,"},{"Start":"02:41.355 ","End":"02:45.880","Text":"so really what we get is this whole line,"},{"Start":"02:46.250 ","End":"02:50.280","Text":"it\u0027s this line here from 1-6 including,"},{"Start":"02:50.280 ","End":"02:54.395","Text":"I\u0027ll draw it again maybe better above."},{"Start":"02:54.395 ","End":"02:56.150","Text":"This will be above the 1,"},{"Start":"02:56.150 ","End":"02:58.400","Text":"this will be above the 6,"},{"Start":"02:58.400 ","End":"03:02.880","Text":"it meant to be on the axis and the line joining them."},{"Start":"03:03.710 ","End":"03:07.290","Text":"It\u0027s not to be hovering but like on the axis."},{"Start":"03:07.290 ","End":"03:16.045","Text":"Anyway, what I mean is that this represents that x is between 1 and 6,"},{"Start":"03:16.045 ","End":"03:19.400","Text":"but including the 6 and including the 1,"},{"Start":"03:19.400 ","End":"03:21.415","Text":"because this is in."},{"Start":"03:21.415 ","End":"03:24.890","Text":"We have the less than or equal to,"},{"Start":"03:24.890 ","End":"03:27.965","Text":"less than or equal to so the 0 is in,"},{"Start":"03:27.965 ","End":"03:30.455","Text":"that means that 6 and 1 are included."},{"Start":"03:30.455 ","End":"03:32.845","Text":"Maybe I should write it down here."},{"Start":"03:32.845 ","End":"03:35.990","Text":"1 less than or equal to x,"},{"Start":"03:35.990 ","End":"03:37.850","Text":"less than or equal to 6,"},{"Start":"03:37.850 ","End":"03:40.460","Text":"but that\u0027s just the answer to this part."},{"Start":"03:40.460 ","End":"03:44.665","Text":"Then we have an end and we have the second part."},{"Start":"03:44.665 ","End":"03:47.610","Text":"3x minus one is less than the 11."},{"Start":"03:47.610 ","End":"03:52.760","Text":"Let\u0027s continue, so 3x is going to be less than 12."},{"Start":"03:52.760 ","End":"03:56.060","Text":"Just bring the one over to the other side and divide by 3,"},{"Start":"03:56.060 ","End":"03:57.770","Text":"which is a positive number."},{"Start":"03:57.770 ","End":"04:02.910","Text":"So we get x less than 4."},{"Start":"04:03.980 ","End":"04:06.890","Text":"This one we\u0027ve sketched here,"},{"Start":"04:06.890 ","End":"04:10.205","Text":"Let\u0027s sketch the x minus 4."},{"Start":"04:10.205 ","End":"04:12.845","Text":"Sorry, the x less than four,"},{"Start":"04:12.845 ","End":"04:16.230","Text":"I need to put four on here somewhere."},{"Start":"04:16.430 ","End":"04:21.410","Text":"Let\u0027s say 4 is somewhere here."},{"Start":"04:21.410 ","End":"04:25.235","Text":"Then if I want to draw the line,"},{"Start":"04:25.235 ","End":"04:27.750","Text":"I\u0027ll just erase this,"},{"Start":"04:28.370 ","End":"04:32.130","Text":"and I\u0027ll draw the x less than 4 in a different color,"},{"Start":"04:32.130 ","End":"04:35.170","Text":"it means that it includes the 4,"},{"Start":"04:35.660 ","End":"04:41.775","Text":"and continues onwards to the left indefinitely,"},{"Start":"04:41.775 ","End":"04:47.420","Text":"and that\u0027s this one just like this color was this one."},{"Start":"04:47.420 ","End":"04:54.455","Text":"Now we want the end of these two when I take and it\u0027s the overlap,"},{"Start":"04:54.455 ","End":"04:56.975","Text":"use a different color again."},{"Start":"04:56.975 ","End":"04:58.880","Text":"This overlap here."},{"Start":"04:58.880 ","End":"05:02.935","Text":"Let\u0027s just be careful because they overlap includes this dot,"},{"Start":"05:02.935 ","End":"05:07.475","Text":"but it does not include this meaning 1 is in but 4 is out."},{"Start":"05:07.475 ","End":"05:17.990","Text":"When we combine these two with an end what we get is 1 less than or equal to x,"},{"Start":"05:17.990 ","End":"05:21.610","Text":"less than strictly 4."},{"Start":"05:21.610 ","End":"05:27.320","Text":"This is the combined answer with the end."},{"Start":"05:27.320 ","End":"05:32.460","Text":"I\u0027ll just highlight it and we\u0027re done and then we\u0027re on to part b."},{"Start":"05:35.950 ","End":"05:39.050","Text":"What do we have here?"},{"Start":"05:39.050 ","End":"05:42.350","Text":"Two quadratics."},{"Start":"05:42.350 ","End":"05:46.610","Text":"Let me start with the second one because it\u0027s easier."},{"Start":"05:46.610 ","End":"05:50.030","Text":"Here we have a difference of squares formula."},{"Start":"05:50.030 ","End":"05:55.330","Text":"I hope you notice that 36 is 6^2."},{"Start":"05:55.330 ","End":"06:02.330","Text":"And then I\u0027m assuming you remember the difference of squares formula of a^2 minus b^2."},{"Start":"06:02.330 ","End":"06:07.175","Text":"This is x minus 6 x plus 6,"},{"Start":"06:07.175 ","End":"06:13.590","Text":"from the a^2 minus b^2 formula, less than 0."},{"Start":"06:14.030 ","End":"06:19.380","Text":"If we draw the number line,"},{"Start":"06:19.380 ","End":"06:25.325","Text":"thus, and I put the roots or I should\u0027ve said the roots,"},{"Start":"06:25.325 ","End":"06:26.840","Text":"if x minus 6 is 0,"},{"Start":"06:26.840 ","End":"06:27.950","Text":"then x is 6."},{"Start":"06:27.950 ","End":"06:29.420","Text":"If x plus 6 is 0,"},{"Start":"06:29.420 ","End":"06:30.755","Text":"x is minus 6,"},{"Start":"06:30.755 ","End":"06:36.780","Text":"so the two roots are minus 6 and 6."},{"Start":"06:36.830 ","End":"06:42.710","Text":"We\u0027ll just do this quickly because we have so much work."},{"Start":"06:42.710 ","End":"06:47.645","Text":"Want to tell whether it\u0027s plus or minus in each of these three intervals,"},{"Start":"06:47.645 ","End":"06:49.445","Text":"Let\u0027s just do a quick substitution."},{"Start":"06:49.445 ","End":"06:52.789","Text":"0 is in this interval, I plug in 0,"},{"Start":"06:52.789 ","End":"06:56.480","Text":"I\u0027ve got minus and plus minus 6 plus 6,"},{"Start":"06:56.480 ","End":"06:59.315","Text":"minus times plus is minus."},{"Start":"06:59.315 ","End":"07:01.280","Text":"If I go above 6,"},{"Start":"07:01.280 ","End":"07:03.109","Text":"so say 7 for example,"},{"Start":"07:03.109 ","End":"07:07.690","Text":"this will be positive and this will be positive, so positive,"},{"Start":"07:07.690 ","End":"07:10.820","Text":"and if I take something to the left of minus 6,"},{"Start":"07:10.820 ","End":"07:13.530","Text":"let\u0027s just say minus 10."},{"Start":"07:13.530 ","End":"07:15.570","Text":"Minus 10 minus 6 is negative,"},{"Start":"07:15.570 ","End":"07:17.300","Text":"minus 10 plus 6 is still negative,"},{"Start":"07:17.300 ","End":"07:19.445","Text":"but together they give us a positive."},{"Start":"07:19.445 ","End":"07:25.025","Text":"What we need is less than 0."},{"Start":"07:25.025 ","End":"07:28.280","Text":"Less than 0 means negative."},{"Start":"07:28.280 ","End":"07:30.110","Text":"This is the bit we want,"},{"Start":"07:30.110 ","End":"07:31.910","Text":"and strictly less than,"},{"Start":"07:31.910 ","End":"07:35.300","Text":"which means we want from minus 6 to 6,"},{"Start":"07:35.300 ","End":"07:41.175","Text":"not including minus 6 less than x, less than 6."},{"Start":"07:41.175 ","End":"07:44.835","Text":"That solves this bit."},{"Start":"07:44.835 ","End":"07:52.265","Text":"For this bit, we need to factorize the left-hand thought."},{"Start":"07:52.265 ","End":"07:56.905","Text":"I mean, what\u0027s before the greater than, the quadratic."},{"Start":"07:56.905 ","End":"07:59.590","Text":"I\u0027ll give it to you."},{"Start":"07:59.590 ","End":"08:06.955","Text":"This happens to be x minus 3, x minus 5."},{"Start":"08:06.955 ","End":"08:09.650","Text":"Tell you up, once in a while,"},{"Start":"08:09.650 ","End":"08:11.075","Text":"I\u0027ll show you how we get to it."},{"Start":"08:11.075 ","End":"08:14.545","Text":"I\u0027ll remind you of the trinomial technique,"},{"Start":"08:14.545 ","End":"08:17.254","Text":"just go over this briefly."},{"Start":"08:17.254 ","End":"08:21.830","Text":"What we do with this technique is to ask"},{"Start":"08:21.830 ","End":"08:28.890","Text":"which two numbers can we multiply together to give ac,"},{"Start":"08:28.890 ","End":"08:31.605","Text":"which in this case is 1 times 15."},{"Start":"08:31.605 ","End":"08:34.515","Text":"We add those same two numbers,"},{"Start":"08:34.515 ","End":"08:39.060","Text":"and we want to get the b term which is minus 8."},{"Start":"08:39.410 ","End":"08:41.690","Text":"What we do is this,"},{"Start":"08:41.690 ","End":"08:45.350","Text":"first of all, don\u0027t pay too much attention to the signs."},{"Start":"08:45.350 ","End":"08:48.210","Text":"We know that the product is 15 is positive,"},{"Start":"08:48.210 ","End":"08:50.810","Text":"so they\u0027re both positive or both negative."},{"Start":"08:50.810 ","End":"08:53.510","Text":"Because the sum is negative, they\u0027re both negative."},{"Start":"08:53.510 ","End":"08:56.435","Text":"But effectively it\u0027s an addition because when you add negatives,"},{"Start":"08:56.435 ","End":"08:58.295","Text":"if you know the minus, it\u0027s an addition."},{"Start":"08:58.295 ","End":"09:02.120","Text":"I\u0027m really asking what two numbers add together to give 8,"},{"Start":"09:02.120 ","End":"09:05.215","Text":"and multiply together to give 15."},{"Start":"09:05.215 ","End":"09:08.580","Text":"Few moments reflection gives you 5 and 3,"},{"Start":"09:08.580 ","End":"09:11.515","Text":"5 times 3 is 15 and 5 plus 3 is 8."},{"Start":"09:11.515 ","End":"09:13.205","Text":"But because of this minus,"},{"Start":"09:13.205 ","End":"09:17.325","Text":"we need a minus 5 times minus 3."},{"Start":"09:17.325 ","End":"09:19.530","Text":"Should have be brackets here, never mind."},{"Start":"09:19.530 ","End":"09:25.770","Text":"Minus 5 plus minus 3 gives us the minus 8,"},{"Start":"09:25.770 ","End":"09:27.540","Text":"so these are the two numbers."},{"Start":"09:27.540 ","End":"09:29.495","Text":"What we do with these two numbers,"},{"Start":"09:29.495 ","End":"09:32.320","Text":"we normally go back here,"},{"Start":"09:32.320 ","End":"09:36.700","Text":"we would say x^2 minus 8x plus 15,"},{"Start":"09:36.700 ","End":"09:39.385","Text":"and then we would decompose the middle,"},{"Start":"09:39.385 ","End":"09:41.035","Text":"this minus 8,"},{"Start":"09:41.035 ","End":"09:44.390","Text":"according to the numbers here,"},{"Start":"09:44.390 ","End":"09:51.710","Text":"so we\u0027d write it as x^2 minus 5x minus 3x plus 15,"},{"Start":"09:51.710 ","End":"09:54.940","Text":"then this is equal to from the first pair,"},{"Start":"09:54.940 ","End":"09:58.930","Text":"we take out x and we\u0027ve got x minus 5."},{"Start":"09:58.930 ","End":"10:04.305","Text":"From the second pair, I take out minus 3 and I get x minus 5."},{"Start":"10:04.305 ","End":"10:09.055","Text":"Finally I combine them by taking x minus 5 out."},{"Start":"10:09.055 ","End":"10:12.610","Text":"What we\u0027re left with is x minus 3,"},{"Start":"10:12.800 ","End":"10:15.340","Text":"and that\u0027s how we got that."},{"Start":"10:15.340 ","End":"10:18.505","Text":"In the other way is using the formula, the quadratic."},{"Start":"10:18.505 ","End":"10:22.340","Text":"Let\u0027s continue, this was just a refresher."},{"Start":"10:22.790 ","End":"10:25.250","Text":"Now we need a sketch,"},{"Start":"10:25.250 ","End":"10:28.780","Text":"I\u0027m going to just copy this one then modify it."},{"Start":"10:28.780 ","End":"10:30.490","Text":"Here\u0027s the copy,"},{"Start":"10:30.490 ","End":"10:34.435","Text":"what I don\u0027t need, this stuff."},{"Start":"10:34.435 ","End":"10:39.035","Text":"What I do need the roots which are 3 and 5,"},{"Start":"10:39.035 ","End":"10:41.545","Text":"I guess I should have said if x minus 3 is 0,"},{"Start":"10:41.545 ","End":"10:43.750","Text":"then x is 3, if x minus 5 is 0,"},{"Start":"10:43.750 ","End":"10:46.570","Text":"x is 5, so the two roots of 3 and 5."},{"Start":"10:46.570 ","End":"10:50.275","Text":"I didn\u0027t erase the plus, minus, plus,"},{"Start":"10:50.275 ","End":"10:55.185","Text":"because if you check it actually comes out to be the same."},{"Start":"10:55.185 ","End":"10:57.800","Text":"We can quickly do that in our heads."},{"Start":"10:57.800 ","End":"11:04.665","Text":"If x is, let\u0027s say 4,4 minus 3 is positive and 4 minus 5 is negative and so on."},{"Start":"11:04.665 ","End":"11:06.180","Text":"Check with x equals 6,"},{"Start":"11:06.180 ","End":"11:07.440","Text":"check with x equals 2,"},{"Start":"11:07.440 ","End":"11:09.615","Text":"this comes out positive, positive."},{"Start":"11:09.615 ","End":"11:17.570","Text":"Just like before, the difference is that here we had less than 0,"},{"Start":"11:17.570 ","End":"11:20.080","Text":"so we focused on the negative."},{"Start":"11:20.080 ","End":"11:23.580","Text":"But now we have bigger than 0,"},{"Start":"11:23.580 ","End":"11:27.290","Text":"so we focus on the positive which is here and here,"},{"Start":"11:27.290 ","End":"11:33.780","Text":"and so we don\u0027t get that instead we get left of 3 and right of 5,"},{"Start":"11:33.780 ","End":"11:36.195","Text":"x less than 3,"},{"Start":"11:36.195 ","End":"11:41.170","Text":"or x bigger than 5."},{"Start":"11:41.560 ","End":"11:45.560","Text":"Now we\u0027ve got the two pieces separately."},{"Start":"11:45.560 ","End":"11:51.158","Text":"I want to now combine them using the and or techniques."},{"Start":"11:51.158 ","End":"11:53.320","Text":"Here\u0027s another number line."},{"Start":"11:53.320 ","End":"11:54.820","Text":"This time we have to put all the points,"},{"Start":"11:54.820 ","End":"11:55.975","Text":"we\u0027re going to combine these."},{"Start":"11:55.975 ","End":"11:57.670","Text":"Let\u0027s see. We need a 3,"},{"Start":"11:57.670 ","End":"11:59.965","Text":"a 5, a minus 6, and a 6."},{"Start":"11:59.965 ","End":"12:02.635","Text":"Let\u0027s start off with the minus 6."},{"Start":"12:02.635 ","End":"12:05.950","Text":"Then let\u0027s have a 3,"},{"Start":"12:05.950 ","End":"12:10.465","Text":"a 5, and a 6,"},{"Start":"12:10.465 ","End":"12:12.325","Text":"that\u0027s all these points."},{"Start":"12:12.325 ","End":"12:21.025","Text":"Now, the picture for this one is here, I drew it for you."},{"Start":"12:21.025 ","End":"12:22.540","Text":"X less than 3,"},{"Start":"12:22.540 ","End":"12:25.075","Text":"[inaudible] circle because it doesn\u0027t include 3."},{"Start":"12:25.075 ","End":"12:26.380","Text":"X is bigger than 5,"},{"Start":"12:26.380 ","End":"12:27.969","Text":"it goes on indefinitely."},{"Start":"12:27.969 ","End":"12:30.850","Text":"That\u0027s this inequality."},{"Start":"12:30.850 ","End":"12:36.040","Text":"What we need to do is combine this one with this one."},{"Start":"12:36.040 ","End":"12:40.330","Text":"Now this one, I\u0027ll do in a different color."},{"Start":"12:40.330 ","End":"12:42.730","Text":"There we are. That\u0027s this one,"},{"Start":"12:42.730 ","End":"12:45.445","Text":"I\u0027ll just mark it in and this is the one in this color."},{"Start":"12:45.445 ","End":"12:49.045","Text":"Between minus 6 and 6, not inclusive."},{"Start":"12:49.045 ","End":"12:53.230","Text":"Now, we need the end of these two,"},{"Start":"12:53.230 ","End":"12:55.630","Text":"which means where do they overlap?"},{"Start":"12:55.630 ","End":"12:58.930","Text":"I see we have a bit here that overlaps,"},{"Start":"12:58.930 ","End":"13:03.920","Text":"and we also have a bit here, maybe it\u0027s here."},{"Start":"13:04.080 ","End":"13:06.610","Text":"Let\u0027s just write it out."},{"Start":"13:06.610 ","End":"13:12.640","Text":"This of course, the overlap is between minus 6 and 3."},{"Start":"13:12.640 ","End":"13:14.500","Text":"This one goes on indefinitely,"},{"Start":"13:14.500 ","End":"13:17.035","Text":"but this one stops at minus 6."},{"Start":"13:17.035 ","End":"13:21.430","Text":"So we have that this end is"},{"Start":"13:21.430 ","End":"13:27.820","Text":"minus 6 less than x, less than 3."},{"Start":"13:27.820 ","End":"13:32.455","Text":"We also have the extra bit which makes it an or,"},{"Start":"13:32.455 ","End":"13:35.410","Text":"because x can be either here or here."},{"Start":"13:35.410 ","End":"13:41.740","Text":"We have 5 less than x, less than 6."},{"Start":"13:41.740 ","End":"13:52.010","Text":"I\u0027ll just highlight this and declare that this is the answer for part b."},{"Start":"13:52.050 ","End":"13:58.405","Text":"Moving on to part c. This time"},{"Start":"13:58.405 ","End":"14:05.335","Text":"we spend too much time on parts a and b. I\u0027ll just do the factoring for you."},{"Start":"14:05.335 ","End":"14:12.410","Text":"This quadratic factorizes as x minus 10,"},{"Start":"14:12.410 ","End":"14:18.105","Text":"x minus 9 is still bigger or equal to 0."},{"Start":"14:18.105 ","End":"14:21.315","Text":"This combines with the operator."},{"Start":"14:21.315 ","End":"14:23.655","Text":"It\u0027s called an operator,"},{"Start":"14:23.655 ","End":"14:26.630","Text":"logical operator. Doesn\u0027t matter anyway."},{"Start":"14:26.630 ","End":"14:33.730","Text":"This one factorizes as x minus 2,"},{"Start":"14:33.730 ","End":"14:35.500","Text":"x plus 1,"},{"Start":"14:35.500 ","End":"14:38.890","Text":"and it\u0027s less than or equal to 0."},{"Start":"14:38.890 ","End":"14:41.289","Text":"As for the roots,"},{"Start":"14:41.289 ","End":"14:44.860","Text":"the roots here are 10,"},{"Start":"14:44.860 ","End":"14:50.080","Text":"and 9, and the roots here are 2 and minus 1."},{"Start":"14:50.080 ","End":"14:53.200","Text":"These are the things that make these brackets 0,"},{"Start":"14:53.200 ","End":"14:54.490","Text":"10 makes this one 0,"},{"Start":"14:54.490 ","End":"14:57.620","Text":"9 makes this one 0, and so on."},{"Start":"14:59.280 ","End":"15:06.020","Text":"There is a difference between bigger or equal to 0 and less than or equal to 0."},{"Start":"15:06.630 ","End":"15:11.065","Text":"Let\u0027s do a diagram for each."},{"Start":"15:11.065 ","End":"15:13.960","Text":"You know what? I believe in recycling."},{"Start":"15:13.960 ","End":"15:17.110","Text":"I\u0027m going to copy these two pictures down here."},{"Start":"15:17.110 ","End":"15:19.030","Text":"I need to change the numbers."},{"Start":"15:19.030 ","End":"15:24.865","Text":"These go, and instead I have 9 and 10,"},{"Start":"15:24.865 ","End":"15:26.920","Text":"just get them in the right order,"},{"Start":"15:26.920 ","End":"15:30.970","Text":"and here minus 1 and 2."},{"Start":"15:30.970 ","End":"15:35.215","Text":"The other difference is that here we have bigger or equal to 0."},{"Start":"15:35.215 ","End":"15:42.295","Text":"It\u0027s plus, but also these points here will be included."},{"Start":"15:42.295 ","End":"15:46.600","Text":"Here, less than or equal to 0,"},{"Start":"15:46.600 ","End":"15:52.540","Text":"so the minus, but also the points where it\u0027s 0."},{"Start":"15:52.540 ","End":"16:01.870","Text":"The essential difference is that here I get,"},{"Start":"16:01.870 ","End":"16:04.870","Text":"instead of x less than 3, x bigger than 5."},{"Start":"16:04.870 ","End":"16:12.115","Text":"In this question it comes out to x less than or equal to 9,"},{"Start":"16:12.115 ","End":"16:19.915","Text":"or x bigger than 5 in our case is x bigger or equal to,"},{"Start":"16:19.915 ","End":"16:21.445","Text":"because we also have the equal,"},{"Start":"16:21.445 ","End":"16:26.755","Text":"and it\u0027s 10 and brackets."},{"Start":"16:26.755 ","End":"16:30.470","Text":"This time I have an OR."},{"Start":"16:30.680 ","End":"16:35.940","Text":"What I have is just like before between the 2 values,"},{"Start":"16:35.940 ","End":"16:38.945","Text":"except that I have to take the equals case also."},{"Start":"16:38.945 ","End":"16:42.965","Text":"Minus 1 less than or equal to x,"},{"Start":"16:42.965 ","End":"16:45.505","Text":"less than or equal to 2."},{"Start":"16:45.505 ","End":"16:48.115","Text":"Now to combine these two,"},{"Start":"16:48.115 ","End":"16:52.750","Text":"I\u0027m going to use another diagram."},{"Start":"16:52.750 ","End":"16:55.675","Text":"Here I drew the diagram,"},{"Start":"16:55.675 ","End":"17:02.320","Text":"the first 1 with the OR is the blue one and this bit is the purple one."},{"Start":"17:02.320 ","End":"17:03.940","Text":"When I OR them together,"},{"Start":"17:03.940 ","End":"17:08.365","Text":"it\u0027s like a squished them one on top of the other or however way you want to look at it."},{"Start":"17:08.365 ","End":"17:11.110","Text":"But because this is included in this,"},{"Start":"17:11.110 ","End":"17:14.020","Text":"we just get what was written here."},{"Start":"17:14.020 ","End":"17:19.645","Text":"The answer is just x less than or equal to"},{"Start":"17:19.645 ","End":"17:25.765","Text":"9 or x bigger or equal to 10,"},{"Start":"17:25.765 ","End":"17:29.390","Text":"and that is the answer."},{"Start":"17:29.760 ","End":"17:34.540","Text":"To be consistent I\u0027ll shade it just like I did with the others."},{"Start":"17:34.540 ","End":"17:40.300","Text":"That\u0027s part c. Moving on to last part,"},{"Start":"17:40.300 ","End":"17:43.015","Text":"d. What do we have here?"},{"Start":"17:43.015 ","End":"17:46.645","Text":"Again, two quadratic inequalities."},{"Start":"17:46.645 ","End":"17:53.305","Text":"I notice that in both cases it\u0027s without the equals, the strict inequalities."},{"Start":"17:53.305 ","End":"17:56.950","Text":"These are similar to ones we\u0027ve done before."},{"Start":"17:56.950 ","End":"18:00.100","Text":"This one we used the difference of squares."},{"Start":"18:00.100 ","End":"18:02.635","Text":"Last time it was x^2 minus something else."},{"Start":"18:02.635 ","End":"18:05.515","Text":"But this is x minus 1,"},{"Start":"18:05.515 ","End":"18:10.250","Text":"x plus 1 bigger than 0."},{"Start":"18:13.770 ","End":"18:16.885","Text":"This is 0 when x is 1,"},{"Start":"18:16.885 ","End":"18:19.030","Text":"this is 0 when x is minus 1,"},{"Start":"18:19.030 ","End":"18:20.215","Text":"those are the two roots."},{"Start":"18:20.215 ","End":"18:23.695","Text":"For this one, I\u0027ll factorize it for you."},{"Start":"18:23.695 ","End":"18:25.900","Text":"X minus 4,"},{"Start":"18:25.900 ","End":"18:30.355","Text":"x plus 1 less than 0."},{"Start":"18:30.355 ","End":"18:35.575","Text":"That means that the roots for this one,"},{"Start":"18:35.575 ","End":"18:38.050","Text":"4, for this one minus 1."},{"Start":"18:38.050 ","End":"18:39.520","Text":"That\u0027s what make this 0,"},{"Start":"18:39.520 ","End":"18:41.780","Text":"these are the two roots."},{"Start":"18:42.450 ","End":"18:47.215","Text":"We can use a couple of sketches."},{"Start":"18:47.215 ","End":"18:53.600","Text":"While you weren\u0027t looking I just threw in a couple of axes, number lines."},{"Start":"18:55.080 ","End":"18:58.075","Text":"I\u0027m going to label them both."},{"Start":"18:58.075 ","End":"19:00.790","Text":"Here I need to label 4 and minus 1."},{"Start":"19:00.790 ","End":"19:02.710","Text":"So this one will be minus 1,"},{"Start":"19:02.710 ","End":"19:04.540","Text":"this one will be 4 to the right of it."},{"Start":"19:04.540 ","End":"19:07.210","Text":"Here I need minus 1 on the left,"},{"Start":"19:07.210 ","End":"19:10.015","Text":"and I need 1 on the right."},{"Start":"19:10.015 ","End":"19:13.960","Text":"Here I need less than 0."},{"Start":"19:13.960 ","End":"19:20.950","Text":"Let me just say that if we take sample values and substitute in,"},{"Start":"19:20.950 ","End":"19:23.080","Text":"we\u0027ve seen this dozens of times."},{"Start":"19:23.080 ","End":"19:30.175","Text":"I\u0027ll just save time by saying that here it comes out to be negative."},{"Start":"19:30.175 ","End":"19:31.900","Text":"I\u0027ll just give an example."},{"Start":"19:31.900 ","End":"19:36.174","Text":"If x is 0, then we have negative times positive is negative."},{"Start":"19:36.174 ","End":"19:37.990","Text":"Here we\u0027ll get x=5,"},{"Start":"19:37.990 ","End":"19:39.550","Text":"here we\u0027ll get x equals minus 2,"},{"Start":"19:39.550 ","End":"19:42.250","Text":"so we get basically plus, minus, plus."},{"Start":"19:42.250 ","End":"19:44.320","Text":"Similarly here we\u0027ll also,"},{"Start":"19:44.320 ","End":"19:47.770","Text":"when x is 0,"},{"Start":"19:47.770 ","End":"19:49.060","Text":"which is between the two."},{"Start":"19:49.060 ","End":"19:51.415","Text":"We get minus, plus,"},{"Start":"19:51.415 ","End":"19:54.745","Text":"minus and here we get plus and here we get plus."},{"Start":"19:54.745 ","End":"19:56.680","Text":"But what are we interested in?"},{"Start":"19:56.680 ","End":"19:59.920","Text":"Here we\u0027re interested in the bigger than 0,"},{"Start":"19:59.920 ","End":"20:01.645","Text":"which is the positive."},{"Start":"20:01.645 ","End":"20:04.330","Text":"Here we\u0027re interested in the negative,"},{"Start":"20:04.330 ","End":"20:05.710","Text":"which is this bit."},{"Start":"20:05.710 ","End":"20:11.785","Text":"So over here we get minus 1 less than x,"},{"Start":"20:11.785 ","End":"20:13.930","Text":"less than 4,"},{"Start":"20:13.930 ","End":"20:15.580","Text":"not including, as I said,"},{"Start":"20:15.580 ","End":"20:17.470","Text":"because we don\u0027t have the equal here."},{"Start":"20:17.470 ","End":"20:26.350","Text":"Here, we get x is less than minus 1 or x is bigger than 1."},{"Start":"20:26.350 ","End":"20:29.544","Text":"We get this bit and this bit, not including."},{"Start":"20:29.544 ","End":"20:34.495","Text":"Now, we have to take this and combine the with OR."},{"Start":"20:34.495 ","End":"20:36.310","Text":"So basically we have three OR\u0027s,"},{"Start":"20:36.310 ","End":"20:38.845","Text":"this OR, this OR, this."},{"Start":"20:38.845 ","End":"20:45.510","Text":"We can do these on one number line."},{"Start":"20:45.510 ","End":"20:47.865","Text":"I\u0027ve room over here."},{"Start":"20:47.865 ","End":"20:49.800","Text":"Let\u0027s see what points do we need?"},{"Start":"20:49.800 ","End":"20:51.510","Text":"Minus 1 is in common,"},{"Start":"20:51.510 ","End":"20:53.730","Text":"so we need minus 1, 1,"},{"Start":"20:53.730 ","End":"20:56.425","Text":"and 4. Very well."},{"Start":"20:56.425 ","End":"20:59.590","Text":"Minus 1, 1, and 4,"},{"Start":"20:59.590 ","End":"21:03.535","Text":"nothing has to be to scale but the order is very important."},{"Start":"21:03.535 ","End":"21:05.710","Text":"Now, let\u0027s see."},{"Start":"21:05.710 ","End":"21:12.074","Text":"Even though these two really belong together, they don\u0027t."},{"Start":"21:12.074 ","End":"21:16.930","Text":"I can undo the brackets and say this or this or this and consider it as three pieces."},{"Start":"21:16.930 ","End":"21:21.380","Text":"Let\u0027s see. X between minus 1 and 4."},{"Start":"21:21.380 ","End":"21:23.820","Text":"I do the minus 1,"},{"Start":"21:23.820 ","End":"21:25.260","Text":"I do the 4,"},{"Start":"21:25.260 ","End":"21:28.980","Text":"and then I join them with a line."},{"Start":"21:28.980 ","End":"21:32.535","Text":"Next one, x less than minus 1."},{"Start":"21:32.535 ","End":"21:35.950","Text":"Here\u0027s the minus 1,"},{"Start":"21:35.950 ","End":"21:40.795","Text":"and x less than gives me this."},{"Start":"21:40.795 ","End":"21:45.860","Text":"The final bit, x bigger than 1,"},{"Start":"21:46.470 ","End":"21:54.310","Text":"so this and here."},{"Start":"21:54.310 ","End":"22:03.084","Text":"What we do with OR\u0027ing all these things is that we see anything,"},{"Start":"22:03.084 ","End":"22:07.225","Text":"take everything that\u0027s under any one or more of these."},{"Start":"22:07.225 ","End":"22:09.400","Text":"I once gave an example"},{"Start":"22:09.400 ","End":"22:15.750","Text":"of these are like roofs or shades and the sun is shining from above."},{"Start":"22:15.750 ","End":"22:17.959","Text":"Let\u0027s see what\u0027s in the shadow."},{"Start":"22:17.959 ","End":"22:19.570","Text":"Well, basically,"},{"Start":"22:19.570 ","End":"22:22.120","Text":"these three lines cover everything."},{"Start":"22:22.120 ","End":"22:24.385","Text":"But if you really look closely,"},{"Start":"22:24.385 ","End":"22:27.760","Text":"there\u0027s a gap, nothing covers minus 1."},{"Start":"22:27.760 ","End":"22:29.500","Text":"Everything is covered,"},{"Start":"22:29.500 ","End":"22:32.230","Text":"but there\u0027s a gap through here."},{"Start":"22:32.230 ","End":"22:42.520","Text":"Really, this or this or this is just all x except minus 1,"},{"Start":"22:42.520 ","End":"22:46.910","Text":"so I\u0027ll write this as x not equal to minus 1."},{"Start":"22:48.000 ","End":"22:51.250","Text":"It\u0027s just everything with a hole here."},{"Start":"22:51.250 ","End":"22:55.945","Text":"Some people like to write this as x less than minus 1,"},{"Start":"22:55.945 ","End":"22:58.615","Text":"or x bigger than minus 1."},{"Start":"22:58.615 ","End":"23:00.910","Text":"It\u0027s possible to write it this way,"},{"Start":"23:00.910 ","End":"23:03.865","Text":"but I prefer this, it\u0027s more compact."},{"Start":"23:03.865 ","End":"23:09.160","Text":"I\u0027ll go with this and highlight it."},{"Start":"23:09.160 ","End":"23:11.935","Text":"That is the answer to part d,"},{"Start":"23:11.935 ","End":"23:15.110","Text":"which is the last part. So we are done."}],"Thumbnail":null,"ID":5066},{"Watched":false,"Name":"Exercise 5","Duration":"21m 33s","ChapterTopicVideoID":5064,"CourseChapterTopicPlaylistID":3876,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.250","Text":"In this exercise, we have,"},{"Start":"00:02.250 ","End":"00:05.295","Text":"let\u0027s see, 5 inequalities to solve."},{"Start":"00:05.295 ","End":"00:09.150","Text":"In all of them, the degree is going to be greater than 2."},{"Start":"00:09.150 ","End":"00:11.610","Text":"Mostly we\u0027ve been doing quadratic polynomials."},{"Start":"00:11.610 ","End":"00:13.935","Text":"Now this time we have higher order."},{"Start":"00:13.935 ","End":"00:17.295","Text":"Let\u0027s begin with the first one,"},{"Start":"00:17.295 ","End":"00:20.820","Text":"where it\u0027s already factored."},{"Start":"00:20.820 ","End":"00:25.515","Text":"The roots are x=1,"},{"Start":"00:25.515 ","End":"00:27.315","Text":"2, or 3."},{"Start":"00:27.315 ","End":"00:30.540","Text":"These are the things that make one of these factors 0,"},{"Start":"00:30.540 ","End":"00:34.380","Text":"and we\u0027ll do it with the diagram."},{"Start":"00:34.380 ","End":"00:37.315","Text":"I\u0027ll sketch the number line."},{"Start":"00:37.315 ","End":"00:40.605","Text":"On the number line, I\u0027ll mark these 3 points."},{"Start":"00:40.605 ","End":"00:42.320","Text":"We have to do them in the correct order,"},{"Start":"00:42.320 ","End":"00:45.840","Text":"but that\u0027s as easy as 1, 2, 3."},{"Start":"00:45.840 ","End":"00:48.045","Text":"We\u0027ll have 1 first,"},{"Start":"00:48.045 ","End":"00:49.665","Text":"then we\u0027ll have 2,"},{"Start":"00:49.665 ","End":"00:52.195","Text":"and then we\u0027ll have 3."},{"Start":"00:52.195 ","End":"01:01.475","Text":"Now we want to see in which interval this polynomial is positive and where it\u0027s negative."},{"Start":"01:01.475 ","End":"01:06.260","Text":"Don\u0027t always assume that the sign alternates as is often the case."},{"Start":"01:06.260 ","End":"01:09.510","Text":"We have to check each integral separately."},{"Start":"01:09.510 ","End":"01:12.640","Text":"Here we have 1, 2, 3, 4 intervals."},{"Start":"01:12.730 ","End":"01:16.125","Text":"I think we\u0027ll do this in our heads."},{"Start":"01:16.125 ","End":"01:19.239","Text":"I\u0027ll use the snake method."},{"Start":"01:19.239 ","End":"01:22.850","Text":"I\u0027ll choose here the number 0."},{"Start":"01:22.850 ","End":"01:24.800","Text":"If I put 0 in,"},{"Start":"01:24.800 ","End":"01:29.930","Text":"then I get minus, minus, minus."},{"Start":"01:29.930 ","End":"01:33.930","Text":"It\u0027s a minus, so it\u0027s somewhere below the axis."},{"Start":"01:33.930 ","End":"01:37.575","Text":"I don\u0027t even have to plot the 0 or exact the value,"},{"Start":"01:37.575 ","End":"01:42.139","Text":"it\u0027s just schematic to see where we are positive and negative."},{"Start":"01:42.139 ","End":"01:44.740","Text":"Between 1 and 2,"},{"Start":"01:44.740 ","End":"01:47.000","Text":"I\u0027ll choose, say,1 1/2."},{"Start":"01:47.000 ","End":"01:49.130","Text":"That makes this positive,"},{"Start":"01:49.130 ","End":"01:56.110","Text":"but the other two become negative and so what we altogether get is positive."},{"Start":"01:56.110 ","End":"02:00.220","Text":"If I choose x=2 1/2,"},{"Start":"02:00.220 ","End":"02:02.470","Text":"then this is positive,"},{"Start":"02:02.470 ","End":"02:05.840","Text":"this is positive, but this is negative,"},{"Start":"02:05.840 ","End":"02:08.955","Text":"so somewhere, say, here."},{"Start":"02:08.955 ","End":"02:12.870","Text":"Then to the right of 3, I\u0027ll choose 4,"},{"Start":"02:12.870 ","End":"02:14.550","Text":"4 minus 1, 4 minus 2,"},{"Start":"02:14.550 ","End":"02:16.935","Text":"4 minus 3, they\u0027re all positive."},{"Start":"02:16.935 ","End":"02:22.600","Text":"We have a positive number above the 4, something like this."},{"Start":"02:22.600 ","End":"02:24.940","Text":"Then we connect them."},{"Start":"02:24.940 ","End":"02:29.845","Text":"I should really go through these points here."},{"Start":"02:29.845 ","End":"02:32.380","Text":"The roots are where this polynomial is 0."},{"Start":"02:32.380 ","End":"02:34.750","Text":"This is precise. These 3 points are precise."},{"Start":"02:34.750 ","End":"02:36.550","Text":"The rest of it is just a sketch."},{"Start":"02:36.550 ","End":"02:39.450","Text":"We draw what we call the snake."},{"Start":"02:39.450 ","End":"02:42.335","Text":"Something goes through here,"},{"Start":"02:42.335 ","End":"02:44.150","Text":"and then through here,"},{"Start":"02:44.150 ","End":"02:46.940","Text":"then here, then here, then here."},{"Start":"02:46.940 ","End":"02:50.480","Text":"It\u0027s not an accurate representation of what the graph looks like,"},{"Start":"02:50.480 ","End":"02:54.590","Text":"but it is accurate as far as where it\u0027s above the axis and below the axis."},{"Start":"02:54.590 ","End":"02:56.225","Text":"Now what do we want?"},{"Start":"02:56.225 ","End":"03:01.170","Text":"We want above the axis."},{"Start":"03:01.170 ","End":"03:03.965","Text":"Greater than 0 means above the axis."},{"Start":"03:03.965 ","End":"03:10.040","Text":"I can either highlight these parts or we can use shading,"},{"Start":"03:10.040 ","End":"03:11.675","Text":"it doesn\u0027t really matter."},{"Start":"03:11.675 ","End":"03:13.760","Text":"I\u0027ll use the shading."},{"Start":"03:13.760 ","End":"03:19.730","Text":"Important thing is to see where the shading hits the x-axis."},{"Start":"03:19.730 ","End":"03:24.935","Text":"What we have is between 1 and 2,"},{"Start":"03:24.935 ","End":"03:28.250","Text":"but not including these points,"},{"Start":"03:28.250 ","End":"03:32.420","Text":"are where the polynomial is 0."},{"Start":"03:32.420 ","End":"03:33.470","Text":"We don\u0027t want those,"},{"Start":"03:33.470 ","End":"03:36.240","Text":"so it\u0027s between 1 and 2."},{"Start":"03:36.240 ","End":"03:40.195","Text":"I\u0027ll write it 1 less than x, less than 2."},{"Start":"03:40.195 ","End":"03:45.310","Text":"Then there\u0027s this bit which is x bigger than 3."},{"Start":"03:45.310 ","End":"03:49.187","Text":"From 3 onwards, it stays positive all the way to infinity,"},{"Start":"03:49.187 ","End":"03:52.315","Text":"so we write x bigger than 3."},{"Start":"03:52.315 ","End":"03:55.276","Text":"The word or comes in-between"},{"Start":"03:55.276 ","End":"03:59.275","Text":"because each x can be either here or here, and it will be good."},{"Start":"03:59.275 ","End":"04:02.635","Text":"That answers part A."},{"Start":"04:02.635 ","End":"04:04.730","Text":"When I highlight the answer,"},{"Start":"04:04.730 ","End":"04:08.470","Text":"it looks nicer. Onto part B."},{"Start":"04:08.470 ","End":"04:10.795","Text":"In part B,"},{"Start":"04:10.795 ","End":"04:14.240","Text":"we might think, someone hasn\u0027t factorized it all the way."},{"Start":"04:14.240 ","End":"04:15.290","Text":"That wasn\u0027t nice of them."},{"Start":"04:15.290 ","End":"04:17.465","Text":"Here, they gave us all the factors."},{"Start":"04:17.465 ","End":"04:20.210","Text":"It turns out that this can\u0027t be factorized,"},{"Start":"04:20.210 ","End":"04:22.220","Text":"but at this point, we don\u0027t know that."},{"Start":"04:22.220 ","End":"04:25.220","Text":"If you went ahead and tried to do it,"},{"Start":"04:25.220 ","End":"04:27.740","Text":"either using the method of"},{"Start":"04:27.740 ","End":"04:33.185","Text":"the trinomial decomposition or using the formula to find the roots,"},{"Start":"04:33.185 ","End":"04:36.455","Text":"you will find that it doesn\u0027t work."},{"Start":"04:36.455 ","End":"04:40.205","Text":"Here\u0027s a suggestion I often like to make."},{"Start":"04:40.205 ","End":"04:42.950","Text":"If you think there may not be any roots,"},{"Start":"04:42.950 ","End":"04:49.160","Text":"just check the b^2 minus 4ac where these,"},{"Start":"04:49.160 ","End":"04:50.987","Text":"from the quadratic equation,"},{"Start":"04:50.987 ","End":"04:54.875","Text":"is equal to, and I\u0027m just talking about this part here,"},{"Start":"04:54.875 ","End":"05:01.690","Text":"is 1^2 minus 4 times 1 times 1."},{"Start":"05:01.690 ","End":"05:03.660","Text":"It\u0027s equal to 1 minus 4,"},{"Start":"05:03.660 ","End":"05:04.875","Text":"which is minus 3."},{"Start":"05:04.875 ","End":"05:06.755","Text":"Minus 3 is not important."},{"Start":"05:06.755 ","End":"05:11.745","Text":"What\u0027s important is that it\u0027s less than 0."},{"Start":"05:11.745 ","End":"05:13.820","Text":"Negative 3 has no square root."},{"Start":"05:13.820 ","End":"05:17.060","Text":"Whenever you have the b^2 minus 4ac as negative,"},{"Start":"05:17.060 ","End":"05:19.130","Text":"you know that this doesn\u0027t decompose,"},{"Start":"05:19.130 ","End":"05:20.915","Text":"it can\u0027t be factorized."},{"Start":"05:20.915 ","End":"05:23.910","Text":"We just have to leave it as is."},{"Start":"05:24.190 ","End":"05:28.325","Text":"That means the only root can come from this part."},{"Start":"05:28.325 ","End":"05:31.613","Text":"Clearly the root is x=0,"},{"Start":"05:31.613 ","End":"05:37.040","Text":"so x=0 is the only place where this polynomial will become 0."},{"Start":"05:37.040 ","End":"05:40.177","Text":"This is never 0, so only when x is 0."},{"Start":"05:40.177 ","End":"05:42.830","Text":"Maybe I should write the word roots."},{"Start":"05:42.830 ","End":"05:45.875","Text":"These are the roots of the polynomial,"},{"Start":"05:45.875 ","End":"05:49.550","Text":"and these are the root,"},{"Start":"05:49.550 ","End":"05:53.735","Text":"no s. Let\u0027s draw the picture."},{"Start":"05:53.735 ","End":"05:59.135","Text":"Here\u0027s a number line and here\u0027s the single root 0."},{"Start":"05:59.135 ","End":"06:00.920","Text":"All we have to do is check,"},{"Start":"06:00.920 ","End":"06:01.940","Text":"there\u0027s only 2 intervals,"},{"Start":"06:01.940 ","End":"06:03.440","Text":"left of 0 and right of 0,"},{"Start":"06:03.440 ","End":"06:05.830","Text":"which is positive and which is negative."},{"Start":"06:05.830 ","End":"06:08.910","Text":"At 0 itself where 0,"},{"Start":"06:08.910 ","End":"06:11.235","Text":"but let\u0027s see."},{"Start":"06:11.235 ","End":"06:13.290","Text":"Try minus 1."},{"Start":"06:13.290 ","End":"06:17.065","Text":"If we put x as minus 1,"},{"Start":"06:17.065 ","End":"06:18.815","Text":"we\u0027ll get, let\u0027s see,"},{"Start":"06:18.815 ","End":"06:22.490","Text":"in our heads, minus 1^2 is 1,"},{"Start":"06:22.490 ","End":"06:25.405","Text":"and then minus 1 plus 1."},{"Start":"06:25.405 ","End":"06:29.940","Text":"This is positive, minus 1 is negative."},{"Start":"06:29.940 ","End":"06:35.655","Text":"In this part, we are below the axis,"},{"Start":"06:35.655 ","End":"06:38.370","Text":"minus 1 somewhere here."},{"Start":"06:38.370 ","End":"06:40.920","Text":"If x is, let\u0027s say,"},{"Start":"06:40.920 ","End":"06:42.930","Text":"1, that\u0027s going to be easier."},{"Start":"06:42.930 ","End":"06:47.445","Text":"1 plus 1 plus 1 is 3 positive, 1 is positive."},{"Start":"06:47.445 ","End":"06:50.500","Text":"Here we\u0027re going to be above."},{"Start":"06:52.130 ","End":"06:56.240","Text":"The so-called snake, which"},{"Start":"06:56.240 ","End":"06:59.690","Text":"is just a scheme to show where everything is positive or negative,"},{"Start":"06:59.690 ","End":"07:01.535","Text":"is something like this."},{"Start":"07:01.535 ","End":"07:04.085","Text":"Then we look, what are we interested in?"},{"Start":"07:04.085 ","End":"07:07.565","Text":"We are interested in greater than 0,"},{"Start":"07:07.565 ","End":"07:11.070","Text":"means above the axis."},{"Start":"07:12.590 ","End":"07:15.295","Text":"If I shade it,"},{"Start":"07:15.295 ","End":"07:19.000","Text":"the greater than 0 part, it\u0027s here."},{"Start":"07:19.000 ","End":"07:20.890","Text":"But when it\u0027s strictly greater,"},{"Start":"07:20.890 ","End":"07:22.410","Text":"we don\u0027t include the zeros."},{"Start":"07:22.410 ","End":"07:26.030","Text":"It\u0027s everything from the 0 onto the right,"},{"Start":"07:27.920 ","End":"07:32.280","Text":"i.e x bigger than 0."},{"Start":"07:32.280 ","End":"07:35.535","Text":"This is the answer to part B."},{"Start":"07:35.535 ","End":"07:43.150","Text":"Next, after B comes C. We have 1 factor."},{"Start":"07:43.150 ","End":"07:46.240","Text":"Let\u0027s see if the other one factors or not."},{"Start":"07:46.240 ","End":"07:49.690","Text":"Here, I see a quadratic and I have this thing where I"},{"Start":"07:49.690 ","End":"07:53.620","Text":"don\u0027t like quadratics with negative a."},{"Start":"07:53.620 ","End":"07:56.745","Text":"It\u0027s just a personal thing,"},{"Start":"07:56.745 ","End":"07:58.310","Text":"I find it makes less mistakes,"},{"Start":"07:58.310 ","End":"08:00.770","Text":"and I\u0027m also used to seeing a positive a. I\u0027m going to"},{"Start":"08:00.770 ","End":"08:04.084","Text":"take the minus outside the brackets."},{"Start":"08:04.084 ","End":"08:05.450","Text":"You have to do this,"},{"Start":"08:05.450 ","End":"08:10.130","Text":"but I\u0027m going to solve exercises where I like a to be positive."},{"Start":"08:10.130 ","End":"08:12.140","Text":"I just reverse all the signs."},{"Start":"08:12.140 ","End":"08:14.690","Text":"Minus, minus plus becomes plus,"},{"Start":"08:14.690 ","End":"08:18.330","Text":"plus, minus. Same thing."},{"Start":"08:18.430 ","End":"08:23.845","Text":"Now I\u0027d like to factorize this if possible."},{"Start":"08:23.845 ","End":"08:28.455","Text":"If you check the b^2 minus 4ac,"},{"Start":"08:28.455 ","End":"08:30.045","Text":"you\u0027ll find that it\u0027s positive,"},{"Start":"08:30.045 ","End":"08:33.750","Text":"so this should be able to be factorized."},{"Start":"08:33.750 ","End":"08:37.160","Text":"To save time, I\u0027m going to factor it for you."},{"Start":"08:37.160 ","End":"08:44.210","Text":"This one turns out to be 2x minus 1 times x plus 2."},{"Start":"08:44.210 ","End":"08:47.210","Text":"Now if I didn\u0027t show you how I got there,"},{"Start":"08:47.210 ","End":"08:49.625","Text":"at least I can show you that it actually works out,"},{"Start":"08:49.625 ","End":"08:51.860","Text":"2x times x is 2x^2."},{"Start":"08:51.860 ","End":"08:54.235","Text":"We have here minus x plus 4x,"},{"Start":"08:54.235 ","End":"08:55.765","Text":"that gives us plus 3x,"},{"Start":"08:55.765 ","End":"08:57.775","Text":"minus 1 times 2 is minus 2."},{"Start":"08:57.775 ","End":"08:59.970","Text":"At least we see it\u0027s correct."},{"Start":"08:59.970 ","End":"09:03.935","Text":"Just very time-consuming to do all these,"},{"Start":"09:03.935 ","End":"09:06.950","Text":"less than or equal to 0."},{"Start":"09:06.950 ","End":"09:12.200","Text":"At this point, I can either keep the minus and just work"},{"Start":"09:12.200 ","End":"09:17.072","Text":"with an extra minus and have 4 things here to check the sign of,"},{"Start":"09:17.072 ","End":"09:18.800","Text":"or I could move the minus,"},{"Start":"09:18.800 ","End":"09:20.630","Text":"multiply both sides by minus 1."},{"Start":"09:20.630 ","End":"09:23.420","Text":"You know what, I\u0027ll multiply both sides or divide,"},{"Start":"09:23.420 ","End":"09:25.670","Text":"but both sides doesn\u0027t make any difference."},{"Start":"09:25.670 ","End":"09:30.410","Text":"I\u0027ll multiply by minus 1 on both sides."},{"Start":"09:30.410 ","End":"09:33.720","Text":"But then I have to reverse."},{"Start":"09:35.080 ","End":"09:39.455","Text":"It\u0027s greater than or equal to 0,"},{"Start":"09:39.455 ","End":"09:41.735","Text":"0 times minus 1, of course is 0."},{"Start":"09:41.735 ","End":"09:47.315","Text":"All I have to do is now get rid of this minus and I have 2x minus 1,"},{"Start":"09:47.315 ","End":"09:49.125","Text":"x plus 2,"},{"Start":"09:49.125 ","End":"09:52.445","Text":"x plus 1, bigger or equal to 0."},{"Start":"09:52.445 ","End":"09:56.855","Text":"Now I want the roots of this polynomial."},{"Start":"09:56.855 ","End":"09:59.180","Text":"The roots are, of course,"},{"Start":"09:59.180 ","End":"10:01.415","Text":"where the polynomial is 0,"},{"Start":"10:01.415 ","End":"10:04.880","Text":"so any one of these 3 factors has to be 0."},{"Start":"10:04.880 ","End":"10:06.905","Text":"If this one is 0,"},{"Start":"10:06.905 ","End":"10:10.625","Text":"we get that x is 1/2."},{"Start":"10:10.625 ","End":"10:12.620","Text":"If this is 0,"},{"Start":"10:12.620 ","End":"10:15.350","Text":"we get that x is minus 2."},{"Start":"10:15.350 ","End":"10:19.175","Text":"If this is 0, we get x is minus 1."},{"Start":"10:19.175 ","End":"10:22.625","Text":"Now we have the important numbers we need for the sketch."},{"Start":"10:22.625 ","End":"10:24.880","Text":"I\u0027ll bring the sketch over here."},{"Start":"10:24.880 ","End":"10:26.460","Text":"Here\u0027s a number line,"},{"Start":"10:26.460 ","End":"10:27.920","Text":"and I\u0027ll put these 3 numbers on,"},{"Start":"10:27.920 ","End":"10:30.635","Text":"but you have to put them in the correct order."},{"Start":"10:30.635 ","End":"10:35.910","Text":"First or leftmost is minus 2,"},{"Start":"10:35.910 ","End":"10:38.505","Text":"then we have a minus 1,"},{"Start":"10:38.505 ","End":"10:40.540","Text":"and then we have 1/2."},{"Start":"10:40.540 ","End":"10:42.409","Text":"Scale is not important."},{"Start":"10:42.409 ","End":"10:45.090","Text":"The order is very important."},{"Start":"10:45.890 ","End":"10:48.680","Text":"At these points exactly,"},{"Start":"10:48.680 ","End":"10:51.260","Text":"the polynomial is 0,"},{"Start":"10:51.260 ","End":"10:55.865","Text":"so I\u0027ll just actually put solid points on the axis."},{"Start":"10:55.865 ","End":"10:58.625","Text":"Then we have 4 intervals,"},{"Start":"10:58.625 ","End":"11:03.050","Text":"and we have to see what the sign is in each of the intervals."},{"Start":"11:03.050 ","End":"11:06.600","Text":"We take sample points."},{"Start":"11:07.190 ","End":"11:10.380","Text":"Just for a change, I don\u0027t usually do this."},{"Start":"11:10.380 ","End":"11:12.630","Text":"Actually, let\u0027s write the sample points."},{"Start":"11:12.630 ","End":"11:14.930","Text":"From here, I\u0027ll take 0,"},{"Start":"11:14.930 ","End":"11:16.550","Text":"my favorite number,"},{"Start":"11:16.550 ","End":"11:18.545","Text":"substitute at any rate."},{"Start":"11:18.545 ","End":"11:23.450","Text":"Here I\u0027ll take 1 of the representative between these 2,"},{"Start":"11:23.450 ","End":"11:24.830","Text":"I can\u0027t take a whole number."},{"Start":"11:24.830 ","End":"11:28.505","Text":"I have to take, let\u0027s say, minus 1 1/2."},{"Start":"11:28.505 ","End":"11:31.195","Text":"Here I can take minus 3."},{"Start":"11:31.195 ","End":"11:34.995","Text":"Now, for each of them, we\u0027ll substitute,"},{"Start":"11:34.995 ","End":"11:38.270","Text":"and we have to substitute in this one,"},{"Start":"11:38.270 ","End":"11:40.760","Text":"not in the original or anything because we\u0027ve already"},{"Start":"11:40.760 ","End":"11:43.490","Text":"modified it by multiplying both sides."},{"Start":"11:43.490 ","End":"11:47.770","Text":"I\u0027m going to be comparing to bigger or equal to 0."},{"Start":"11:49.830 ","End":"11:52.330","Text":"For minus 3,"},{"Start":"11:52.330 ","End":"11:57.235","Text":"I get, obviously negative here."},{"Start":"11:57.235 ","End":"12:01.240","Text":"Minus 3 plus 2, still negative."},{"Start":"12:01.240 ","End":"12:04.330","Text":"Minus 3 plus 1, negative,"},{"Start":"12:04.330 ","End":"12:07.180","Text":"negative times negative times negative is positive,"},{"Start":"12:07.180 ","End":"12:11.395","Text":"positive is above the axis."},{"Start":"12:11.395 ","End":"12:13.600","Text":"I guess I never really said this, but yeah,"},{"Start":"12:13.600 ","End":"12:18.910","Text":"of course at the upwards is plus and downwards is minus as if there was a y-axis also."},{"Start":"12:18.910 ","End":"12:20.620","Text":"I don\u0027t know why I never mentioned that,"},{"Start":"12:20.620 ","End":"12:22.855","Text":"but yeah, the thing everyone assumes, yeah."},{"Start":"12:22.855 ","End":"12:26.590","Text":"Plus is up right above the axis."},{"Start":"12:26.590 ","End":"12:31.585","Text":"Next, 1 minus 11/2,"},{"Start":"12:31.585 ","End":"12:33.595","Text":"twice that is minus 3 minus 1,"},{"Start":"12:33.595 ","End":"12:38.720","Text":"negative, minus 11/2 plus 2, positive."},{"Start":"12:39.510 ","End":"12:49.100","Text":"Minus 11/2 plus 1 is negative."},{"Start":"12:49.230 ","End":"12:52.540","Text":"Take 2 on the last bit."},{"Start":"12:52.540 ","End":"12:57.850","Text":"If we substitute minus 3 in here, we get, well,"},{"Start":"12:57.850 ","End":"12:59.755","Text":"this one\u0027s negative minus 3 plus 2,"},{"Start":"12:59.755 ","End":"13:01.180","Text":"negative minus 3 plus 1,"},{"Start":"13:01.180 ","End":"13:05.605","Text":"negative, negative, negative, negative is negative."},{"Start":"13:05.605 ","End":"13:08.570","Text":"Here I\u0027ve got some negative value."},{"Start":"13:08.580 ","End":"13:13.450","Text":"Next, minus 11/2, I double it,"},{"Start":"13:13.450 ","End":"13:15.190","Text":"it\u0027s still negative and take away 1,"},{"Start":"13:15.190 ","End":"13:19.435","Text":"is still negative, minus 11/2 plus 2, positive."},{"Start":"13:19.435 ","End":"13:21.730","Text":"Minus 11/2 plus 1, negative."},{"Start":"13:21.730 ","End":"13:30.145","Text":"Minus plus minus gives us a plus above the 1 minus 11/2."},{"Start":"13:30.145 ","End":"13:33.040","Text":"Next, 0, that\u0027s always the easiest."},{"Start":"13:33.040 ","End":"13:35.920","Text":"Minus 1 times 2 times 1,"},{"Start":"13:35.920 ","End":"13:40.550","Text":"minus plus, plus, minus."},{"Start":"13:41.670 ","End":"13:45.020","Text":"When x is 1,"},{"Start":"13:46.980 ","End":"13:51.115","Text":"we have, yeah,"},{"Start":"13:51.115 ","End":"13:53.890","Text":"twice 1 minus 1 is positive,"},{"Start":"13:53.890 ","End":"13:55.689","Text":"1 plus 2, positive,"},{"Start":"13:55.689 ","End":"13:57.325","Text":"1 plus 1, positive,"},{"Start":"13:57.325 ","End":"14:01.700","Text":"3 positives, so it\u0027s a positive."},{"Start":"14:03.840 ","End":"14:09.040","Text":"Now let\u0027s sketch this so-called snake."},{"Start":"14:09.040 ","End":"14:11.440","Text":"Something like this."},{"Start":"14:11.440 ","End":"14:15.670","Text":"This, this, this, this,"},{"Start":"14:15.670 ","End":"14:17.140","Text":"it goes on forever."},{"Start":"14:17.140 ","End":"14:19.150","Text":"This goes on forever."},{"Start":"14:19.150 ","End":"14:24.384","Text":"What we\u0027re interested in is bigger or equal to 0,"},{"Start":"14:24.384 ","End":"14:26.605","Text":"bigger or equal to 0."},{"Start":"14:26.605 ","End":"14:28.705","Text":"It\u0027s here."},{"Start":"14:28.705 ","End":"14:36.520","Text":"But to emphasize, we\u0027re including these 2 points and [inaudible] emphasize that,"},{"Start":"14:36.520 ","End":"14:38.680","Text":"make them extra big, I guess."},{"Start":"14:38.680 ","End":"14:44.500","Text":"Including these. Again,"},{"Start":"14:44.500 ","End":"14:47.590","Text":"bigger or equal to 0 is above the axis,"},{"Start":"14:47.590 ","End":"14:50.600","Text":"but including this one."},{"Start":"14:51.210 ","End":"14:59.650","Text":"This part gives us minus 2 less than or equal to x,"},{"Start":"14:59.650 ","End":"15:02.200","Text":"less than or equal to minus 1,"},{"Start":"15:02.200 ","End":"15:03.580","Text":"and this part,"},{"Start":"15:03.580 ","End":"15:12.160","Text":"we write an OR is x bigger or equal to 1/2."},{"Start":"15:12.160 ","End":"15:15.985","Text":"Now you could write 1/2 less than or equal to x,"},{"Start":"15:15.985 ","End":"15:19.075","Text":"just optionally, put whatever you prefer,"},{"Start":"15:19.075 ","End":"15:21.355","Text":"I\u0027ll leave it this way."},{"Start":"15:21.355 ","End":"15:25.510","Text":"This is the answer to part c,"},{"Start":"15:25.510 ","End":"15:36.260","Text":"and so we now can move on to part d. This is a lot of work in this exercise."},{"Start":"15:38.130 ","End":"15:45.895","Text":"Let\u0027s get to it. What we want to do with this polynomial is find it\u0027s roots,"},{"Start":"15:45.895 ","End":"15:51.310","Text":"and clearly one of the roots is,"},{"Start":"15:51.310 ","End":"15:56.140","Text":"where x is equal to minus 3 because that\u0027s when this is 0."},{"Start":"15:56.140 ","End":"15:57.940","Text":"Now if you look at the other one,"},{"Start":"15:57.940 ","End":"16:00.595","Text":"this is never 0."},{"Start":"16:00.595 ","End":"16:07.510","Text":"You could try the b^2 minus 4ac method and you\u0027ll find that that\u0027s negative,"},{"Start":"16:07.510 ","End":"16:08.590","Text":"but it\u0027s easier than that,"},{"Start":"16:08.590 ","End":"16:10.510","Text":"x^2 is at least 0."},{"Start":"16:10.510 ","End":"16:13.195","Text":"I mean, something squared can never be negative."},{"Start":"16:13.195 ","End":"16:16.585","Text":"If I have something bigger or equal to 0 and add 6 to it,"},{"Start":"16:16.585 ","End":"16:21.430","Text":"it\u0027s at least 6 when it\u0027s certainly positive, it\u0027s never 0."},{"Start":"16:21.430 ","End":"16:29.170","Text":"This is the only route this bit has no roots."},{"Start":"16:29.170 ","End":"16:32.600","Text":"On the number line,"},{"Start":"16:32.790 ","End":"16:36.160","Text":"we only have 1 number to back,"},{"Start":"16:36.160 ","End":"16:38.725","Text":"and that is the minus 3."},{"Start":"16:38.725 ","End":"16:43.975","Text":"We want to see which of these 2 parts is positive or negative."},{"Start":"16:43.975 ","End":"16:47.800","Text":"I\u0027ll choose a sample point here of 0."},{"Start":"16:47.800 ","End":"16:52.620","Text":"If I put in 0, then this is positive."},{"Start":"16:52.620 ","End":"16:54.240","Text":"We know is always positive."},{"Start":"16:54.240 ","End":"16:56.460","Text":"Really we just have to look at x plus 3."},{"Start":"16:56.460 ","End":"17:00.445","Text":"Anyway, so this is positive."},{"Start":"17:00.445 ","End":"17:03.270","Text":"Somewhere up here."},{"Start":"17:03.480 ","End":"17:06.850","Text":"If I put x equals,"},{"Start":"17:06.850 ","End":"17:08.740","Text":"let\u0027s say minus 4,"},{"Start":"17:08.740 ","End":"17:12.310","Text":"minus 4 plus 3 is negative and this we know is positive,"},{"Start":"17:12.310 ","End":"17:18.385","Text":"so here we are negative at this point itself, it\u0027s 0."},{"Start":"17:18.385 ","End":"17:25.420","Text":"Draw a snake, some kind snake method."},{"Start":"17:25.420 ","End":"17:32.080","Text":"The important thing is to find out where it is above the axis."},{"Start":"17:32.080 ","End":"17:34.825","Text":"I mean, greater than 0 means above the axis,"},{"Start":"17:34.825 ","End":"17:37.450","Text":"which is this part here,"},{"Start":"17:37.450 ","End":"17:38.980","Text":"and going on forever."},{"Start":"17:38.980 ","End":"17:43.430","Text":"Sometimes they write a little arrow there to say it doesn\u0027t stop."},{"Start":"17:43.980 ","End":"17:51.535","Text":"But we don\u0027t include the minus 3 here because we want strictly above the axis,"},{"Start":"17:51.535 ","End":"17:59.090","Text":"and so the answer is x bigger than minus 3."},{"Start":"17:59.730 ","End":"18:04.270","Text":"That does it for part d. Finally,"},{"Start":"18:04.270 ","End":"18:06.880","Text":"we get to part e,"},{"Start":"18:06.880 ","End":"18:13.075","Text":"and part e we have a less than or equal to change."},{"Start":"18:13.075 ","End":"18:15.610","Text":"This one is the linear factor."},{"Start":"18:15.610 ","End":"18:18.775","Text":"We would like to factorize this to get roots."},{"Start":"18:18.775 ","End":"18:21.550","Text":"Turns out that this doesn\u0027t factor."},{"Start":"18:21.550 ","End":"18:24.580","Text":"You need to take my word for it or you can try it yourselves."},{"Start":"18:24.580 ","End":"18:29.275","Text":"You know what, let\u0027s just check the b^2 minus 4ac for that."},{"Start":"18:29.275 ","End":"18:31.735","Text":"Just for the first term."},{"Start":"18:31.735 ","End":"18:41.710","Text":"Let\u0027s see, is 8^2 minus 4 times 1 times 20,"},{"Start":"18:41.710 ","End":"18:46.990","Text":"which is 64 minus 80,"},{"Start":"18:46.990 ","End":"18:50.590","Text":"I would say that is negative."},{"Start":"18:50.590 ","End":"18:53.530","Text":"When the b^2 minus 4ac is negative,"},{"Start":"18:53.530 ","End":"18:54.880","Text":"you can\u0027t solve the quadratic,"},{"Start":"18:54.880 ","End":"18:56.930","Text":"so there\u0027s no roots."},{"Start":"18:57.000 ","End":"19:04.210","Text":"In fact, you can even say straight away whether it\u0027s always positive or always negative."},{"Start":"19:04.210 ","End":"19:06.610","Text":"If we put in x equals 0,"},{"Start":"19:06.610 ","End":"19:10.330","Text":"we get 20, which is positive."},{"Start":"19:10.330 ","End":"19:12.835","Text":"This is always positive."},{"Start":"19:12.835 ","End":"19:19.240","Text":"Really the only thing that matters as far as the sign goes is this,"},{"Start":"19:19.240 ","End":"19:25.705","Text":"and it is 0 when [inaudible] 5 over to the other side divide by 3."},{"Start":"19:25.705 ","End":"19:31.420","Text":"When x is 5/3."},{"Start":"19:31.420 ","End":"19:33.385","Text":"Actually that\u0027s the only route."},{"Start":"19:33.385 ","End":"19:38.080","Text":"The only route we have is 5/3."},{"Start":"19:38.080 ","End":"19:40.880","Text":"Let\u0027s do a sketch."},{"Start":"19:41.070 ","End":"19:47.630","Text":"Number line 1 single point 5/3."},{"Start":"19:47.630 ","End":"19:55.705","Text":"At 5/3 itself because it\u0027s a root the polynomial is 0."},{"Start":"19:55.705 ","End":"20:00.460","Text":"Beyond 5/3, for example, well,"},{"Start":"20:00.460 ","End":"20:08.440","Text":"let me just say that this improper fraction is 1 and 2/3 as a mixed number."},{"Start":"20:08.440 ","End":"20:14.575","Text":"For example, I could take 2 here for this interval,"},{"Start":"20:14.575 ","End":"20:16.375","Text":"if I put in 2,"},{"Start":"20:16.375 ","End":"20:21.115","Text":"I already said this is always positive and 3 times 2 minus 5,"},{"Start":"20:21.115 ","End":"20:22.750","Text":"6 minus 5 is 1."},{"Start":"20:22.750 ","End":"20:23.920","Text":"Is positive."},{"Start":"20:23.920 ","End":"20:26.530","Text":"Here we are positive,"},{"Start":"20:26.530 ","End":"20:29.575","Text":"and on the other side,"},{"Start":"20:29.575 ","End":"20:33.145","Text":"I will take 1 that will be less than,"},{"Start":"20:33.145 ","End":"20:34.585","Text":"if I put in 1,"},{"Start":"20:34.585 ","End":"20:39.655","Text":"3 times 1 minus 5 is minus 2, which is negative."},{"Start":"20:39.655 ","End":"20:45.710","Text":"This is always positive [inaudible] positive or negative is negative, here."},{"Start":"20:48.690 ","End":"20:51.790","Text":"We don\u0027t know exactly how it goes."},{"Start":"20:51.790 ","End":"20:56.140","Text":"It\u0027s just a schematic sketch to show where positive and negative,"},{"Start":"20:56.140 ","End":"20:58.195","Text":"we have to look what do we want?"},{"Start":"20:58.195 ","End":"21:01.135","Text":"We want less than or equal to 0."},{"Start":"21:01.135 ","End":"21:05.380","Text":"Means below the axis or on the axis."},{"Start":"21:05.380 ","End":"21:10.930","Text":"It\u0027s this part here, including this."},{"Start":"21:10.930 ","End":"21:13.720","Text":"If we write it in terms of x,"},{"Start":"21:13.720 ","End":"21:20.780","Text":"this just translates as x less than or equal to 5/3."},{"Start":"21:20.780 ","End":"21:23.170","Text":"You know what, I\u0027ll put it as a mixed number,"},{"Start":"21:23.170 ","End":"21:26.740","Text":"1 and 2/3. That\u0027s less important."},{"Start":"21:26.740 ","End":"21:28.420","Text":"Anyway, this is the answer."},{"Start":"21:28.420 ","End":"21:33.890","Text":"I\u0027ve highlighted it, and that\u0027s the last one for this exercise finally."}],"Thumbnail":null,"ID":5067},{"Watched":false,"Name":"Exercise 6","Duration":"27m 19s","ChapterTopicVideoID":5065,"CourseChapterTopicPlaylistID":3876,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.355","Text":"Here we have a set of five inequalities to solve."},{"Start":"00:05.355 ","End":"00:10.320","Text":"They\u0027re all with polynomials of degree greater than 2,"},{"Start":"00:10.320 ","End":"00:14.260","Text":"degree 3, 4, and so on."},{"Start":"00:14.270 ","End":"00:19.095","Text":"We\u0027ll solve them essentially using the snake method."},{"Start":"00:19.095 ","End":"00:25.530","Text":"Let\u0027s get to the first one and one of our first tasks is to find"},{"Start":"00:25.530 ","End":"00:33.010","Text":"the roots of the polynomial and to do that we want to factorize it as much as possible."},{"Start":"00:33.010 ","End":"00:37.660","Text":"Here the first thing I can do is take x outside the brackets,"},{"Start":"00:37.660 ","End":"00:42.980","Text":"so I\u0027ve got x times x^2 minus 25,"},{"Start":"00:42.980 ","End":"00:45.710","Text":"bigger or equal to 0."},{"Start":"00:45.710 ","End":"00:52.595","Text":"This you should immediately recognize as a difference of squares, x^2 minus 5^2."},{"Start":"00:52.595 ","End":"00:56.265","Text":"I\u0027m not even going to write the formula,"},{"Start":"00:56.265 ","End":"00:58.685","Text":"this is something you should memorize already."},{"Start":"00:58.685 ","End":"01:04.320","Text":"We get x^2 minus 25,"},{"Start":"01:04.320 ","End":"01:07.185","Text":"as x minus 5,"},{"Start":"01:07.185 ","End":"01:11.250","Text":"x plus 5, or x plus 5,"},{"Start":"01:11.250 ","End":"01:15.630","Text":"x minus 5, that\u0027s not bigger or equal to 0."},{"Start":"01:15.630 ","End":"01:19.430","Text":"Difference of squares, that\u0027s the name of the formula we use."},{"Start":"01:19.430 ","End":"01:25.645","Text":"This is already completely factorized into linear factors so we can easily see the roots."},{"Start":"01:25.645 ","End":"01:29.570","Text":"The roots are whatever makes this thing 0,"},{"Start":"01:29.570 ","End":"01:32.780","Text":"which means making any one of them 0,"},{"Start":"01:32.780 ","End":"01:36.245","Text":"so we\u0027ve got x equals,"},{"Start":"01:36.245 ","End":"01:39.305","Text":"could be 0, that would make that 0."},{"Start":"01:39.305 ","End":"01:41.010","Text":"Could be 5,"},{"Start":"01:41.010 ","End":"01:42.885","Text":"that would make this one 0,"},{"Start":"01:42.885 ","End":"01:44.605","Text":"could be minus 5."},{"Start":"01:44.605 ","End":"01:46.550","Text":"These are the only possibilities."},{"Start":"01:46.550 ","End":"01:48.920","Text":"To get this thing to be 0,"},{"Start":"01:48.920 ","End":"01:51.110","Text":"has to be one of these."},{"Start":"01:51.110 ","End":"01:55.735","Text":"That\u0027s enough for us to get started on the sketch."},{"Start":"01:55.735 ","End":"01:58.215","Text":"Here\u0027s a number line,"},{"Start":"01:58.215 ","End":"02:00.545","Text":"and on it we have to place these numbers,"},{"Start":"02:00.545 ","End":"02:04.460","Text":"but in increasing order from left to right of course."},{"Start":"02:04.460 ","End":"02:12.080","Text":"We have first of all minus 5 and then 0 and then plus 5"},{"Start":"02:12.080 ","End":"02:15.620","Text":"and we want to know in each of"},{"Start":"02:15.620 ","End":"02:21.215","Text":"these four intervals whether the function is positive or negative."},{"Start":"02:21.215 ","End":"02:26.735","Text":"I meant to say whether the polynomial is positive or negative."},{"Start":"02:26.735 ","End":"02:29.330","Text":"At these points themselves,"},{"Start":"02:29.330 ","End":"02:37.010","Text":"we know it\u0027s 0 and the usual technique will take a sample point in each and substitute."},{"Start":"02:39.080 ","End":"02:42.240","Text":"What we get is,"},{"Start":"02:42.240 ","End":"02:46.795","Text":"let\u0027s see, count your 0 as the sample point."},{"Start":"02:46.795 ","End":"02:50.660","Text":"Let\u0027s choose, let\u0027s say for this interval,"},{"Start":"02:50.660 ","End":"02:53.185","Text":"I\u0027ll choose x equals 1."},{"Start":"02:53.185 ","End":"02:56.040","Text":"To put x equals 1,"},{"Start":"02:56.040 ","End":"02:59.175","Text":"I think I\u0027ll mark them, I\u0027ll choose 1 for here,"},{"Start":"02:59.175 ","End":"03:01.230","Text":"I\u0027ll choose minus 1 here,"},{"Start":"03:01.230 ","End":"03:03.705","Text":"I\u0027ll choose 6 here,"},{"Start":"03:03.705 ","End":"03:08.580","Text":"and I\u0027ll choose minus 6 here and then we\u0027ll see what we get."},{"Start":"03:08.580 ","End":"03:12.420","Text":"As I was saying 1 gives us 1 is positive,"},{"Start":"03:12.420 ","End":"03:14.460","Text":"1 minus 5 is negative,"},{"Start":"03:14.460 ","End":"03:16.940","Text":"1 plus 5 is positive,"},{"Start":"03:16.940 ","End":"03:19.255","Text":"two positives and a negative,"},{"Start":"03:19.255 ","End":"03:21.030","Text":"so here it\u0027s negative,"},{"Start":"03:21.030 ","End":"03:23.330","Text":"so I\u0027ll put a point somewhere here."},{"Start":"03:23.330 ","End":"03:27.950","Text":"Let\u0027s say 6 is positive,"},{"Start":"03:27.950 ","End":"03:29.690","Text":"6 minus 5 positive,"},{"Start":"03:29.690 ","End":"03:32.075","Text":"6 plus 5 positive,"},{"Start":"03:32.075 ","End":"03:36.915","Text":"positive here, minus 1."},{"Start":"03:36.915 ","End":"03:39.630","Text":"This is negative, minus 1,"},{"Start":"03:39.630 ","End":"03:40.800","Text":"minus 5 is negative,"},{"Start":"03:40.800 ","End":"03:43.290","Text":"minus 1 plus 5 is positive."},{"Start":"03:43.290 ","End":"03:47.235","Text":"So here it\u0027s going to be positive,"},{"Start":"03:47.235 ","End":"03:52.365","Text":"and then minus 6, it\u0027s negative,"},{"Start":"03:52.365 ","End":"03:58.160","Text":"take away 5 is still negative and even plus 5 from minus 6 is still negative."},{"Start":"03:58.160 ","End":"04:03.150","Text":"Three negatives, hence a negative."},{"Start":"04:03.190 ","End":"04:08.330","Text":"Now, the snake, we don\u0027t have to draw the snake,"},{"Start":"04:08.330 ","End":"04:12.810","Text":"but it\u0027s the method I\u0027m going to use now."},{"Start":"04:18.550 ","End":"04:21.080","Text":"As I mentioned several times,"},{"Start":"04:21.080 ","End":"04:24.680","Text":"this is doesn\u0027t mean that the function looks exactly like this."},{"Start":"04:24.680 ","End":"04:30.335","Text":"This just gives us an idea of where it\u0027s positive and where it\u0027s negative."},{"Start":"04:30.335 ","End":"04:33.020","Text":"Now, what are we interested in?"},{"Start":"04:33.020 ","End":"04:38.690","Text":"We\u0027re interested in places where it\u0027s above the axis or on the axis,"},{"Start":"04:38.690 ","End":"04:43.880","Text":"the bigger than 0 means above the axis equals 0 is exactly on the axis."},{"Start":"04:43.880 ","End":"04:48.210","Text":"If we shade the bits that we\u0027re interested in,"},{"Start":"04:48.920 ","End":"04:53.885","Text":"but you know what, instead of shading just for a change, I\u0027ll use highlighting."},{"Start":"04:53.885 ","End":"04:58.235","Text":"I just got a bit bored with the shading so let\u0027s do the highlighting,"},{"Start":"04:58.235 ","End":"05:04.615","Text":"so bigger than 0 is here,"},{"Start":"05:04.615 ","End":"05:08.960","Text":"bigger than 0 is here and so on"},{"Start":"05:08.960 ","End":"05:14.180","Text":"indefinitely and the thing is that because we have the equals also,"},{"Start":"05:14.180 ","End":"05:17.930","Text":"I\u0027ll just try and emphasize that we do take this point,"},{"Start":"05:17.930 ","End":"05:20.165","Text":"and we do take this point here,"},{"Start":"05:20.165 ","End":"05:22.730","Text":"and we do take this point."},{"Start":"05:22.730 ","End":"05:25.475","Text":"This bit, this I don\u0027t know,"},{"Start":"05:25.475 ","End":"05:26.921","Text":"hump or whatever you want to call it."},{"Start":"05:26.921 ","End":"05:29.440","Text":"No, a snake doesn\u0027t have a hump, never mind."},{"Start":"05:29.440 ","End":"05:35.330","Text":"Whatever it is, it\u0027s minus 5 less than or equal to,"},{"Start":"05:35.330 ","End":"05:45.135","Text":"importantly equal to x less than or equal to 0 and this bit is 5 and to the right,"},{"Start":"05:45.135 ","End":"05:48.720","Text":"is x bigger or equal to 5,"},{"Start":"05:48.720 ","End":"05:53.810","Text":"or you could write 5 less than or equal to x. I\u0027ll do it this way again,"},{"Start":"05:53.810 ","End":"05:58.799","Text":"including this point and that\u0027s our answer."},{"Start":"05:59.530 ","End":"06:04.320","Text":"There we are and on to part b."},{"Start":"06:05.100 ","End":"06:11.350","Text":"In part b, the first thing we do even before trying to factorize or anything,"},{"Start":"06:11.350 ","End":"06:19.190","Text":"is to bring everything to one side because the method relies on comparing with 0,"},{"Start":"06:19.190 ","End":"06:21.330","Text":"finding roots and all that,"},{"Start":"06:21.330 ","End":"06:22.570","Text":"and the snake method."},{"Start":"06:22.570 ","End":"06:28.300","Text":"So I\u0027ll write this as x^4th minus x^2 bigger than 0."},{"Start":"06:28.300 ","End":"06:30.190","Text":"I want 0 on one side,"},{"Start":"06:30.190 ","End":"06:32.415","Text":"the right side, usually."},{"Start":"06:32.415 ","End":"06:35.540","Text":"Now we can start factoring."},{"Start":"06:35.540 ","End":"06:38.920","Text":"Now we start factoring in order to find roots."},{"Start":"06:38.920 ","End":"06:41.920","Text":"x^2 comes out of the brackets,"},{"Start":"06:41.920 ","End":"06:47.690","Text":"so we have x^2 and what we\u0027re left with is x^2 minus 1."},{"Start":"06:47.690 ","End":"06:51.964","Text":"X plus minus 1 using the difference of squared formulas,"},{"Start":"06:51.964 ","End":"06:55.280","Text":"x plus minus 1^2 is x minus 1, x plus one."},{"Start":"06:55.280 ","End":"06:56.960","Text":"We have x^2,"},{"Start":"06:56.960 ","End":"06:58.815","Text":"x minus 1,"},{"Start":"06:58.815 ","End":"07:02.700","Text":"x plus 1 bigger than 0,"},{"Start":"07:02.700 ","End":"07:12.515","Text":"and this also gives us the roots for x that the first one can be 0,"},{"Start":"07:12.515 ","End":"07:16.049","Text":"only when x is 0."},{"Start":"07:16.670 ","End":"07:21.175","Text":"This one is 0 when x is 1,"},{"Start":"07:21.175 ","End":"07:24.470","Text":"and this one is 0 when x is minus 1."},{"Start":"07:24.470 ","End":"07:30.220","Text":"These are our three roots and now we need a number line."},{"Start":"07:30.220 ","End":"07:32.970","Text":"Here we are out of thin air,"},{"Start":"07:32.970 ","End":"07:34.800","Text":"a number line appeared."},{"Start":"07:34.800 ","End":"07:38.460","Text":"Now we want to put the roots on,"},{"Start":"07:38.460 ","End":"07:39.750","Text":"so 0, 1,"},{"Start":"07:39.750 ","End":"07:41.880","Text":"and minus 1 in the correct order,"},{"Start":"07:41.880 ","End":"07:47.585","Text":"minus 1, 0, 1."},{"Start":"07:47.585 ","End":"07:49.720","Text":"Now we have 4 intervals."},{"Start":"07:49.720 ","End":"07:55.450","Text":"We want to know where it\u0027s positive and negative and this is an interesting example"},{"Start":"07:55.450 ","End":"08:01.645","Text":"in the sense that it will show you a case where the signs don\u0027t alternate."},{"Start":"08:01.645 ","End":"08:06.130","Text":"Normally you say, aha if here\u0027s positive and negative then positive and negative."},{"Start":"08:06.130 ","End":"08:08.560","Text":"This is a counterexample to show you that you"},{"Start":"08:08.560 ","End":"08:12.440","Text":"can\u0027t assume that even though it usually is the case."},{"Start":"08:14.250 ","End":"08:16.660","Text":"Let\u0027s see what I mean."},{"Start":"08:16.660 ","End":"08:20.169","Text":"I take sample points."},{"Start":"08:20.169 ","End":"08:22.480","Text":"For this interval on the left,"},{"Start":"08:22.480 ","End":"08:27.925","Text":"I\u0027ll choose minus 2 and if I put x is minus 2,"},{"Start":"08:27.925 ","End":"08:31.801","Text":"in here, I get minus 2 squared,"},{"Start":"08:31.801 ","End":"08:33.400","Text":"well, it\u0027s positive."},{"Start":"08:33.400 ","End":"08:36.505","Text":"Minus 2 minus 1 is negative."},{"Start":"08:36.505 ","End":"08:39.235","Text":"Minus 2 plus 1 also negative."},{"Start":"08:39.235 ","End":"08:41.245","Text":"Positive negative negative,"},{"Start":"08:41.245 ","End":"08:45.595","Text":"so positive at minus 2 somewhere here."},{"Start":"08:45.595 ","End":"08:48.790","Text":"Between here and here don\u0027t have a whole number."},{"Start":"08:48.790 ","End":"08:50.961","Text":"I\u0027ll take x is 1/2,"},{"Start":"08:50.961 ","End":"08:53.140","Text":"I mean minus 1/2, sorry."},{"Start":"08:53.140 ","End":"08:56.965","Text":"Minus 1/2 if I put that in,"},{"Start":"08:56.965 ","End":"09:03.250","Text":"this one is positive because it\u0027s squared minus 1/2 minus 1, clearly negative."},{"Start":"09:03.250 ","End":"09:11.200","Text":"Minus 1/2 plus 1 is plus 1/2 so we have plus minus plus, that means minus."},{"Start":"09:11.200 ","End":"09:14.390","Text":"At 1/2 we have something negative."},{"Start":"09:14.850 ","End":"09:17.875","Text":"I can mark these also here,"},{"Start":"09:17.875 ","End":"09:21.400","Text":"it\u0027s 0 at the roots themselves."},{"Start":"09:21.400 ","End":"09:24.730","Text":"Now, for the next interval, this one,"},{"Start":"09:24.730 ","End":"09:28.705","Text":"I\u0027ll choose a sample, x equals 1/2."},{"Start":"09:28.705 ","End":"09:33.051","Text":"If I put in x equals 1/2 in here,"},{"Start":"09:33.051 ","End":"09:35.365","Text":"1/2 squared, positive,"},{"Start":"09:35.365 ","End":"09:38.890","Text":"1/2 minus 1, negative,"},{"Start":"09:38.890 ","End":"09:44.200","Text":"and 1/2 plus 1, also positive."},{"Start":"09:44.200 ","End":"09:49.490","Text":"Positive negative positive is also negative."},{"Start":"09:51.000 ","End":"09:56.875","Text":"It doesn\u0027t alternate signs."},{"Start":"09:56.875 ","End":"10:01.930","Text":"If you put in x equals 2 or just save you the trouble it comes out positive."},{"Start":"10:01.930 ","End":"10:04.130","Text":"Just to emphasize that,"},{"Start":"10:04.130 ","End":"10:06.165","Text":"because it doesn\u0027t happen often,"},{"Start":"10:06.165 ","End":"10:09.150","Text":"we have plus, minus,"},{"Start":"10:09.150 ","End":"10:14.020","Text":"minus, plus no alternation."},{"Start":"10:15.660 ","End":"10:18.940","Text":"Just wanted to point that out."},{"Start":"10:18.940 ","End":"10:22.030","Text":"Because there\u0027s a temptation to assume that all I have to do"},{"Start":"10:22.030 ","End":"10:24.504","Text":"is check in one interval and then alternate."},{"Start":"10:24.504 ","End":"10:27.205","Text":"No, it doesn\u0027t always work."},{"Start":"10:27.205 ","End":"10:31.240","Text":"Let\u0027s sketch in the snake though it\u0027s not"},{"Start":"10:31.240 ","End":"10:35.200","Text":"really necessary because we know where we are positive and negative."},{"Start":"10:35.200 ","End":"10:39.265","Text":"In fact, you know what, I won\u0027t bother drawing the snake this time."},{"Start":"10:39.265 ","End":"10:41.905","Text":"I can just say what am I interested in?"},{"Start":"10:41.905 ","End":"10:44.830","Text":"I\u0027m interested in the greater than 0,"},{"Start":"10:44.830 ","End":"10:47.080","Text":"bigger than 0, above the axis,"},{"Start":"10:47.080 ","End":"10:49.525","Text":"plus different ways of saying it."},{"Start":"10:49.525 ","End":"10:53.395","Text":"This is what I want and this is what I want."},{"Start":"10:53.395 ","End":"10:57.040","Text":"The pluses strictly plus."},{"Start":"10:57.040 ","End":"11:05.690","Text":"Here it corresponds to this interval x less than minus 1,"},{"Start":"11:06.180 ","End":"11:09.760","Text":"strictly, and for here,"},{"Start":"11:09.760 ","End":"11:18.550","Text":"x bigger than 1 and an or in-between and I\u0027ll highlight it."},{"Start":"11:18.550 ","End":"11:23.245","Text":"I\u0027m certain I think maybe someone feels cheated that he wants his snake."},{"Start":"11:23.245 ","End":"11:26.515","Text":"We\u0027ll draw 1 in."},{"Start":"11:26.515 ","End":"11:29.875","Text":"But look how it goes."},{"Start":"11:29.875 ","End":"11:32.230","Text":"It doesn\u0027t cross the axis here,"},{"Start":"11:32.230 ","End":"11:38.575","Text":"it just grazes and the positive bit was this bit, and this bit."},{"Start":"11:38.575 ","End":"11:42.595","Text":"I hope everyone is satisfied now and we can move on to"},{"Start":"11:42.595 ","End":"11:52.210","Text":"Part c. Here everything\u0027s on the left and zeros on the right."},{"Start":"11:52.210 ","End":"11:56.425","Text":"That\u0027s fine and so what we want to do is factorize to get the roots."},{"Start":"11:56.425 ","End":"12:01.000","Text":"The most obvious thing is the x that can come out the brackets and we\u0027re left with x^2"},{"Start":"12:01.000 ","End":"12:07.735","Text":"minus 6x plus 9 less than or equal to 0."},{"Start":"12:07.735 ","End":"12:10.315","Text":"At this point I want to remind you of"},{"Start":"12:10.315 ","End":"12:15.550","Text":"the special binomial expansion"},{"Start":"12:15.550 ","End":"12:21.010","Text":"that when we have a plus b all squared,"},{"Start":"12:21.010 ","End":"12:26.064","Text":"it\u0027s a^2 plus 2ab plus b^2."},{"Start":"12:26.064 ","End":"12:27.400","Text":"Actually, there\u0027s 2 formulas,"},{"Start":"12:27.400 ","End":"12:31.690","Text":"if there\u0027s a minus here then it\u0027s a minus here and actually that\u0027s the one we\u0027re going to"},{"Start":"12:31.690 ","End":"12:36.490","Text":"use with the minus and also we\u0027re going to use the equality from right-to-left."},{"Start":"12:36.490 ","End":"12:40.150","Text":"If we let a be x and b is 3,"},{"Start":"12:40.150 ","End":"12:41.860","Text":"this is what we get."},{"Start":"12:41.860 ","End":"12:48.490","Text":"This is certainly a^2 and this is certainly b^2."},{"Start":"12:48.490 ","End":"12:50.680","Text":"The middle term has to be verified."},{"Start":"12:50.680 ","End":"12:53.560","Text":"In other words, if I want to say that"},{"Start":"12:53.560 ","End":"12:58.540","Text":"this bit here is x minus 3 squared from the formula,"},{"Start":"12:58.540 ","End":"13:04.645","Text":"the x^2 is obvious and the 3^2 is 9 is obviously just to check on the middle term."},{"Start":"13:04.645 ","End":"13:08.740","Text":"2ab is 2x times 3,"},{"Start":"13:08.740 ","End":"13:10.435","Text":"which is 6x so we\u0027re okay."},{"Start":"13:10.435 ","End":"13:12.535","Text":"Yeah, I verified with that."},{"Start":"13:12.535 ","End":"13:18.220","Text":"Let\u0027s continue then we still have the x here and we have less than or equal to 0."},{"Start":"13:18.220 ","End":"13:20.170","Text":"Now this thing\u0027s been factorized,"},{"Start":"13:20.170 ","End":"13:23.335","Text":"so we want to know what its roots are,"},{"Start":"13:23.335 ","End":"13:26.650","Text":"what are possibilities for x."},{"Start":"13:26.650 ","End":"13:30.760","Text":"For this one, it\u0027s just 0 and for this one,"},{"Start":"13:30.760 ","End":"13:33.235","Text":"clearly x has to be 3."},{"Start":"13:33.235 ","End":"13:35.530","Text":"In fact, the squared doesn\u0027t make any difference,"},{"Start":"13:35.530 ","End":"13:41.090","Text":"x minus 3 squared means that x minus 3 is 0, x is 3."},{"Start":"13:41.220 ","End":"13:45.710","Text":"I\u0027ll make a little sketch now."},{"Start":"13:45.990 ","End":"13:56.095","Text":"Here\u0027s a number line and here are the points 0 and 3."},{"Start":"13:56.095 ","End":"14:00.370","Text":"This time I\u0027m definitely going to save time and not draw the whole snake,"},{"Start":"14:00.370 ","End":"14:02.575","Text":"just the pluses and minuses."},{"Start":"14:02.575 ","End":"14:09.339","Text":"At 0, the polynomial is 0 and the same at 3 that\u0027s what being a root means."},{"Start":"14:09.339 ","End":"14:11.589","Text":"Now let\u0027s take sample points."},{"Start":"14:11.589 ","End":"14:15.955","Text":"I\u0027ll take a sample point of minus 1."},{"Start":"14:15.955 ","End":"14:17.770","Text":"When x is minus 1,"},{"Start":"14:17.770 ","End":"14:23.035","Text":"I have negative and this thing squared is positive."},{"Start":"14:23.035 ","End":"14:26.185","Text":"Here we are negative."},{"Start":"14:26.185 ","End":"14:27.970","Text":"Between 0 and 3,"},{"Start":"14:27.970 ","End":"14:29.965","Text":"let\u0027s say x equals 1,"},{"Start":"14:29.965 ","End":"14:35.949","Text":"1 is positive and 1 minus 3 squared is going to be positive because it\u0027s squared."},{"Start":"14:35.949 ","End":"14:38.695","Text":"Here I have a plus."},{"Start":"14:38.695 ","End":"14:43.420","Text":"For this interval I\u0027ll take x equals 4,"},{"Start":"14:43.420 ","End":"14:48.535","Text":"4 is positive, 4 minus 3 squared also positive."},{"Start":"14:48.535 ","End":"14:53.560","Text":"Once again, we have an example where we don\u0027t alternate."},{"Start":"14:53.560 ","End":"14:55.495","Text":"We have 2 positives in the row."},{"Start":"14:55.495 ","End":"14:58.045","Text":"Now, what do we want?"},{"Start":"14:58.045 ","End":"15:02.815","Text":"What we want is the less than or equal to 0,"},{"Start":"15:02.815 ","End":"15:06.940","Text":"which means on the line or below it,"},{"Start":"15:06.940 ","End":"15:09.130","Text":"meaning it\u0027s 0 or negative,"},{"Start":"15:09.130 ","End":"15:13.825","Text":"so this is good because it\u0027s 0."},{"Start":"15:13.825 ","End":"15:16.780","Text":"This is good also because it\u0027s 0."},{"Start":"15:16.780 ","End":"15:18.940","Text":"We\u0027re collecting zeros and negatives,"},{"Start":"15:18.940 ","End":"15:21.410","Text":"and this is also good."},{"Start":"15:21.540 ","End":"15:26.725","Text":"What this means is that we actually,"},{"Start":"15:26.725 ","End":"15:29.300","Text":"for this bit here,"},{"Start":"15:29.490 ","End":"15:35.030","Text":"we have x less than 0."},{"Start":"15:35.340 ","End":"15:39.340","Text":"But also x equals 0 is good,"},{"Start":"15:39.340 ","End":"15:42.915","Text":"so it\u0027s x less than or equal to 0."},{"Start":"15:42.915 ","End":"15:45.500","Text":"Now, there\u0027s no more negatives,"},{"Start":"15:45.500 ","End":"15:47.285","Text":"but we\u0027re also collecting zeros."},{"Start":"15:47.285 ","End":"15:53.420","Text":"We\u0027ve got an extra bonus point 3 or x equals 3,"},{"Start":"15:53.420 ","End":"15:55.880","Text":"precisely that one extra point."},{"Start":"15:55.880 ","End":"15:57.890","Text":"We have this interval and"},{"Start":"15:57.890 ","End":"16:02.360","Text":"an extra point which is something different than what we\u0027ve seen so far."},{"Start":"16:02.360 ","End":"16:06.455","Text":"Anyway, I\u0027ll highlight that and it\u0027s the answer to Part c,"},{"Start":"16:06.455 ","End":"16:16.459","Text":"and so on to Part d and what do we have here?"},{"Start":"16:16.560 ","End":"16:21.640","Text":"Here we have a degree 3 or cubic polynomial."},{"Start":"16:21.640 ","End":"16:26.065","Text":"The obvious thing to do is to take x outside the brackets,"},{"Start":"16:26.065 ","End":"16:35.800","Text":"leaving us with 4x^2 minus 4x plus 1 less than or equal to 0."},{"Start":"16:35.800 ","End":"16:39.145","Text":"Next, like to factorize this."},{"Start":"16:39.145 ","End":"16:43.360","Text":"Notice that the first term is 2x all squared"},{"Start":"16:43.360 ","End":"16:46.780","Text":"and this is 1 squared so it looks like it\u0027s going to be one"},{"Start":"16:46.780 ","End":"16:53.890","Text":"of those a minus b squared equals a^2 minus 2ab plus b^2,"},{"Start":"16:53.890 ","End":"17:01.280","Text":"one of those cases and indeed we can take our a is 2x."},{"Start":"17:01.280 ","End":"17:03.230","Text":"The b is 1,"},{"Start":"17:03.230 ","End":"17:05.300","Text":"so it\u0027s 2x minus 1 squared."},{"Start":"17:05.300 ","End":"17:09.455","Text":"But we can\u0027t just do that unless we check the middle term."},{"Start":"17:09.455 ","End":"17:12.590","Text":"I mean, I can see that this is 2x all squared and this is 1 squared."},{"Start":"17:12.590 ","End":"17:20.675","Text":"Let\u0027s see, twice ab is twice 2x times 1 is indeed 4x so we are okay that way."},{"Start":"17:20.675 ","End":"17:22.955","Text":"Now we have it factorized."},{"Start":"17:22.955 ","End":"17:25.954","Text":"Now what we want to do is write down the roots,"},{"Start":"17:25.954 ","End":"17:33.750","Text":"which x make this polynomial 0 so for one thing we have x equals 0."},{"Start":"17:34.360 ","End":"17:38.900","Text":"For this bit, if it\u0027s squared 0."},{"Start":"17:38.900 ","End":"17:40.865","Text":"Then the thing itself has to be 0."},{"Start":"17:40.865 ","End":"17:42.650","Text":"2x minus 1 is 0,"},{"Start":"17:42.650 ","End":"17:47.240","Text":"x equals 1/2 and do that in our heads, certainly."},{"Start":"17:47.240 ","End":"17:48.945","Text":"Now we have the roots."},{"Start":"17:48.945 ","End":"17:54.110","Text":"Now do the number line here and on"},{"Start":"17:54.110 ","End":"18:01.410","Text":"the number line we\u0027ll put the roots 0 and 1/2."},{"Start":"18:02.640 ","End":"18:04.840","Text":"We get 3 intervals."},{"Start":"18:04.840 ","End":"18:08.140","Text":"Let\u0027s see where it\u0027s positive and where it\u0027s negative."},{"Start":"18:08.140 ","End":"18:12.085","Text":"The sample point can\u0027t use these points."},{"Start":"18:12.085 ","End":"18:13.900","Text":"I\u0027ll take minus 1."},{"Start":"18:13.900 ","End":"18:16.555","Text":"If x is minus 1,"},{"Start":"18:16.555 ","End":"18:20.439","Text":"then this is negative and this is obviously positive because of the square."},{"Start":"18:20.439 ","End":"18:22.729","Text":"So it\u0027s minus here,"},{"Start":"18:22.729 ","End":"18:24.370","Text":"put a point here."},{"Start":"18:24.370 ","End":"18:28.225","Text":"This is minus 1 and some negative point."},{"Start":"18:28.225 ","End":"18:30.640","Text":"When x is between the 2,"},{"Start":"18:30.640 ","End":"18:32.440","Text":"let\u0027s say a 1/4."},{"Start":"18:32.440 ","End":"18:35.909","Text":"A 1/4 is positive,"},{"Start":"18:35.909 ","End":"18:38.205","Text":"and actually I don\u0027t have to plug in here."},{"Start":"18:38.205 ","End":"18:42.225","Text":"It\u0027s going to be positive because of the square."},{"Start":"18:42.225 ","End":"18:46.515","Text":"Here, we are positive."},{"Start":"18:46.515 ","End":"18:51.070","Text":"Let\u0027s see when x is, let\u0027s say,"},{"Start":"18:51.070 ","End":"18:54.190","Text":"1 sample from here, take x=1,"},{"Start":"18:54.190 ","End":"18:58.135","Text":"1 times something squared is positive."},{"Start":"18:58.135 ","End":"19:02.080","Text":"So again, we have a positive,"},{"Start":"19:02.080 ","End":"19:05.630","Text":"here it\u0027s 0 and here it\u0027s 0."},{"Start":"19:05.630 ","End":"19:08.323","Text":"This time we\u0027ll draw the snake,"},{"Start":"19:08.323 ","End":"19:16.830","Text":"something just schematic here and through here."},{"Start":"19:16.830 ","End":"19:19.930","Text":"But it doesn\u0027t cross the line."},{"Start":"19:22.320 ","End":"19:26.935","Text":"Goes on in both directions forever, somewhere."},{"Start":"19:26.935 ","End":"19:29.845","Text":"Now, let\u0027s see what are we collecting?"},{"Start":"19:29.845 ","End":"19:32.710","Text":"We\u0027re collecting less than or equal to 0."},{"Start":"19:32.710 ","End":"19:37.940","Text":"We are collecting what is below the line or on the line."},{"Start":"19:38.400 ","End":"19:47.710","Text":"I\u0027ll use the shading this time method below the line."},{"Start":"19:47.710 ","End":"19:53.455","Text":"Nothing else below the line and on the line for the equals 0,"},{"Start":"19:53.455 ","End":"19:58.030","Text":"I have, let\u0027s do a different color."},{"Start":"19:58.030 ","End":"19:59.934","Text":"This is on the line,"},{"Start":"19:59.934 ","End":"20:02.960","Text":"and this is on the line."},{"Start":"20:03.870 ","End":"20:11.950","Text":"What we get for this part from 0 including 0 and below,"},{"Start":"20:11.950 ","End":"20:15.280","Text":"so it\u0027s x less than or equal to 0."},{"Start":"20:15.280 ","End":"20:18.220","Text":"From here, the previous exercise,"},{"Start":"20:18.220 ","End":"20:19.840","Text":"I called it a bonus point."},{"Start":"20:19.840 ","End":"20:25.818","Text":"x=1/2 and it\u0027s 0."},{"Start":"20:25.818 ","End":"20:31.015","Text":"Any 0 and negative number is good"},{"Start":"20:31.015 ","End":"20:38.690","Text":"and x=1/2 is just on its own thing which is also good because it makes it 0."},{"Start":"20:40.020 ","End":"20:42.970","Text":"Just highlighted to say, yes,"},{"Start":"20:42.970 ","End":"20:48.070","Text":"this is the answer and we\u0027re done and we go on to the last part which is,"},{"Start":"20:48.070 ","End":"20:50.990","Text":"it\u0027s been a along exercise I admit."},{"Start":"20:52.710 ","End":"20:57.400","Text":"Start out by bringing everything to the left-hand side,"},{"Start":"20:57.400 ","End":"21:01.720","Text":"3x^4 plus 4x^3"},{"Start":"21:01.720 ","End":"21:11.530","Text":"plus x^2\u003e0,"},{"Start":"21:11.530 ","End":"21:17.680","Text":"x^2 comes out of the brackets and what we\u0027re left with"},{"Start":"21:17.680 ","End":"21:26.015","Text":"is 3x^2 plus 4x plus1\u003e"},{"Start":"21:26.015 ","End":"21:28.840","Text":"0."},{"Start":"21:28.840 ","End":"21:33.550","Text":"I\u0027ll save time and factorize this for you since you know to do it so well,"},{"Start":"21:33.550 ","End":"21:36.850","Text":"you should know both methods using the formula for"},{"Start":"21:36.850 ","End":"21:41.920","Text":"the quadratic equation and also with the trinomial reduction method."},{"Start":"21:41.920 ","End":"21:43.420","Text":"Anyway, I\u0027m just going to give you the answer."},{"Start":"21:43.420 ","End":"21:48.130","Text":"This is 3x plus 1x plus 1."},{"Start":"21:48.130 ","End":"21:51.530","Text":"But I will verify it with you,"},{"Start":"21:52.380 ","End":"21:58.180","Text":"3x times x is 3^2 plus x plus 3x plus 4x and 1x1 is 1,"},{"Start":"21:58.180 ","End":"21:59.769","Text":"so at least it\u0027s verified,"},{"Start":"21:59.769 ","End":"22:03.282","Text":"but it\u0027s taking too long otherwise."},{"Start":"22:03.282 ","End":"22:12.369","Text":"Now, we want to find the roots to see what values of x could make this polynomial 0."},{"Start":"22:12.369 ","End":"22:18.445","Text":"Well, for sure, x=0 will make the first bit 0."},{"Start":"22:18.445 ","End":"22:21.895","Text":"For this one, 3x plus 1 is 0,"},{"Start":"22:21.895 ","End":"22:25.314","Text":"x is minus 1/3 easy to do in our heads."},{"Start":"22:25.314 ","End":"22:29.035","Text":"For this one to be 0, we need x is minus 1."},{"Start":"22:29.035 ","End":"22:34.000","Text":"So let\u0027s take a number line here,"},{"Start":"22:34.000 ","End":"22:35.080","Text":"and on it,"},{"Start":"22:35.080 ","End":"22:36.490","Text":"I\u0027ll put these 3 points,"},{"Start":"22:36.490 ","End":"22:39.070","Text":"but it must be in the correct order."},{"Start":"22:39.070 ","End":"22:42.340","Text":"Smallest is minus 1,"},{"Start":"22:42.340 ","End":"22:46.700","Text":"next, minus 1/3,"},{"Start":"22:46.700 ","End":"22:48.679","Text":"and next 0,"},{"Start":"22:48.679 ","End":"22:52.400","Text":"scales not important, but the order is very important."},{"Start":"22:54.390 ","End":"22:56.740","Text":"We need to know on each of"},{"Start":"22:56.740 ","End":"23:02.740","Text":"these 4 intervals whether the polynomial is positive or negative."},{"Start":"23:02.740 ","End":"23:05.860","Text":"Since I did the snake method in the previous one,"},{"Start":"23:05.860 ","End":"23:08.140","Text":"I\u0027ll use the alternative,"},{"Start":"23:08.140 ","End":"23:12.205","Text":"the variation where we just write the signs here."},{"Start":"23:12.205 ","End":"23:14.410","Text":"I deliberately use sometimes one method,"},{"Start":"23:14.410 ","End":"23:17.320","Text":"sometimes another and then you can choose this as"},{"Start":"23:17.320 ","End":"23:23.500","Text":"just a modification where we just write the plus minus."},{"Start":"23:23.500 ","End":"23:27.413","Text":"When x is, say,"},{"Start":"23:27.413 ","End":"23:31.515","Text":"minus 2 then we get,"},{"Start":"23:31.515 ","End":"23:35.336","Text":"let\u0027s see if x is minus 2, this is positive,"},{"Start":"23:35.336 ","End":"23:41.300","Text":"minus 2 here gives minus 6 plus 1."},{"Start":"23:41.300 ","End":"23:46.195","Text":"Definitely negative minus 2 plus 1 is also negative."},{"Start":"23:46.195 ","End":"23:51.430","Text":"Positive, negative, negative so here it is positive."},{"Start":"23:51.430 ","End":"23:54.083","Text":"In this interval,"},{"Start":"23:54.083 ","End":"23:56.875","Text":"what\u0027s between minus 1 and minus 1/3,"},{"Start":"23:56.875 ","End":"24:02.065","Text":"let\u0027s say minus 1/2 will do."},{"Start":"24:02.065 ","End":"24:04.690","Text":"Maybe I\u0027ll indicate the sample points I was using,"},{"Start":"24:04.690 ","End":"24:06.865","Text":"I used minus 2 here."},{"Start":"24:06.865 ","End":"24:12.805","Text":"I\u0027m going to use minus 1/2 here."},{"Start":"24:12.805 ","End":"24:20.305","Text":"Here, let\u0027s say what\u0027s between 0 and minus 1/3,"},{"Start":"24:20.305 ","End":"24:23.800","Text":"minus 1/4 will be good."},{"Start":"24:23.800 ","End":"24:26.065","Text":"Here, I\u0027ll use 1."},{"Start":"24:26.065 ","End":"24:33.250","Text":"Here, I have a plus and for minus 1/2, well,"},{"Start":"24:33.250 ","End":"24:35.852","Text":"this is positive because it\u0027s squared,"},{"Start":"24:35.852 ","End":"24:44.905","Text":"minus 1/2 times 3 is minus 1/2 plus 1 minus 1/2 is negative."},{"Start":"24:44.905 ","End":"24:50.680","Text":"Negative, but minus 1/2 plus 1 is already positive."},{"Start":"24:50.680 ","End":"24:53.050","Text":"We have positive, negative,"},{"Start":"24:53.050 ","End":"24:58.015","Text":"positive, and so at 1/2 it\u0027s negative."},{"Start":"24:58.015 ","End":"25:00.895","Text":"Let\u0027s see now, minus 1/4."},{"Start":"25:00.895 ","End":"25:02.669","Text":"First of all is always positive."},{"Start":"25:02.669 ","End":"25:10.780","Text":"Minus 3/4 plus 1 is already positive,"},{"Start":"25:10.780 ","End":"25:14.650","Text":"and minus 1/4 plus 1 is also positive,"},{"Start":"25:14.650 ","End":"25:18.620","Text":"so here everything\u0027s positive."},{"Start":"25:20.280 ","End":"25:24.340","Text":"We are positive,"},{"Start":"25:24.340 ","End":"25:26.050","Text":"and on the last one,"},{"Start":"25:26.050 ","End":"25:30.205","Text":"when x=1 positive, yeah,"},{"Start":"25:30.205 ","End":"25:34.610","Text":"all positive, so it\u0027s also positive."},{"Start":"25:37.110 ","End":"25:44.410","Text":"That\u0027s again an example where there\u0027s no alternation, at least not always."},{"Start":"25:44.410 ","End":"25:46.480","Text":"What are we collecting?"},{"Start":"25:46.480 ","End":"25:50.420","Text":"We\u0027re collecting strictly positive."},{"Start":"25:51.120 ","End":"25:54.166","Text":"That\u0027s what the 0 means,"},{"Start":"25:54.166 ","End":"25:56.050","Text":"which means we want this."},{"Start":"25:56.050 ","End":"26:00.430","Text":"We want this and we want this."},{"Start":"26:00.430 ","End":"26:04.195","Text":"We have 3 bits."},{"Start":"26:04.195 ","End":"26:09.750","Text":"This bit here is x less than minus"},{"Start":"26:09.750 ","End":"26:15.190","Text":"1 not including because we\u0027re not including on the line."},{"Start":"26:15.190 ","End":"26:23.215","Text":"I could have indicated that here it\u0027s 0 here,"},{"Start":"26:23.215 ","End":"26:26.800","Text":"it\u0027s 0, the root, of course, it\u0027s always 0."},{"Start":"26:26.800 ","End":"26:31.780","Text":"But we don\u0027t want 0 is only strictly positive."},{"Start":"26:31.780 ","End":"26:38.710","Text":"Back here, we don\u0027t plus this plus gives us that x"},{"Start":"26:38.710 ","End":"26:45.160","Text":"can be between minus 1/3 and 0,"},{"Start":"26:45.160 ","End":"26:48.100","Text":"there\u0027s an or connecting them,"},{"Start":"26:48.100 ","End":"26:53.005","Text":"or x is from 0 onwards,"},{"Start":"26:53.005 ","End":"26:57.160","Text":"or x is bigger than 0."},{"Start":"26:57.160 ","End":"27:01.255","Text":"We almost could have joined these two together into one long interval,"},{"Start":"27:01.255 ","End":"27:02.920","Text":"but there\u0027s a gap here,"},{"Start":"27:02.920 ","End":"27:04.450","Text":"0 is not included,"},{"Start":"27:04.450 ","End":"27:06.950","Text":"so it\u0027s written separately."},{"Start":"27:07.650 ","End":"27:14.200","Text":"This is the answer to the final part E,"},{"Start":"27:14.200 ","End":"27:15.738","Text":"and I\u0027m highlighting it,"},{"Start":"27:15.738 ","End":"27:19.310","Text":"and we are done."}],"Thumbnail":null,"ID":5068},{"Watched":false,"Name":"Exercise 7","Duration":"14m 32s","ChapterTopicVideoID":5066,"CourseChapterTopicPlaylistID":3876,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.680","Text":"In this exercise we have to solve 2 sets of inequalities."},{"Start":"00:04.680 ","End":"00:09.360","Text":"The first one has 2 separate inequalities combined with"},{"Start":"00:09.360 ","End":"00:14.520","Text":"and the other one has 2 separate inequalities combined with or so there\u0027s more work."},{"Start":"00:14.520 ","End":"00:15.870","Text":"It\u0027s not just 2 exercises,"},{"Start":"00:15.870 ","End":"00:18.885","Text":"it\u0027s is really 4 exercises at least here."},{"Start":"00:18.885 ","End":"00:23.075","Text":"Anyway we\u0027ll start with the first one."},{"Start":"00:23.075 ","End":"00:26.785","Text":"Let\u0027s go with the easy bit first,"},{"Start":"00:26.785 ","End":"00:30.240","Text":"3x plus 1 bigger than 0,"},{"Start":"00:30.240 ","End":"00:31.500","Text":"we do it in our heads."},{"Start":"00:31.500 ","End":"00:37.115","Text":"There\u0027s no need for any diagrams or plotting number lines and so on."},{"Start":"00:37.115 ","End":"00:39.380","Text":"Just bring 1 to the other side,"},{"Start":"00:39.380 ","End":"00:40.460","Text":"divide by 3,"},{"Start":"00:40.460 ","End":"00:46.480","Text":"which is positive and we get x is bigger than minus 1/3."},{"Start":"00:46.480 ","End":"00:49.050","Text":"That\u0027s just part of the answer,"},{"Start":"00:49.050 ","End":"00:51.015","Text":"that\u0027s the answer for this."},{"Start":"00:51.015 ","End":"00:53.545","Text":"Now let\u0027s look at this one."},{"Start":"00:53.545 ","End":"00:55.850","Text":"Well, it\u0027s already factorized,"},{"Start":"00:55.850 ","End":"00:58.160","Text":"which makes the work a bit easier."},{"Start":"00:58.160 ","End":"01:00.560","Text":"Let\u0027s see the roots of this."},{"Start":"01:00.560 ","End":"01:02.885","Text":"We can tell straight away,"},{"Start":"01:02.885 ","End":"01:05.615","Text":"I\u0027m thinking maybe I\u0027ll use a different color."},{"Start":"01:05.615 ","End":"01:09.320","Text":"The roots are x equals,"},{"Start":"01:09.320 ","End":"01:12.720","Text":"let\u0027s see for this bit we get 2,"},{"Start":"01:12.940 ","End":"01:18.375","Text":"for this bit we get 1.5."},{"Start":"01:18.375 ","End":"01:26.550","Text":"Just bring the 1 over and divide by 2 and for the last bit we get 2/3."},{"Start":"01:26.550 ","End":"01:31.420","Text":"This one will do with the diagram and snake."},{"Start":"01:31.420 ","End":"01:33.860","Text":"Here\u0027s our number line."},{"Start":"01:33.860 ","End":"01:35.120","Text":"Let\u0027s put in the points."},{"Start":"01:35.120 ","End":"01:37.670","Text":"Remember it\u0027s very important to put them in the right order."},{"Start":"01:37.670 ","End":"01:40.670","Text":"It doesn\u0027t have to be to scale or anything."},{"Start":"01:40.670 ","End":"01:47.720","Text":"The lowest one is 1/2 the next one is 2/3,"},{"Start":"01:47.720 ","End":"01:51.130","Text":"and the next one is 2."},{"Start":"01:51.130 ","End":"01:57.260","Text":"We want to know in each of these 4 separate intervals what the sign is."},{"Start":"01:57.260 ","End":"02:00.710","Text":"Sometimes I like to write in the sample points,"},{"Start":"02:00.710 ","End":"02:03.260","Text":"sometimes not here because we\u0027re working with fractions,"},{"Start":"02:03.260 ","End":"02:05.750","Text":"not immediately obvious I will."},{"Start":"02:05.750 ","End":"02:07.910","Text":"On this side, no problem,"},{"Start":"02:07.910 ","End":"02:10.340","Text":"I\u0027ll use 3 as a sample point."},{"Start":"02:10.340 ","End":"02:14.855","Text":"On this side, I\u0027ll use minus 1 as a sample point."},{"Start":"02:14.855 ","End":"02:19.210","Text":"I need to choose something here and here with fractions."},{"Start":"02:19.210 ","End":"02:24.825","Text":"Let\u0027s see 1/2 is 0.5, this is 0.677."},{"Start":"02:24.825 ","End":"02:28.325","Text":"How about I take 0.6,"},{"Start":"02:28.325 ","End":"02:31.798","Text":"I\u0027ll call it 3/5,"},{"Start":"02:31.798 ","End":"02:34.155","Text":"or could work in decimals also doesn\u0027t matter."},{"Start":"02:34.155 ","End":"02:37.950","Text":"Between 2/3 and 2 that\u0027s an easy one,"},{"Start":"02:37.950 ","End":"02:41.565","Text":"3/4 is going to be in between."},{"Start":"02:41.565 ","End":"02:46.910","Text":"Now I\u0027ll take these 4 sample points and substitute them in here,"},{"Start":"02:46.910 ","End":"02:49.429","Text":"but I only care about the sign."},{"Start":"02:49.429 ","End":"02:54.225","Text":"Minus 1, this one is negative minus 1 minus 2,"},{"Start":"02:54.225 ","End":"02:56.820","Text":"twice minus 1 minus 1, negative."},{"Start":"02:56.820 ","End":"02:59.280","Text":"3 times minus 1 minus 1, negative."},{"Start":"02:59.280 ","End":"03:06.805","Text":"Negative, negative, negative is negative so we are here below the axis negative."},{"Start":"03:06.805 ","End":"03:09.360","Text":"When it\u0027s 3/5,"},{"Start":"03:09.360 ","End":"03:12.240","Text":"well 3/5 minus 2 is obviously negative,"},{"Start":"03:12.240 ","End":"03:15.135","Text":"twice 3/5 is 6/5,"},{"Start":"03:15.135 ","End":"03:25.485","Text":"which is more than 1 so that\u0027s positive and 3 times 3/5 is 9/5,"},{"Start":"03:25.485 ","End":"03:29.645","Text":"which is less than 2 because is 2 is 10/5."},{"Start":"03:29.645 ","End":"03:31.505","Text":"This is negative."},{"Start":"03:31.505 ","End":"03:34.705","Text":"If I remember we had,"},{"Start":"03:34.705 ","End":"03:37.245","Text":"let\u0027s see, I forgot already."},{"Start":"03:37.245 ","End":"03:42.735","Text":"This was negative and this was positive,"},{"Start":"03:42.735 ","End":"03:47.145","Text":"and this was a negative."},{"Start":"03:47.145 ","End":"03:49.415","Text":"There are 2 negatives there,"},{"Start":"03:49.415 ","End":"03:53.055","Text":"which makes it positive at 3/5."},{"Start":"03:53.055 ","End":"03:55.935","Text":"Next sample point, 3/4."},{"Start":"03:55.935 ","End":"03:57.750","Text":"This one\u0027s negative."},{"Start":"03:57.750 ","End":"04:06.850","Text":"Twice 3/4 is 1.5 and that makes this one positive and 3 times 3/4 is 2 and 1/4."},{"Start":"04:06.850 ","End":"04:09.815","Text":"That\u0027s also positive, so negative, positive,"},{"Start":"04:09.815 ","End":"04:14.890","Text":"positive, that makes this one negative."},{"Start":"04:14.890 ","End":"04:19.710","Text":"When x is 3, clearly positive,"},{"Start":"04:19.710 ","End":"04:27.250","Text":"they all positive so that is going to be a positive above here."},{"Start":"04:27.770 ","End":"04:34.880","Text":"Let\u0027s do the snake and be sure to pass through the roots."},{"Start":"04:35.790 ","End":"04:39.150","Text":"I\u0027ll mark them on, here it\u0027s 0,"},{"Start":"04:39.150 ","End":"04:41.950","Text":"here it\u0027s 0, and here it\u0027s 0."},{"Start":"04:41.950 ","End":"04:44.960","Text":"These are the roots, these are not the sample points."},{"Start":"04:51.920 ","End":"04:54.810","Text":"Like I said, just schematic,"},{"Start":"04:54.810 ","End":"04:57.910","Text":"it doesn\u0027t mean that that\u0027s what the function actually looks like,"},{"Start":"04:57.910 ","End":"05:01.285","Text":"but it does go on indefinitely in both directions."},{"Start":"05:01.285 ","End":"05:03.490","Text":"Now I want you to see what we\u0027re collecting."},{"Start":"05:03.490 ","End":"05:05.875","Text":"We\u0027re collecting negatives today."},{"Start":"05:05.875 ","End":"05:10.375","Text":"Less than 0, negative meaning below the axis,"},{"Start":"05:10.375 ","End":"05:12.499","Text":"not including the axis."},{"Start":"05:12.499 ","End":"05:14.765","Text":"Sometimes I use highlighting,"},{"Start":"05:14.765 ","End":"05:16.445","Text":"sometimes I use shading."},{"Start":"05:16.445 ","End":"05:19.313","Text":"Let\u0027s go for highlighting this time,"},{"Start":"05:19.313 ","End":"05:23.060","Text":"so less than 0 below the axis is,"},{"Start":"05:28.520 ","End":"05:34.530","Text":"this bit not including and I\u0027m going to stop short of it."},{"Start":"05:34.530 ","End":"05:39.680","Text":"Then negative I have here also and here,"},{"Start":"05:39.680 ","End":"05:42.725","Text":"and just stop short of the axis."},{"Start":"05:42.725 ","End":"05:47.225","Text":"If I look at what this represents,"},{"Start":"05:47.225 ","End":"05:53.015","Text":"on the x-axis, this bit here,"},{"Start":"05:53.015 ","End":"05:55.454","Text":"not including that,"},{"Start":"05:55.454 ","End":"06:00.380","Text":"and here we have from here to here not including,"},{"Start":"06:01.850 ","End":"06:10.690","Text":"this bit means x is less than 1.5, not including."},{"Start":"06:10.690 ","End":"06:19.200","Text":"Here we\u0027ve got, I\u0027ll put an \"or\" x between 2/3 and 2,"},{"Start":"06:19.200 ","End":"06:24.780","Text":"so we write that as 2/3 less than x, less than 2."},{"Start":"06:24.780 ","End":"06:30.115","Text":"I\u0027ll put it in brackets because we now have this one."},{"Start":"06:30.115 ","End":"06:33.985","Text":"Just copy it, never mind the same color."},{"Start":"06:33.985 ","End":"06:40.060","Text":"This and this."},{"Start":"06:40.060 ","End":"06:47.329","Text":"Now we need to separate diagram for the and/or of 2 things."},{"Start":"06:47.329 ","End":"06:52.831","Text":"I\u0027m going to draw a different kind of diagram for doing and\u0027s and or\u0027s,"},{"Start":"06:52.831 ","End":"06:56.260","Text":"so let\u0027s draw each of the pieces."},{"Start":"06:56.260 ","End":"06:59.470","Text":"First of all x bigger than minus 1/3."},{"Start":"06:59.470 ","End":"07:04.045","Text":"Well, even before that I should write the important points on."},{"Start":"07:04.045 ","End":"07:05.916","Text":"Actually I need 4 points."},{"Start":"07:05.916 ","End":"07:07.810","Text":"I need this, this, this,"},{"Start":"07:07.810 ","End":"07:10.096","Text":"and the minus 1/3,"},{"Start":"07:10.096 ","End":"07:11.485","Text":"so let\u0027s put them on."},{"Start":"07:11.485 ","End":"07:20.675","Text":"Minus 1/3, and then 1.5, and then 2/3."},{"Start":"07:20.675 ","End":"07:26.440","Text":"Then 2. Now I\u0027m going to draw this inequality."},{"Start":"07:26.440 ","End":"07:28.975","Text":"X bigger than minus 1/3."},{"Start":"07:28.975 ","End":"07:31.210","Text":"Minus 1/3 is not included,"},{"Start":"07:31.210 ","End":"07:33.250","Text":"so it\u0027s a hollow circle,"},{"Start":"07:33.250 ","End":"07:36.220","Text":"and everything to the right are greater than."},{"Start":"07:36.220 ","End":"07:38.200","Text":"That\u0027s this 1."},{"Start":"07:38.200 ","End":"07:42.895","Text":"Now this inequality is made up of 2 bits,"},{"Start":"07:42.895 ","End":"07:48.130","Text":"less than 1/2. Let\u0027s see."},{"Start":"07:48.130 ","End":"07:50.485","Text":"The 1/2 is here,"},{"Start":"07:50.485 ","End":"07:54.190","Text":"and everything to the left and continuing."},{"Start":"07:54.190 ","End":"07:57.880","Text":"The other bit between 2/3 and 2,"},{"Start":"07:57.880 ","End":"08:02.620","Text":"I need 2/3 and I need the 2,"},{"Start":"08:02.620 ","End":"08:05.530","Text":"and the line between them."},{"Start":"08:05.530 ","End":"08:09.970","Text":"This inequality and this inequality are sketched and because of the,"},{"Start":"08:09.970 ","End":"08:14.170","Text":"and we need to take the overlap,"},{"Start":"08:14.170 ","End":"08:17.875","Text":"that would be this bit here,"},{"Start":"08:17.875 ","End":"08:20.065","Text":"and this bit here."},{"Start":"08:20.065 ","End":"08:22.840","Text":"If we just write it in terms of x,"},{"Start":"08:22.840 ","End":"08:26.860","Text":"what we have for this bit is minus 1/3,"},{"Start":"08:26.860 ","End":"08:31.180","Text":"less than x, less than 1.5."},{"Start":"08:31.180 ","End":"08:40.900","Text":"We have the other bits. We write an OR 2/3 less than x, less than 2."},{"Start":"08:40.900 ","End":"08:43.165","Text":"Everything is strict inequality,"},{"Start":"08:43.165 ","End":"08:46.280","Text":"because we don\u0027t include the points."},{"Start":"08:47.250 ","End":"08:52.090","Text":"This is the answer for Part A,"},{"Start":"08:52.090 ","End":"08:56.215","Text":"answer on to Part B,"},{"Start":"08:56.215 ","End":"08:59.335","Text":"which is also made up of 2 bits."},{"Start":"08:59.335 ","End":"09:03.050","Text":"This time we have OR instead of AND."},{"Start":"09:03.420 ","End":"09:06.340","Text":"Let\u0027s do each piece separately."},{"Start":"09:06.340 ","End":"09:10.180","Text":"We start with this 1x^2 less than 2x."},{"Start":"09:10.180 ","End":"09:14.890","Text":"X^2 minus 2x is less than 0."},{"Start":"09:14.890 ","End":"09:17.290","Text":"This is easy to factorize,"},{"Start":"09:17.290 ","End":"09:19.030","Text":"because we just take x out,"},{"Start":"09:19.030 ","End":"09:21.580","Text":"x, x minus 2,"},{"Start":"09:21.580 ","End":"09:24.175","Text":"is less than 0."},{"Start":"09:24.175 ","End":"09:27.415","Text":"Then the roots of this equation,"},{"Start":"09:27.415 ","End":"09:33.430","Text":"that x is equal to either 0 for this bit,"},{"Start":"09:33.430 ","End":"09:36.414","Text":"or 2 what makes this bit 0."},{"Start":"09:36.414 ","End":"09:38.495","Text":"Those are the 2 roots."},{"Start":"09:38.495 ","End":"09:44.715","Text":"A quick sketch here. Here\u0027s 0."},{"Start":"09:44.715 ","End":"09:50.860","Text":"Here\u0027s 2, L2 samples in each interval."},{"Start":"09:50.860 ","End":"09:54.220","Text":"I\u0027ll choose a sample of minus here."},{"Start":"09:54.220 ","End":"09:55.570","Text":"I\u0027ll choose 1 here,"},{"Start":"09:55.570 ","End":"09:57.445","Text":"and I\u0027ll choose 3 here."},{"Start":"09:57.445 ","End":"09:59.320","Text":"Let\u0027s not do the full snake."},{"Start":"09:59.320 ","End":"10:00.955","Text":"We\u0027ll just write the sign,"},{"Start":"10:00.955 ","End":"10:03.100","Text":"when x is minus 1,"},{"Start":"10:03.100 ","End":"10:06.940","Text":"negative, negative, so plus."},{"Start":"10:06.940 ","End":"10:09.295","Text":"When x is 1,"},{"Start":"10:09.295 ","End":"10:14.185","Text":"positive, 1 minus 2 negative, so negative."},{"Start":"10:14.185 ","End":"10:16.150","Text":"When x is 3,"},{"Start":"10:16.150 ","End":"10:20.695","Text":"positive 3 minus 2 positive, so positive."},{"Start":"10:20.695 ","End":"10:22.480","Text":"What are we interested in?"},{"Start":"10:22.480 ","End":"10:24.490","Text":"We\u0027re interested in less than 0."},{"Start":"10:24.490 ","End":"10:27.504","Text":"Less than 0, means negative."},{"Start":"10:27.504 ","End":"10:32.275","Text":"This is the bit we care about so everything between 0 and 2 not inclusive,"},{"Start":"10:32.275 ","End":"10:36.475","Text":"0 less than x, less than 2."},{"Start":"10:36.475 ","End":"10:39.235","Text":"This part before the OR."},{"Start":"10:39.235 ","End":"10:41.575","Text":"Now let\u0027s look at this 1."},{"Start":"10:41.575 ","End":"10:44.230","Text":"This 1 has already been factorized,"},{"Start":"10:44.230 ","End":"10:46.120","Text":"so we can easily see what the roots are."},{"Start":"10:46.120 ","End":"10:50.395","Text":"The roots are x equals, let\u0027s see."},{"Start":"10:50.395 ","End":"10:52.690","Text":"3 for this bit,"},{"Start":"10:52.690 ","End":"10:54.730","Text":"6 for this bit,"},{"Start":"10:54.730 ","End":"10:56.470","Text":"and 9 for this bit."},{"Start":"10:56.470 ","End":"10:59.980","Text":"That\u0027s what makes each of them 0 respectively."},{"Start":"10:59.980 ","End":"11:05.665","Text":"Once again, we\u0027ll do a sketch number line."},{"Start":"11:05.665 ","End":"11:11.090","Text":"The roots are already in the correct order, 3,6,9."},{"Start":"11:11.550 ","End":"11:14.695","Text":"I\u0027ll choose some sample points."},{"Start":"11:14.695 ","End":"11:17.665","Text":"How about 0 here,"},{"Start":"11:17.665 ","End":"11:25.225","Text":"4 here, 8 here and 10 here?"},{"Start":"11:25.225 ","End":"11:31.690","Text":"Now again, let\u0027s go for the short adjust the plus minus is rather than the full snake."},{"Start":"11:31.690 ","End":"11:36.550","Text":"When x is 0, we have minus,"},{"Start":"11:36.550 ","End":"11:40.945","Text":"minus and minus 0, where minus."},{"Start":"11:40.945 ","End":"11:43.990","Text":"At 4, 4 minus 3 is positive,"},{"Start":"11:43.990 ","End":"11:47.560","Text":"but the other 2 are negative, so it\u0027s positive."},{"Start":"11:47.560 ","End":"11:50.610","Text":"If x is 8, this is positive,"},{"Start":"11:50.610 ","End":"11:53.985","Text":"this is positive, but 8 minus 9 is already negative."},{"Start":"11:53.985 ","End":"11:55.830","Text":"Here we are negative,"},{"Start":"11:55.830 ","End":"11:57.570","Text":"and you can easily see it."},{"Start":"11:57.570 ","End":"12:02.855","Text":"The other 1, we are also going to have all of them positive, so it\u0027s positive."},{"Start":"12:02.855 ","End":"12:05.785","Text":"Question is what are we interested in?"},{"Start":"12:05.785 ","End":"12:08.890","Text":"We\u0027re interested in bigger than 0,"},{"Start":"12:08.890 ","End":"12:12.550","Text":"and bigger than 0 means positive so this is what we want,"},{"Start":"12:12.550 ","End":"12:16.150","Text":"and this is also what we\u0027re collecting and so altogether,"},{"Start":"12:16.150 ","End":"12:19.990","Text":"what we have because strict inequality not inclusive."},{"Start":"12:19.990 ","End":"12:25.420","Text":"We\u0027ve got 3 less than x,"},{"Start":"12:25.420 ","End":"12:26.980","Text":"less than 6,"},{"Start":"12:26.980 ","End":"12:32.785","Text":"or just from 9 onwards,"},{"Start":"12:32.785 ","End":"12:36.680","Text":"or x bigger than 9."},{"Start":"12:39.690 ","End":"12:47.110","Text":"Now, this part and this part they can be combined using OR."},{"Start":"12:47.110 ","End":"12:52.735","Text":"We\u0027ll do a third diagram for the OR."},{"Start":"12:52.735 ","End":"12:55.975","Text":"I have to put all the interesting points here,"},{"Start":"12:55.975 ","End":"12:57.910","Text":"I mean all the ones that are in this color here,"},{"Start":"12:57.910 ","End":"13:00.040","Text":"the 3,6,9 from here and the 0,2 from"},{"Start":"13:00.040 ","End":"13:03.580","Text":"here but the order is clear because all these are less than all these."},{"Start":"13:03.580 ","End":"13:06.955","Text":"I\u0027m just going to put 5 points in 0"},{"Start":"13:06.955 ","End":"13:14.440","Text":"to 3,6 and 9."},{"Start":"13:14.440 ","End":"13:16.960","Text":"The first inequality, 0 less than x,"},{"Start":"13:16.960 ","End":"13:19.070","Text":"less than 2,"},{"Start":"13:19.560 ","End":"13:24.220","Text":"here\u0027s 0, here\u0027s 2,"},{"Start":"13:24.220 ","End":"13:27.770","Text":"and here\u0027s what\u0027s between them."},{"Start":"13:28.230 ","End":"13:31.885","Text":"For this bit, we need"},{"Start":"13:31.885 ","End":"13:39.970","Text":"between 3 and 6 like that,"},{"Start":"13:39.970 ","End":"13:48.460","Text":"and from 9 onwards, going on indefinitely."},{"Start":"13:48.460 ","End":"13:52.630","Text":"Actually, I didn\u0027t really have to draw a sketch in"},{"Start":"13:52.630 ","End":"13:56.110","Text":"this case because when we have OR and ORs,"},{"Start":"13:56.110 ","End":"13:58.165","Text":"we just take the combination."},{"Start":"13:58.165 ","End":"14:01.000","Text":"Everything that\u0027s covered by at least 1 of these,"},{"Start":"14:01.000 ","End":"14:06.220","Text":"it\u0027s just this or this or this but it\u0027s good practice to have the sketch."},{"Start":"14:06.220 ","End":"14:12.310","Text":"In other words, the answer is 0 less than x,"},{"Start":"14:12.310 ","End":"14:13.765","Text":"less than 2,"},{"Start":"14:13.765 ","End":"14:17.740","Text":"or 3 less than x,"},{"Start":"14:17.740 ","End":"14:20.290","Text":"less than 6 for this bit,"},{"Start":"14:20.290 ","End":"14:26.090","Text":"or x bigger than 9."},{"Start":"14:26.310 ","End":"14:30.040","Text":"This is the answer for part B,"},{"Start":"14:30.040 ","End":"14:32.570","Text":"and we are done."}],"Thumbnail":null,"ID":5069}],"ID":3876},{"Name":"Rational Inequalities","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Rational Inequalities","Duration":"20m 14s","ChapterTopicVideoID":5080,"CourseChapterTopicPlaylistID":3877,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.540","Text":"In this clip, we\u0027ll be talking about inequalities with rational expressions."},{"Start":"00:06.540 ","End":"00:11.025","Text":"It\u0027s going to be very similar to inequalities with polynomials."},{"Start":"00:11.025 ","End":"00:16.200","Text":"Make sure you really understand how to work with polynomials."},{"Start":"00:16.200 ","End":"00:18.510","Text":"Is going to be only a small difference,"},{"Start":"00:18.510 ","End":"00:20.160","Text":"but an essential difference."},{"Start":"00:20.160 ","End":"00:22.920","Text":"Let\u0027s start right away with an example."},{"Start":"00:22.920 ","End":"00:25.485","Text":"I\u0027ll call it example 1,"},{"Start":"00:25.485 ","End":"00:31.067","Text":"and it will be x minus"},{"Start":"00:31.067 ","End":"00:40.399","Text":"2 over x^2-25 and bigger or equal to 0."},{"Start":"00:40.399 ","End":"00:43.130","Text":"The first step, as with polynomials,"},{"Start":"00:43.130 ","End":"00:48.050","Text":"is to bring everything on one side and have the inequality compared to 0."},{"Start":"00:48.050 ","End":"00:50.345","Text":"Well, this is done already."},{"Start":"00:50.345 ","End":"00:58.144","Text":"Step 2, to factorize the polynomials on the top and on the bottom as far as possible."},{"Start":"00:58.144 ","End":"01:00.080","Text":"There is still some work to do."},{"Start":"01:00.080 ","End":"01:03.155","Text":"The top, we have to leave as it is, factorize."},{"Start":"01:03.155 ","End":"01:08.600","Text":"The bottom, we could write as instead of x^2-25,"},{"Start":"01:08.600 ","End":"01:13.670","Text":"we can write x-5 times x+5,"},{"Start":"01:13.670 ","End":"01:17.675","Text":"just using the difference of squares formula."},{"Start":"01:17.675 ","End":"01:21.230","Text":"It\u0027s still bigger or equal to 0."},{"Start":"01:21.230 ","End":"01:25.070","Text":"Now we draw a number line here,"},{"Start":"01:25.070 ","End":"01:29.000","Text":"and normally with polynomials that place the roots on the number line."},{"Start":"01:29.000 ","End":"01:32.420","Text":"But here we distinguish between two kinds of roots."},{"Start":"01:32.420 ","End":"01:37.550","Text":"We have the roots of the numerator,"},{"Start":"01:37.550 ","End":"01:42.080","Text":"and we have the roots of the denominator."},{"Start":"01:42.080 ","End":"01:44.030","Text":"I distinguish between them."},{"Start":"01:44.030 ","End":"01:46.085","Text":"In the numerator,"},{"Start":"01:46.085 ","End":"01:50.210","Text":"obviously the root is x=2."},{"Start":"01:50.210 ","End":"01:53.870","Text":"Because that\u0027s the only thing that will make the numerator 0."},{"Start":"01:53.870 ","End":"01:56.780","Text":"But with the denominator I have from here plus 5,"},{"Start":"01:56.780 ","End":"01:59.360","Text":"and from here, minus 5,"},{"Start":"01:59.360 ","End":"02:02.210","Text":"so I have 5 and minus 5."},{"Start":"02:02.210 ","End":"02:07.455","Text":"I\u0027ll mark them on now, but let\u0027s see,"},{"Start":"02:07.455 ","End":"02:15.790","Text":"minus 5, let\u0027s say 5 and say 2."},{"Start":"02:15.980 ","End":"02:25.775","Text":"The way I distinguish them is the roots of the numerator in this case 2 or as before,"},{"Start":"02:25.775 ","End":"02:29.420","Text":"I indicate that with a solid dot,"},{"Start":"02:29.420 ","End":"02:32.885","Text":"which means that the expression at this point is 0."},{"Start":"02:32.885 ","End":"02:34.430","Text":"I mean, if I put x=2,"},{"Start":"02:34.430 ","End":"02:37.760","Text":"the numerator is 0, so the whole thing is 0."},{"Start":"02:37.760 ","End":"02:43.355","Text":"But, the denominator I indicate differently,"},{"Start":"02:43.355 ","End":"02:47.240","Text":"I choose to do this with a hollow dot,"},{"Start":"02:47.240 ","End":"02:54.950","Text":"which in this case means that the expression is not defined."},{"Start":"02:54.950 ","End":"02:59.030","Text":"There\u0027s quite a difference between being 0 and being undefined."},{"Start":"02:59.030 ","End":"03:03.800","Text":"Because I can\u0027t put x=5 or minus 5 into this expression."},{"Start":"03:03.800 ","End":"03:06.645","Text":"That would make the denominator 0."},{"Start":"03:06.645 ","End":"03:08.600","Text":"Make sure you really distinguish."},{"Start":"03:08.600 ","End":"03:11.030","Text":"You could do it with different colors or another scheme,"},{"Start":"03:11.030 ","End":"03:14.360","Text":"but I\u0027ve chosen the solid dot for roots to the numerator,"},{"Start":"03:14.360 ","End":"03:18.050","Text":"hollow dots for roots of the denominator."},{"Start":"03:18.050 ","End":"03:24.080","Text":"But as before, we still get this line divided into intervals."},{"Start":"03:24.080 ","End":"03:26.760","Text":"We have four pieces."},{"Start":"03:26.760 ","End":"03:33.930","Text":"I\u0027ll use some dotted lines to separate this bit, this bit and this bit."},{"Start":"03:33.930 ","End":"03:35.855","Text":"Just like with polynomials,"},{"Start":"03:35.855 ","End":"03:40.460","Text":"the expression won\u0027t change sign within an interval."},{"Start":"03:40.460 ","End":"03:44.090","Text":"It will be always positive or always negative in each interval."},{"Start":"03:44.090 ","End":"03:46.610","Text":"It will always be defined inside the interval,"},{"Start":"03:46.610 ","End":"03:50.515","Text":"the only places undefined are these two points here."},{"Start":"03:50.515 ","End":"03:54.155","Text":"We choose sample points like before."},{"Start":"03:54.155 ","End":"03:57.320","Text":"Let\u0027s say I\u0027ll choose minus 6,"},{"Start":"03:57.320 ","End":"04:00.425","Text":"here I\u0027ll choose 0,"},{"Start":"04:00.425 ","End":"04:08.805","Text":"here I\u0027ll choose in 3 and here I\u0027ll choose 6."},{"Start":"04:08.805 ","End":"04:12.800","Text":"Then we substitute in the expression easier."},{"Start":"04:12.800 ","End":"04:18.020","Text":"Maybe just substitute in here and see if we get positive or negative."},{"Start":"04:18.020 ","End":"04:20.780","Text":"If x is minus 6,"},{"Start":"04:20.780 ","End":"04:22.790","Text":"then this is negative."},{"Start":"04:22.790 ","End":"04:25.925","Text":"They are all negative."},{"Start":"04:25.925 ","End":"04:29.225","Text":"Three negatives means negative."},{"Start":"04:29.225 ","End":"04:34.920","Text":"I\u0027ll put a dot below the line."},{"Start":"04:35.360 ","End":"04:38.880","Text":"If I take x=0,"},{"Start":"04:38.880 ","End":"04:44.430","Text":"then this is negative but positive,"},{"Start":"04:44.430 ","End":"04:47.055","Text":"so that is positive."},{"Start":"04:47.055 ","End":"04:48.450","Text":"At the 0 I put a dot"},{"Start":"04:48.450 ","End":"04:55.055","Text":"above because we couldn\u0027t use the other technique with the pluses and minuses."},{"Start":"04:55.055 ","End":"04:58.430","Text":"I\u0027m going to do it with this snake method this time."},{"Start":"04:58.430 ","End":"05:01.670","Text":"With 3, we get 3-2 is positive,"},{"Start":"05:01.670 ","End":"05:06.875","Text":"3-5 is negative and then positive."},{"Start":"05:06.875 ","End":"05:12.435","Text":"Altogether where negative and with 6,"},{"Start":"05:12.435 ","End":"05:15.075","Text":"they\u0027re all positive,"},{"Start":"05:15.075 ","End":"05:19.075","Text":"so that would make that positive."},{"Start":"05:19.075 ","End":"05:21.320","Text":"Now I draw the snake,"},{"Start":"05:21.320 ","End":"05:23.990","Text":"which as you remember, is not the same as the graph."},{"Start":"05:23.990 ","End":"05:28.610","Text":"It\u0027s just to show where we\u0027re positive, negative like here,"},{"Start":"05:28.610 ","End":"05:34.770","Text":"we\u0027re negative, here it\u0027s undefined, it\u0027s like a gap."},{"Start":"05:35.740 ","End":"05:38.435","Text":"Here we\u0027re positive."},{"Start":"05:38.435 ","End":"05:40.940","Text":"Here, it\u0027s actually 0."},{"Start":"05:40.940 ","End":"05:44.610","Text":"The expression is defined and equal to 0."},{"Start":"05:45.790 ","End":"05:50.975","Text":"Here it\u0027s not defined, a gap."},{"Start":"05:50.975 ","End":"05:54.770","Text":"Here, it\u0027s positive and it keeps being positive"},{"Start":"05:54.770 ","End":"05:59.700","Text":"forever and here it keeps being negative forever."},{"Start":"06:00.620 ","End":"06:03.210","Text":"That\u0027s the picture."},{"Start":"06:03.210 ","End":"06:06.865","Text":"Now we need to see what do we want?"},{"Start":"06:06.865 ","End":"06:10.460","Text":"What we want is bigger or equal to 0,"},{"Start":"06:10.460 ","End":"06:17.370","Text":"which means that I want positives and 0."},{"Start":"06:17.370 ","End":"06:20.735","Text":"I can use highlighting or shading."},{"Start":"06:20.735 ","End":"06:24.110","Text":"I\u0027ll go with highlighting. Positive or 0."},{"Start":"06:24.110 ","End":"06:30.420","Text":"Well here it\u0027s positive, here is 0."},{"Start":"06:30.420 ","End":"06:33.255","Text":"We\u0027re fine with 0 too,"},{"Start":"06:33.255 ","End":"06:36.675","Text":"sorry, don\u0027t continue there."},{"Start":"06:36.675 ","End":"06:38.700","Text":"Not this point either,"},{"Start":"06:38.700 ","End":"06:41.090","Text":"it\u0027s not a 0, where it\u0027s undefined,"},{"Start":"06:41.090 ","End":"06:43.680","Text":"we continue with positive"},{"Start":"06:49.840 ","End":"06:54.440","Text":"and so on forever."},{"Start":"06:54.440 ","End":"07:03.200","Text":"If I interpret this as to which parts of the x-axis corresponds to,"},{"Start":"07:03.200 ","End":"07:05.900","Text":"what I get is a hollow dot here,"},{"Start":"07:05.900 ","End":"07:08.430","Text":"this is not included."},{"Start":"07:08.800 ","End":"07:12.840","Text":"Then up to 2,"},{"Start":"07:12.840 ","End":"07:17.980","Text":"but including 2 because we want, remember positive 0."},{"Start":"07:20.960 ","End":"07:25.290","Text":"Then here we have some more positives."},{"Start":"07:27.590 ","End":"07:34.880","Text":"Hollow dot, and then continuing onwards forever."},{"Start":"07:34.880 ","End":"07:45.090","Text":"What this gives us is the answer is x between 2 and minus 5,"},{"Start":"07:45.090 ","End":"07:54.190","Text":"but including the 2 or x bigger than 5."},{"Start":"07:54.590 ","End":"07:58.050","Text":"That\u0027s the answer."},{"Start":"07:58.050 ","End":"08:00.570","Text":"Let\u0027s take another example."},{"Start":"08:00.570 ","End":"08:02.250","Text":"Example 2."},{"Start":"08:02.250 ","End":"08:06.100","Text":"This time, let\u0027s take"},{"Start":"08:07.580 ","End":"08:20.680","Text":"x^2-3x+2/2x-4."},{"Start":"08:20.680 ","End":"08:26.930","Text":"What we want is bigger than 0, strict inequality."},{"Start":"08:26.930 ","End":"08:30.410","Text":"This time we still have 0 on the right,"},{"Start":"08:30.410 ","End":"08:33.545","Text":"but we have to factorize on the left."},{"Start":"08:33.545 ","End":"08:35.450","Text":"That\u0027s not being done for us."},{"Start":"08:35.450 ","End":"08:39.065","Text":"I\u0027ll just tell you the factorization."},{"Start":"08:39.065 ","End":"08:41.240","Text":"We won\u0027t waste time on that."},{"Start":"08:41.240 ","End":"08:46.500","Text":"This comes out to x-2 times"},{"Start":"08:46.500 ","End":"08:52.370","Text":"x-1 because you can check by multiplying out x times x is x^2,"},{"Start":"08:52.370 ","End":"08:56.825","Text":"-2x-x is -3x and plus 2."},{"Start":"08:56.825 ","End":"08:59.545","Text":"On the denominator,"},{"Start":"08:59.545 ","End":"09:01.890","Text":"we can take the 2 out."},{"Start":"09:01.890 ","End":"09:10.125","Text":"We have 2(x-2) and this is bigger than 0."},{"Start":"09:10.125 ","End":"09:12.290","Text":"As in the previous one,"},{"Start":"09:12.290 ","End":"09:14.315","Text":"I just copied this bit."},{"Start":"09:14.315 ","End":"09:18.910","Text":"We want the roots of the numerator and the roots of the denominator separately."},{"Start":"09:18.910 ","End":"09:23.440","Text":"In the numerator, the roots are 2 and 1."},{"Start":"09:23.440 ","End":"09:27.085","Text":"In the denominator,"},{"Start":"09:27.085 ","End":"09:29.680","Text":"the root is 2."},{"Start":"09:29.680 ","End":"09:31.960","Text":"Notice that there\u0027s a repetition."},{"Start":"09:31.960 ","End":"09:35.280","Text":"Looking back here, we see that"},{"Start":"09:35.280 ","End":"09:38.925","Text":"there\u0027s an x minus 2 in the numerator and the denominator,"},{"Start":"09:38.925 ","End":"09:47.770","Text":"and we can cancel it but we have to note that because it\u0027s a root of the denominator,"},{"Start":"09:47.770 ","End":"09:50.845","Text":"we still can\u0027t have x=2."},{"Start":"09:50.845 ","End":"09:54.520","Text":"This cancellation only work provided that we still"},{"Start":"09:54.520 ","End":"10:01.100","Text":"keep the restriction that x is not equal to 2."},{"Start":"10:02.910 ","End":"10:05.125","Text":"If that\u0027s the case,"},{"Start":"10:05.125 ","End":"10:12.063","Text":"I need a bit more space here, what we have,"},{"Start":"10:12.063 ","End":"10:18.010","Text":"is x minus 1 over 2 is bigger than 0,"},{"Start":"10:18.010 ","End":"10:23.755","Text":"but we still have that restriction that x cannot equal 2."},{"Start":"10:23.755 ","End":"10:26.350","Text":"Here\u0027s a number line,"},{"Start":"10:26.350 ","End":"10:32.335","Text":"but I have to mark on the roots all of them,"},{"Start":"10:32.335 ","End":"10:36.850","Text":"even the 2, just because I canceled it doesn\u0027t mean that I exclude it,"},{"Start":"10:36.850 ","End":"10:42.475","Text":"so I put here 1 and 2."},{"Start":"10:42.475 ","End":"10:46.120","Text":"Now, because 2 appears in the denominator,"},{"Start":"10:46.120 ","End":"10:47.620","Text":"it\u0027s a forbidden value,"},{"Start":"10:47.620 ","End":"10:54.290","Text":"so it gets a hallow dot even though it appears as a root of enumerator also."},{"Start":"10:54.510 ","End":"11:03.100","Text":"2 is the hollow dot and 1 is a solid dot because that\u0027s just the root of the numerator."},{"Start":"11:03.100 ","End":"11:04.930","Text":"Now we have, again,"},{"Start":"11:04.930 ","End":"11:08.350","Text":"3 regions, 3 intervals."},{"Start":"11:08.350 ","End":"11:14.620","Text":"Put some dotted lines here and here and once again, sample points."},{"Start":"11:14.620 ","End":"11:17.035","Text":"I\u0027ll take 0 here,"},{"Start":"11:17.035 ","End":"11:19.960","Text":"and here I\u0027ll take,"},{"Start":"11:19.960 ","End":"11:25.100","Text":"let\u0027s see, 1.5 and here 3."},{"Start":"11:25.620 ","End":"11:29.890","Text":"If I plug in 0, I get negative,"},{"Start":"11:29.890 ","End":"11:31.750","Text":"I won\u0027t use the snake method here,"},{"Start":"11:31.750 ","End":"11:34.570","Text":"I\u0027ll just put the minuses and pluses."},{"Start":"11:34.570 ","End":"11:38.350","Text":"Actually, you can see here that this inequality,"},{"Start":"11:38.350 ","End":"11:39.400","Text":"if I multiply it by 2,"},{"Start":"11:39.400 ","End":"11:42.325","Text":"is just x bigger than 1."},{"Start":"11:42.325 ","End":"11:44.290","Text":"Less than 1 it\u0027s negative,"},{"Start":"11:44.290 ","End":"11:46.240","Text":"bigger than 1, it\u0027s positive,"},{"Start":"11:46.240 ","End":"11:51.865","Text":"or I could have substituted the values, it doesn\u0027t matter."},{"Start":"11:51.865 ","End":"11:55.000","Text":"Either way, you get to minus, plus,"},{"Start":"11:55.000 ","End":"11:58.495","Text":"plus and here it\u0027s 0 and here it\u0027s undefined."},{"Start":"11:58.495 ","End":"12:05.905","Text":"Now what we\u0027re interested in is the bigger than 0 or originally from here,"},{"Start":"12:05.905 ","End":"12:08.245","Text":"which means we only want positives."},{"Start":"12:08.245 ","End":"12:09.850","Text":"I want this and this,"},{"Start":"12:09.850 ","End":"12:13.540","Text":"I don\u0027t want undefined and I don\u0027t want 0."},{"Start":"12:13.540 ","End":"12:18.835","Text":"If we write that in terms of x,"},{"Start":"12:18.835 ","End":"12:23.320","Text":"then we get that x could be,"},{"Start":"12:23.320 ","End":"12:27.550","Text":"well, first of all, sketch it from here to here."},{"Start":"12:27.550 ","End":"12:31.120","Text":"I excluded this because we only want the positive,"},{"Start":"12:31.120 ","End":"12:32.830","Text":"the positive doesn\u0027t include this,"},{"Start":"12:32.830 ","End":"12:38.785","Text":"and also from here onwards."},{"Start":"12:38.785 ","End":"12:41.260","Text":"Now there\u0027s 2 ways to describe this,"},{"Start":"12:41.260 ","End":"12:44.395","Text":"one way I could describe it is to say,"},{"Start":"12:44.395 ","End":"12:50.390","Text":"x is between 2 and 1,"},{"Start":"12:50.940 ","End":"12:56.110","Text":"or x is bigger than 2,"},{"Start":"12:56.110 ","End":"12:59.270","Text":"and that would be fine as an answer."},{"Start":"12:59.730 ","End":"13:03.670","Text":"Another possibility would be to say,"},{"Start":"13:03.670 ","End":"13:06.985","Text":"it\u0027s from 1 onwards,"},{"Start":"13:06.985 ","End":"13:09.700","Text":"with the exception of a little hole here."},{"Start":"13:09.700 ","End":"13:19.060","Text":"You could say x is bigger than 1 but is end,"},{"Start":"13:19.060 ","End":"13:22.975","Text":"x not equal to 2."},{"Start":"13:22.975 ","End":"13:27.025","Text":"You can either say it\u0027s this line segment from 1 onwards,"},{"Start":"13:27.025 ","End":"13:32.995","Text":"but a O in it, or you could say it\u0027s this bit and this bit combined with an O."},{"Start":"13:32.995 ","End":"13:36.410","Text":"Both of these are acceptable."},{"Start":"13:37.860 ","End":"13:40.810","Text":"I\u0027ll pick 1 of them at random,"},{"Start":"13:40.810 ","End":"13:44.500","Text":"let\u0027s say the first one and call this the answer."},{"Start":"13:44.500 ","End":"13:47.320","Text":"This is the alternative."},{"Start":"13:47.320 ","End":"13:49.675","Text":"That\u0027s number 2,"},{"Start":"13:49.675 ","End":"13:52.310","Text":"let\u0027s go on and do another one."},{"Start":"13:52.620 ","End":"13:58.570","Text":"Number 3, x minus 2"},{"Start":"13:58.570 ","End":"14:09.715","Text":"over 2x plus 5 is bigger or equal to minus 1."},{"Start":"14:09.715 ","End":"14:16.210","Text":"This time it\u0027s different because we don\u0027t have a 0 on the right-hand side,"},{"Start":"14:16.210 ","End":"14:18.640","Text":"so we have to do a little bit of work,"},{"Start":"14:18.640 ","End":"14:21.715","Text":"some algebra to get the 0 on the right."},{"Start":"14:21.715 ","End":"14:24.910","Text":"What I\u0027ll do is add 1 to both sides,"},{"Start":"14:24.910 ","End":"14:29.245","Text":"so we get x minus 2 over"},{"Start":"14:29.245 ","End":"14:36.475","Text":"2x plus 5 plus 1 bigger or equal to 0."},{"Start":"14:36.475 ","End":"14:40.075","Text":"Now I\u0027ll put this with a common denominator."},{"Start":"14:40.075 ","End":"14:50.340","Text":"The common denominator will be 2x plus 5 and here we have x minus 2,"},{"Start":"14:50.340 ","End":"14:56.010","Text":"and the 1 we can write as 2x plus 5 over 2x plus 5."},{"Start":"14:56.010 ","End":"14:58.190","Text":"This is what I have here,"},{"Start":"14:58.190 ","End":"15:00.039","Text":"bigger or equal to 0."},{"Start":"15:00.039 ","End":"15:02.320","Text":"I\u0027ll continue over here."},{"Start":"15:02.320 ","End":"15:09.565","Text":"Collecting like terms what I have is 3x minus 2 plus 5 is plus 3,"},{"Start":"15:09.565 ","End":"15:18.040","Text":"3x plus 3 over 2x plus 5,"},{"Start":"15:18.040 ","End":"15:20.260","Text":"bigger or equal to 0,"},{"Start":"15:20.260 ","End":"15:25.825","Text":"and I can take the 3 out as well."},{"Start":"15:25.825 ","End":"15:32.740","Text":"3 times x plus 1 over,"},{"Start":"15:32.740 ","End":"15:40.660","Text":"and I\u0027ll also take 2 out of here because then I can easier see the root."},{"Start":"15:40.660 ","End":"15:46.330","Text":"Twice x plus 2.5 or 5 over 2,"},{"Start":"15:46.330 ","End":"15:51.890","Text":"whichever, is bigger or equal to 0."},{"Start":"15:52.710 ","End":"15:56.980","Text":"I need the roots separately for the numerator and the denominator."},{"Start":"15:56.980 ","End":"15:58.570","Text":"For the numerator,"},{"Start":"15:58.570 ","End":"16:03.820","Text":"there\u0027s only 1 root and that is minus 1,"},{"Start":"16:03.820 ","End":"16:06.985","Text":"and for the denominator,"},{"Start":"16:06.985 ","End":"16:12.865","Text":"there\u0027s also only 1 root and that is minus 2.5."},{"Start":"16:12.865 ","End":"16:16.240","Text":"The difference is that this is an illegal value,"},{"Start":"16:16.240 ","End":"16:19.510","Text":"I can\u0027t substitute it into the expression."},{"Start":"16:19.510 ","End":"16:26.275","Text":"This root of the numerator is a value which makes the whole expression 0."},{"Start":"16:26.275 ","End":"16:30.175","Text":"Let\u0027s draw the number line."},{"Start":"16:30.175 ","End":"16:34.675","Text":"Number line and they\u0027ll put these 2 points in,"},{"Start":"16:34.675 ","End":"16:40.450","Text":"minus 2.5 is to the left of minus 1."},{"Start":"16:40.450 ","End":"16:47.020","Text":"Now separate the intervals with some dotted lines and I have 3 intervals."},{"Start":"16:47.020 ","End":"16:49.540","Text":"I\u0027ll take sample points,"},{"Start":"16:49.540 ","End":"16:56.620","Text":"minus 3, minus 2 and 0."},{"Start":"16:56.620 ","End":"16:59.080","Text":"Last time I didn\u0027t use this snake,"},{"Start":"16:59.080 ","End":"17:01.855","Text":"this time I will for a change."},{"Start":"17:01.855 ","End":"17:07.040","Text":"Sample points at minus 3,"},{"Start":"17:07.050 ","End":"17:13.000","Text":"I get this thing is negative and this thing is negative."},{"Start":"17:13.000 ","End":"17:15.070","Text":"By the way, the 3 over 2 I can ignore they\u0027re"},{"Start":"17:15.070 ","End":"17:17.695","Text":"both positive so they won\u0027t change the sign."},{"Start":"17:17.695 ","End":"17:20.200","Text":"Negative of a negative is a positive,"},{"Start":"17:20.200 ","End":"17:25.450","Text":"so I put a point above the axis for this interval,"},{"Start":"17:25.450 ","End":"17:29.815","Text":"minus 2 plus 1 is negative,"},{"Start":"17:29.815 ","End":"17:32.170","Text":"minus 2 plus 2.5 is positive,"},{"Start":"17:32.170 ","End":"17:36.279","Text":"negative of a positive is negative,"},{"Start":"17:36.279 ","End":"17:37.360","Text":"so minus 2,"},{"Start":"17:37.360 ","End":"17:40.645","Text":"I go down below the axis."},{"Start":"17:40.645 ","End":"17:46.135","Text":"At 0, both of these are positive."},{"Start":"17:46.135 ","End":"17:48.520","Text":"As I said, the 3 over 2 is always positive,"},{"Start":"17:48.520 ","End":"17:51.550","Text":"so got something positive here."},{"Start":"17:51.550 ","End":"17:53.575","Text":"These 2 special points,"},{"Start":"17:53.575 ","End":"18:00.100","Text":"the root of the numerator minus 1 is a solid dot,"},{"Start":"18:00.100 ","End":"18:03.370","Text":"and the denominator,"},{"Start":"18:03.370 ","End":"18:09.385","Text":"a hollow dot, because this is a place where the expression is undefined."},{"Start":"18:09.385 ","End":"18:13.530","Text":"The snake will look something like this,"},{"Start":"18:13.530 ","End":"18:15.970","Text":"I don\u0027t what happens here,"},{"Start":"18:15.970 ","End":"18:18.670","Text":"it picks up again."},{"Start":"18:18.670 ","End":"18:22.705","Text":"Positive here, 0 here, positive here,"},{"Start":"18:22.705 ","End":"18:27.470","Text":"indefinitely positive, indefinitely positive here."},{"Start":"18:29.580 ","End":"18:33.550","Text":"Now you want to see what we\u0027re collecting,"},{"Start":"18:33.550 ","End":"18:38.710","Text":"we are collecting bigger or equal to."},{"Start":"18:38.710 ","End":"18:42.850","Text":"Bigger or equal to 0 means positive or 0,"},{"Start":"18:42.850 ","End":"18:45.835","Text":"now, where are we positive or 0?"},{"Start":"18:45.835 ","End":"18:48.460","Text":"I don\u0027t use the shading method this time,"},{"Start":"18:48.460 ","End":"18:51.665","Text":"positive above the axis here,"},{"Start":"18:51.665 ","End":"18:54.410","Text":"but not including this point."},{"Start":"18:54.410 ","End":"19:02.390","Text":"Positive also here and including the 0,"},{"Start":"19:02.390 ","End":"19:04.705","Text":"let\u0027s just note that."},{"Start":"19:04.705 ","End":"19:07.355","Text":"If we sketch it,"},{"Start":"19:07.355 ","End":"19:10.654","Text":"the interval we have"},{"Start":"19:10.654 ","End":"19:18.830","Text":"from minus 2.5 and downwards forever,"},{"Start":"19:18.830 ","End":"19:22.390","Text":"it\u0027s positive from minus 1,"},{"Start":"19:22.390 ","End":"19:25.655","Text":"including the minus 1, this is 0."},{"Start":"19:25.655 ","End":"19:29.300","Text":"Here, it\u0027s positive and it goes on indefinitely,"},{"Start":"19:29.300 ","End":"19:36.200","Text":"and if I was to write this in terms of x,"},{"Start":"19:36.200 ","End":"19:42.455","Text":"this is x less than minus 2.5,"},{"Start":"19:42.455 ","End":"19:48.785","Text":"or x bigger or equal to minus 1."},{"Start":"19:48.785 ","End":"19:52.680","Text":"That\u0027s the answer and I\u0027ll highlight it."},{"Start":"19:53.150 ","End":"20:01.100","Text":"I\u0027m done for this section on rational inequalities because there"},{"Start":"20:01.100 ","End":"20:08.515","Text":"are many more solved example exercises outside of the tutorial,"},{"Start":"20:08.515 ","End":"20:13.890","Text":"and I suggest you go and look at those. That\u0027s it then."}],"Thumbnail":null,"ID":5070},{"Watched":false,"Name":"Exercise 1","Duration":"20m 41s","ChapterTopicVideoID":5070,"CourseChapterTopicPlaylistID":3877,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.250","Text":"In this exercise, we have 5 inequalities to solve."},{"Start":"00:05.250 ","End":"00:08.480","Text":"Each of them involves what we call rational functions,"},{"Start":"00:08.480 ","End":"00:11.325","Text":"or polynomial over a polynomial."},{"Start":"00:11.325 ","End":"00:15.240","Text":"Let\u0027s start with part a,"},{"Start":"00:15.240 ","End":"00:17.220","Text":"and what we do in this case,"},{"Start":"00:17.220 ","End":"00:20.385","Text":"is very similar to the case of polynomials."},{"Start":"00:20.385 ","End":"00:23.250","Text":"With polynomials, we find the roots,"},{"Start":"00:23.250 ","End":"00:25.830","Text":"which means the values of x,"},{"Start":"00:25.830 ","End":"00:27.430","Text":"which makes something 0,"},{"Start":"00:27.430 ","End":"00:28.785","Text":"one of the factors."},{"Start":"00:28.785 ","End":"00:30.390","Text":"Here we do the same,"},{"Start":"00:30.390 ","End":"00:32.819","Text":"but they\u0027re not called roots."},{"Start":"00:32.819 ","End":"00:36.330","Text":"We have a factor here, 2x minus 1."},{"Start":"00:36.330 ","End":"00:37.890","Text":"What makes it 0?"},{"Start":"00:37.890 ","End":"00:40.110","Text":"X equals 1,1/2,"},{"Start":"00:40.110 ","End":"00:42.375","Text":"I\u0027ll just write it next to the factor."},{"Start":"00:42.375 ","End":"00:45.615","Text":"Here what makes x minus 5, 0?"},{"Start":"00:45.615 ","End":"00:50.235","Text":"Well, it\u0027s clear that\u0027s 5, 1/2 and 5,"},{"Start":"00:50.235 ","End":"00:51.710","Text":"they\u0027re not roots anymore,"},{"Start":"00:51.710 ","End":"00:57.275","Text":"but they are the places which make some factors 0 on the top, or on the bottom."},{"Start":"00:57.275 ","End":"01:01.640","Text":"Now, in the first example you\u0027ll see why it makes a difference if it\u0027s on top,"},{"Start":"01:01.640 ","End":"01:03.260","Text":"or it\u0027s on bottom."},{"Start":"01:03.260 ","End":"01:09.395","Text":"Now, both these numbers 1/2 and 5 placed on the number line in the correct order,"},{"Start":"01:09.395 ","End":"01:14.865","Text":"say 1/2 is here and 5 is here,"},{"Start":"01:14.865 ","End":"01:16.745","Text":"but there\u0027s a difference."},{"Start":"01:16.745 ","End":"01:21.365","Text":"In the case of the numerator being 0,"},{"Start":"01:21.365 ","End":"01:25.465","Text":"I put it as a solid dot."},{"Start":"01:25.465 ","End":"01:30.545","Text":"Something like this which shows me that the polynomial is 0 here."},{"Start":"01:30.545 ","End":"01:32.995","Text":"But in the case of 5,"},{"Start":"01:32.995 ","End":"01:35.885","Text":"I actually remove a point,"},{"Start":"01:35.885 ","End":"01:39.410","Text":"and put a hollow circle,"},{"Start":"01:39.410 ","End":"01:45.170","Text":"let\u0027s say, because x cannot be 5 in this polynomial,"},{"Start":"01:45.170 ","End":"01:48.185","Text":"we\u0027re not allowed to divide by 0,"},{"Start":"01:48.185 ","End":"01:50.825","Text":"so 5 makes no sense."},{"Start":"01:50.825 ","End":"01:56.480","Text":"This is one system of indicating that this number is completely ruled out."},{"Start":"01:56.480 ","End":"01:58.390","Text":"You cannot substitute it."},{"Start":"01:58.390 ","End":"02:02.270","Text":"That\u0027s the difference between numerator 0 and denominator 0."},{"Start":"02:02.270 ","End":"02:06.470","Text":"In this case, it is not really called the root,"},{"Start":"02:06.470 ","End":"02:08.975","Text":"but it\u0027s the number that makes the thing 0,"},{"Start":"02:08.975 ","End":"02:11.825","Text":"and this just makes the thing undefined."},{"Start":"02:11.825 ","End":"02:16.610","Text":"But, we proceed as usual in the intervals in between,"},{"Start":"02:16.610 ","End":"02:18.290","Text":"we can take sample values."},{"Start":"02:18.290 ","End":"02:26.505","Text":"For example, here I could choose 0, between 1/2,"},{"Start":"02:26.505 ","End":"02:28.845","Text":"and 5 I could choose 1,"},{"Start":"02:28.845 ","End":"02:31.185","Text":"and after 5 I could choose 6,"},{"Start":"02:31.185 ","End":"02:36.270","Text":"and I can still substitute these and say, well, if x is 0,"},{"Start":"02:36.270 ","End":"02:39.210","Text":"in the numerator I\u0027ve got negative,"},{"Start":"02:39.210 ","End":"02:42.465","Text":"and here also negative, so it\u0027s positive."},{"Start":"02:42.465 ","End":"02:48.085","Text":"I just put at any point above the line just to show that I have positive there."},{"Start":"02:48.085 ","End":"02:51.300","Text":"When x is 1,"},{"Start":"02:51.300 ","End":"02:55.335","Text":"I would say twice 1 minus 1 is positive,"},{"Start":"02:55.335 ","End":"02:57.120","Text":"1 minus 5 is negative,"},{"Start":"02:57.120 ","End":"02:59.085","Text":"positive over negative is negative,"},{"Start":"02:59.085 ","End":"03:04.925","Text":"so we put some point below the number line to say negative."},{"Start":"03:04.925 ","End":"03:07.620","Text":"Then the other sample 0.6,"},{"Start":"03:07.620 ","End":"03:09.390","Text":"twice 6 minus 1 clearly,"},{"Start":"03:09.390 ","End":"03:12.205","Text":"positive 6 minus 5 also positive,"},{"Start":"03:12.205 ","End":"03:15.605","Text":"positive over positive is positive,"},{"Start":"03:15.605 ","End":"03:22.355","Text":"and we can still draw a snake so-called."},{"Start":"03:22.355 ","End":"03:32.535","Text":"However, I don\u0027t make it go through this hollow point,"},{"Start":"03:32.535 ","End":"03:37.270","Text":"may be close to it, you could even boarder on it."},{"Start":"03:38.810 ","End":"03:40.995","Text":"Something like this,"},{"Start":"03:40.995 ","End":"03:44.540","Text":"and optionally, I like to draw arrows."},{"Start":"03:44.540 ","End":"03:51.080","Text":"It\u0027s not standard to show that it goes on indefinitely in both directions."},{"Start":"03:51.080 ","End":"03:54.740","Text":"Now we examine as before what we want,"},{"Start":"03:54.740 ","End":"03:57.050","Text":"we want less than or equal to 0."},{"Start":"03:57.050 ","End":"04:02.165","Text":"Less than or equal to 0 means below the line, or on the line."},{"Start":"04:02.165 ","End":"04:07.205","Text":"Now, below, if I use the shading method rather than the highlighting,"},{"Start":"04:07.205 ","End":"04:09.045","Text":"this is what I get."},{"Start":"04:09.045 ","End":"04:11.120","Text":"Where we have to be careful,"},{"Start":"04:11.120 ","End":"04:13.580","Text":"is with these 2 points."},{"Start":"04:13.580 ","End":"04:18.875","Text":"Although we want to allow points on the line,"},{"Start":"04:18.875 ","End":"04:21.620","Text":"this one is okay to take,"},{"Start":"04:21.620 ","End":"04:27.950","Text":"but this one we can never take it because we can\u0027t define the polynomial at this point."},{"Start":"04:27.950 ","End":"04:34.370","Text":"What we end up with is 1/2 less than or equal to,"},{"Start":"04:34.370 ","End":"04:38.255","Text":"I\u0027m emphasizing this less than or equal to x,"},{"Start":"04:38.255 ","End":"04:41.715","Text":"but strictly less than 5."},{"Start":"04:41.715 ","End":"04:43.490","Text":"In the case of polynomials,"},{"Start":"04:43.490 ","End":"04:44.900","Text":"we would\u0027ve taken the 5,"},{"Start":"04:44.900 ","End":"04:46.460","Text":"but with rational functions,"},{"Start":"04:46.460 ","End":"04:48.590","Text":"if it\u0027s a denominator 0,"},{"Start":"04:48.590 ","End":"04:50.000","Text":"not a numerator 0,"},{"Start":"04:50.000 ","End":"04:51.890","Text":"then we can\u0027t take it."},{"Start":"04:51.890 ","End":"04:54.529","Text":"That\u0027s the essential difference,"},{"Start":"04:54.529 ","End":"04:57.380","Text":"and if you followed what I said here,"},{"Start":"04:57.380 ","End":"05:01.415","Text":"you\u0027ll have no problem with the rest of the rational functions."},{"Start":"05:01.415 ","End":"05:05.680","Text":"Now let\u0027s go on to the next one, which is b."},{"Start":"05:05.680 ","End":"05:10.460","Text":"The first thing we want to do is factorize both numerator, and denominator."},{"Start":"05:10.460 ","End":"05:12.515","Text":"In the numerator, there\u0027s nothing to be done."},{"Start":"05:12.515 ","End":"05:13.970","Text":"In the denominator,"},{"Start":"05:13.970 ","End":"05:19.820","Text":"we can use the difference of squares formulas because this is x^2- 4^2."},{"Start":"05:19.820 ","End":"05:29.565","Text":"We can write it as (x-4)(x+4) bigger than 0,"},{"Start":"05:29.565 ","End":"05:38.550","Text":"and the places where one of the factors is 0 when x=4,"},{"Start":"05:38.550 ","End":"05:44.070","Text":"that would make this one 0 and x=-4,"},{"Start":"05:44.070 ","End":"05:46.410","Text":"which would make this one 0."},{"Start":"05:46.410 ","End":"05:52.675","Text":"The only interesting points for the sketch are 4 and -4,"},{"Start":"05:52.675 ","End":"05:58.445","Text":"here\u0027s a number line and on it I place my interesting points -4 and 4."},{"Start":"05:58.445 ","End":"06:04.910","Text":"In this case, both of these places are 0s of the denominator,"},{"Start":"06:04.910 ","End":"06:13.805","Text":"so they get a missing point or hollow circle, like so."},{"Start":"06:13.805 ","End":"06:17.735","Text":"Okay, it didn\u0027t come out quite right, but forgive me."},{"Start":"06:17.735 ","End":"06:20.990","Text":"Now we choose sample points and substitute."},{"Start":"06:20.990 ","End":"06:23.135","Text":"We can do this in our heads if we want."},{"Start":"06:23.135 ","End":"06:28.478","Text":"If we take -5,"},{"Start":"06:28.478 ","End":"06:29.835","Text":"and we substitute it,"},{"Start":"06:29.835 ","End":"06:36.670","Text":"-5-4 is negative, -5+4 is negative."},{"Start":"06:38.180 ","End":"06:44.250","Text":"Numerator is positive because basically get 2 negatives makes it a positive,"},{"Start":"06:44.250 ","End":"06:46.685","Text":"so we get a point somewhere here."},{"Start":"06:46.685 ","End":"06:49.925","Text":"If I take something in-between,"},{"Start":"06:49.925 ","End":"06:52.680","Text":"-4 and 4, for example,"},{"Start":"06:52.680 ","End":"07:00.293","Text":"say 0, 0-4 is negative, 0+4 is positive."},{"Start":"07:00.293 ","End":"07:05.100","Text":"Negative times positive is negative,"},{"Start":"07:05.100 ","End":"07:08.625","Text":"so at 0 we are below,"},{"Start":"07:08.625 ","End":"07:13.470","Text":"and then I could take x=5,"},{"Start":"07:13.470 ","End":"07:17.420","Text":"5-4 is positive, 5+4 is positive,"},{"Start":"07:17.420 ","End":"07:21.980","Text":"positive times positive is positive, something like this."},{"Start":"07:21.980 ","End":"07:26.165","Text":"Then I can draw the sketch,"},{"Start":"07:26.165 ","End":"07:32.090","Text":"which is symbolic that here we are positive,"},{"Start":"07:32.090 ","End":"07:35.850","Text":"here we are negative,"},{"Start":"07:35.850 ","End":"07:39.620","Text":"and here we are positive, and so on,"},{"Start":"07:39.620 ","End":"07:41.845","Text":"and definitely with each of these,"},{"Start":"07:41.845 ","End":"07:43.160","Text":"but they never cross again."},{"Start":"07:43.160 ","End":"07:45.155","Text":"They never cross the number line again, of course,"},{"Start":"07:45.155 ","End":"07:50.455","Text":"because we found everywhere where the numerator or denominator is 0."},{"Start":"07:50.455 ","End":"07:52.970","Text":"Now we just have to see what are we searching for."},{"Start":"07:52.970 ","End":"07:56.389","Text":"We\u0027re searching for bigger than 0,"},{"Start":"07:56.389 ","End":"08:02.385","Text":"which means strictly above the line."},{"Start":"08:02.385 ","End":"08:08.670","Text":"It\u0027s this bit here and this bit here,"},{"Start":"08:08.670 ","End":"08:11.180","Text":"and in any event,"},{"Start":"08:11.180 ","End":"08:15.020","Text":"we wouldn\u0027t take these 2 points because we have to be strictly above 0."},{"Start":"08:15.020 ","End":"08:17.930","Text":"The fact that they\u0027re illegal, doesn\u0027t matter."},{"Start":"08:17.930 ","End":"08:23.100","Text":"We still end up with x less than -4,"},{"Start":"08:23.100 ","End":"08:26.940","Text":"or x greater than 4,"},{"Start":"08:26.940 ","End":"08:31.480","Text":"and onto the next one."},{"Start":"08:32.890 ","End":"08:35.930","Text":"This one\u0027s already factorized."},{"Start":"08:35.930 ","End":"08:38.810","Text":"The only factor with x in it is this one."},{"Start":"08:38.810 ","End":"08:43.505","Text":"The only place it can be 0 is when x equals 1,"},{"Start":"08:43.505 ","End":"08:49.339","Text":"so x=1 is a 0 of the numerator or denominator."},{"Start":"08:49.339 ","End":"08:50.780","Text":"It happens to be denominator,"},{"Start":"08:50.780 ","End":"08:54.725","Text":"which means that we\u0027ll draw the hollow circle rather than the solid circle."},{"Start":"08:54.725 ","End":"08:59.330","Text":"But hollow circle in this context"},{"Start":"08:59.330 ","End":"09:03.845","Text":"means something different than it does when we\u0027re drawing intervals."},{"Start":"09:03.845 ","End":"09:07.865","Text":"It just means there\u0027s a missing value."},{"Start":"09:07.865 ","End":"09:13.625","Text":"Perhaps I should indicate it with a hollow square or something. Why not?"},{"Start":"09:13.625 ","End":"09:17.595","Text":"Let\u0027s do that. Anyway,"},{"Start":"09:17.595 ","End":"09:27.100","Text":"I put the value x equals 1 here and it\u0027s not defined at 1 because it\u0027s a denominator 0."},{"Start":"09:28.190 ","End":"09:31.010","Text":"I\u0027ll change my mind about what I said before."},{"Start":"09:31.010 ","End":"09:34.190","Text":"I\u0027ll keep using a circle, but in this case,"},{"Start":"09:34.190 ","End":"09:40.280","Text":"it just means that the missing point in the domain of definition of the function,"},{"Start":"09:40.280 ","End":"09:44.680","Text":"the values we can substitute here to not include 1."},{"Start":"09:44.680 ","End":"09:50.860","Text":"Before we look at the intervals and see which is positive, which is negative."},{"Start":"09:50.860 ","End":"09:55.950","Text":"For those who don\u0027t like the snake we\u0027ll do it the other way this time."},{"Start":"09:56.230 ","End":"10:00.140","Text":"Just in general, you take a sample value,"},{"Start":"10:00.140 ","End":"10:04.880","Text":"for example, I can take 0 here and 2 here."},{"Start":"10:04.880 ","End":"10:06.740","Text":"If I substitute 0,"},{"Start":"10:06.740 ","End":"10:15.150","Text":"I get 1 over minus 3 times minus 1 because 0 minus 1 is minus 1 minus,"},{"Start":"10:15.150 ","End":"10:17.085","Text":"so one of the minuses is minus 1,"},{"Start":"10:17.085 ","End":"10:22.535","Text":"there\u0027s 2 negatives, which makes it a positive so here it\u0027s positive."},{"Start":"10:22.535 ","End":"10:25.265","Text":"If I put x=2,"},{"Start":"10:25.265 ","End":"10:28.310","Text":"2-1 is positive,"},{"Start":"10:28.310 ","End":"10:29.450","Text":"and this is negative,"},{"Start":"10:29.450 ","End":"10:30.890","Text":"and because of this negative,"},{"Start":"10:30.890 ","End":"10:33.455","Text":"everything comes out to be negative."},{"Start":"10:33.455 ","End":"10:38.815","Text":"I won\u0027t bother with the sketch of a snake,"},{"Start":"10:38.815 ","End":"10:41.840","Text":"and what do we want?"},{"Start":"10:41.840 ","End":"10:43.865","Text":"We want less than 0."},{"Start":"10:43.865 ","End":"10:47.925","Text":"Less than 0 means minus,"},{"Start":"10:47.925 ","End":"10:49.905","Text":"so this is the bit we want,"},{"Start":"10:49.905 ","End":"10:54.615","Text":"and this bit curves from 1 onto the right."},{"Start":"10:54.615 ","End":"10:59.350","Text":"We just get x bigger than 1."},{"Start":"11:03.930 ","End":"11:07.555","Text":"Part d, just scroll down a bit,"},{"Start":"11:07.555 ","End":"11:11.080","Text":"so here we are."},{"Start":"11:11.080 ","End":"11:15.610","Text":"In part d, the numerator is factorized,"},{"Start":"11:15.610 ","End":"11:19.060","Text":"the denominator still needs to be factorized."},{"Start":"11:19.060 ","End":"11:28.000","Text":"X minus 1 over x^2 minus 9 is x^2 minus 3^2."},{"Start":"11:28.000 ","End":"11:30.355","Text":"Using the difference of squares formula,"},{"Start":"11:30.355 ","End":"11:32.380","Text":"it\u0027s x minus 3,"},{"Start":"11:32.380 ","End":"11:37.705","Text":"x plus 3 is bigger than 0."},{"Start":"11:37.705 ","End":"11:41.485","Text":"The points we want to sketch that makes something 0 are here,"},{"Start":"11:41.485 ","End":"11:43.405","Text":"the value x=1,"},{"Start":"11:43.405 ","End":"11:45.760","Text":"here the value x=3,"},{"Start":"11:45.760 ","End":"11:50.320","Text":"and here the value x equals minus 3 makes something 0."},{"Start":"11:50.320 ","End":"11:52.435","Text":"We sketch a number line,"},{"Start":"11:52.435 ","End":"11:54.745","Text":"and I have to place these three points,"},{"Start":"11:54.745 ","End":"11:56.920","Text":"but you have to do it in the correct order."},{"Start":"11:56.920 ","End":"11:59.590","Text":"First we get a minus 3,"},{"Start":"11:59.590 ","End":"12:02.575","Text":"then we get 1,"},{"Start":"12:02.575 ","End":"12:05.905","Text":"and then we get plus 3."},{"Start":"12:05.905 ","End":"12:08.680","Text":"In the case of x=1,"},{"Start":"12:08.680 ","End":"12:12.055","Text":"I put a solid circle,"},{"Start":"12:12.055 ","End":"12:18.090","Text":"in this case to indicate that it gives a value of 0 for the polynomial."},{"Start":"12:18.090 ","End":"12:20.505","Text":"But in the other two cases,"},{"Start":"12:20.505 ","End":"12:25.405","Text":"I make a hole and I make a hole,"},{"Start":"12:25.405 ","End":"12:30.220","Text":"and I use the hollow circles to indicate that this point is missing."},{"Start":"12:30.220 ","End":"12:32.530","Text":"I just have to remove it from the number line as"},{"Start":"12:32.530 ","End":"12:35.695","Text":"far as what I can substitute in the polynomial."},{"Start":"12:35.695 ","End":"12:39.958","Text":"Now we can take sample values."},{"Start":"12:39.958 ","End":"12:43.030","Text":"Let\u0027s say if I put in,"},{"Start":"12:43.030 ","End":"12:46.840","Text":"we\u0027ll do it in our heads, minus 4,"},{"Start":"12:46.840 ","End":"12:49.990","Text":"so this is going to be negative,"},{"Start":"12:49.990 ","End":"12:52.795","Text":"negative and also negative,"},{"Start":"12:52.795 ","End":"12:54.775","Text":"3 negatives is a positive,"},{"Start":"12:54.775 ","End":"12:56.145","Text":"so a positive point."},{"Start":"12:56.145 ","End":"12:59.560","Text":"Let\u0027s to it with the snake method this time."},{"Start":"12:59.560 ","End":"13:01.900","Text":"Between minus 3 and 1,"},{"Start":"13:01.900 ","End":"13:05.305","Text":"I can choose 0, that\u0027s the easiest."},{"Start":"13:05.305 ","End":"13:06.640","Text":"0 minus 1,"},{"Start":"13:06.640 ","End":"13:08.740","Text":"so this is negative, minus 1,"},{"Start":"13:08.740 ","End":"13:10.390","Text":"minus 3 plus 3,"},{"Start":"13:10.390 ","End":"13:12.250","Text":"two negatives all together."},{"Start":"13:12.250 ","End":"13:16.900","Text":"I think I made a mistake in the previous one, sorry."},{"Start":"13:16.900 ","End":"13:19.150","Text":"In the first case, we have three negatives,"},{"Start":"13:19.150 ","End":"13:23.830","Text":"an odd number, so down here."},{"Start":"13:23.830 ","End":"13:27.445","Text":"At 0, we had the positive,"},{"Start":"13:27.445 ","End":"13:29.260","Text":"and now in this interval,"},{"Start":"13:29.260 ","End":"13:32.275","Text":"I\u0027ll choose the 2,"},{"Start":"13:32.275 ","End":"13:34.690","Text":"so 2 minus 1 is positive,"},{"Start":"13:34.690 ","End":"13:36.475","Text":"2 minus 3 is negative,"},{"Start":"13:36.475 ","End":"13:40.701","Text":"then positive, so a single negative, so negative."},{"Start":"13:40.701 ","End":"13:45.670","Text":"Finally, for the last interval,"},{"Start":"13:45.670 ","End":"13:47.485","Text":"I\u0027ll choose x equals 4."},{"Start":"13:47.485 ","End":"13:51.670","Text":"When x is 4, all these three factors are positive,"},{"Start":"13:51.670 ","End":"13:53.545","Text":"so we get,"},{"Start":"13:53.545 ","End":"13:56.990","Text":"again, a positive."},{"Start":"13:58.200 ","End":"14:04.799","Text":"Then this time, I\u0027ll use the snake method of illustration."},{"Start":"14:04.799 ","End":"14:07.180","Text":"Something that goes through here,"},{"Start":"14:07.180 ","End":"14:10.330","Text":"it does leave an extra bit of gap."},{"Start":"14:10.330 ","End":"14:15.728","Text":"It doesn\u0027t really matter as long as you understand that this number is illegal,"},{"Start":"14:15.728 ","End":"14:16.870","Text":"in case we want to take it,"},{"Start":"14:16.870 ","End":"14:19.195","Text":"which we may not want to do anyway."},{"Start":"14:19.195 ","End":"14:25.705","Text":"Here it does go through because that\u0027s a place where the numerator is 0. That\u0027s fine."},{"Start":"14:25.705 ","End":"14:31.135","Text":"Then the gap here and on indefinitely,"},{"Start":"14:31.135 ","End":"14:35.210","Text":"say I put an arrow here and here."},{"Start":"14:36.090 ","End":"14:39.805","Text":"The arrows might think it\u0027s not necessarily standard."},{"Start":"14:39.805 ","End":"14:42.460","Text":"It\u0027s like to say the thing doesn\u0027t stop here,"},{"Start":"14:42.460 ","End":"14:45.190","Text":"it keeps on being positive forever."},{"Start":"14:45.190 ","End":"14:46.510","Text":"In this direction,"},{"Start":"14:46.510 ","End":"14:50.630","Text":"it\u0027s negative, down to minus infinity in this direction."},{"Start":"14:50.910 ","End":"14:52.960","Text":"What are we collecting?"},{"Start":"14:52.960 ","End":"14:57.980","Text":"We\u0027re collecting positives, strictly positives, strictly above."},{"Start":"15:00.330 ","End":"15:03.100","Text":"For a change, instead of shading,"},{"Start":"15:03.100 ","End":"15:04.270","Text":"I\u0027ll use highlighting,"},{"Start":"15:04.270 ","End":"15:07.400","Text":"and these are just minor variations."},{"Start":"15:07.830 ","End":"15:12.460","Text":"Positive is above the axis, not including."},{"Start":"15:12.460 ","End":"15:14.815","Text":"I\u0027ll make sure I stop just before that point."},{"Start":"15:14.815 ","End":"15:20.065","Text":"The point is legal but we\u0027re collecting only positives,"},{"Start":"15:20.065 ","End":"15:24.655","Text":"and here, positive all this, and so on."},{"Start":"15:24.655 ","End":"15:29.035","Text":"What do we get if we translate this into what x"},{"Start":"15:29.035 ","End":"15:34.975","Text":"is for this bit minus 3 less than x, less than 1?"},{"Start":"15:34.975 ","End":"15:39.580","Text":"For this bit, x bigger than 3,"},{"Start":"15:39.580 ","End":"15:42.880","Text":"and this is our answer,"},{"Start":"15:42.880 ","End":"15:46.360","Text":"I\u0027ll highlight it, didn\u0027t highlight the previous one."},{"Start":"15:46.360 ","End":"15:50.270","Text":"Well, I\u0027m sure you\u0027ll manage without it."},{"Start":"15:50.510 ","End":"15:54.640","Text":"Finally, part e,"},{"Start":"15:55.710 ","End":"15:58.990","Text":"we look for the not roots,"},{"Start":"15:58.990 ","End":"16:01.540","Text":"sometimes I\u0027ll call them roots erroneously,"},{"Start":"16:01.540 ","End":"16:05.302","Text":"but I mean the numbers that make something 0,"},{"Start":"16:05.302 ","End":"16:07.780","Text":"when it\u0027s 5 minus 2x, 0,"},{"Start":"16:07.780 ","End":"16:11.380","Text":"when 2x is 5 or x is 2.5,"},{"Start":"16:11.380 ","End":"16:13.990","Text":"when it\u0027s x minus 8^2, 0."},{"Start":"16:13.990 ","End":"16:17.170","Text":"Well, x same as x minus 8 is 0,"},{"Start":"16:17.170 ","End":"16:19.180","Text":"so x is 8."},{"Start":"16:19.180 ","End":"16:27.530","Text":"Put these two routes on the number line."},{"Start":"16:27.530 ","End":"16:29.590","Text":"Just have to be in the correct order,"},{"Start":"16:29.590 ","End":"16:33.760","Text":"2.5 is the leftmost and 8 is the rightmost."},{"Start":"16:33.760 ","End":"16:38.410","Text":"Then I distinguish the fact that here the polynomial 0,"},{"Start":"16:38.410 ","End":"16:40.765","Text":"but here it\u0027s undefined."},{"Start":"16:40.765 ","End":"16:44.990","Text":"X can\u0027t be 8, erase a bit."},{"Start":"16:45.270 ","End":"16:48.280","Text":"Put something hollow here,"},{"Start":"16:48.280 ","End":"16:54.130","Text":"and put something solid here, and let\u0027s see."},{"Start":"16:54.130 ","End":"16:57.220","Text":"We want to find out where it\u0027s plus and where it\u0027s minus."},{"Start":"16:57.220 ","End":"16:59.335","Text":"Let\u0027s take some sample points."},{"Start":"16:59.335 ","End":"17:02.680","Text":"Once in a while I actually write them in, though it\u0027s not necessarily."},{"Start":"17:02.680 ","End":"17:04.600","Text":"Let\u0027s choose 0 here,"},{"Start":"17:04.600 ","End":"17:07.870","Text":"let\u0027s choose 5 here,"},{"Start":"17:07.870 ","End":"17:10.555","Text":"let\u0027s choose 10 here."},{"Start":"17:10.555 ","End":"17:15.265","Text":"When x is 0, we have positive,"},{"Start":"17:15.265 ","End":"17:20.755","Text":"and this thing is bound to be positive because it\u0027s squared, so it\u0027s positive."},{"Start":"17:20.755 ","End":"17:23.755","Text":"When x is 5,"},{"Start":"17:23.755 ","End":"17:26.905","Text":"5 minus 10 is negative,"},{"Start":"17:26.905 ","End":"17:29.365","Text":"and this is still positive,"},{"Start":"17:29.365 ","End":"17:34.195","Text":"so negative of a positive is negative."},{"Start":"17:34.195 ","End":"17:36.235","Text":"When x is 10,"},{"Start":"17:36.235 ","End":"17:38.515","Text":"5 minus twice 10,"},{"Start":"17:38.515 ","End":"17:40.900","Text":"5 minus 20 was negative,"},{"Start":"17:40.900 ","End":"17:42.430","Text":"and this thing is still positive,"},{"Start":"17:42.430 ","End":"17:44.800","Text":"so we have another negative."},{"Start":"17:44.800 ","End":"17:49.585","Text":"By the way, the reason I\u0027m using different methods isn\u0027t just a variety."},{"Start":"17:49.585 ","End":"17:52.150","Text":"I\u0027m just showing you different ways that you could do it"},{"Start":"17:52.150 ","End":"17:55.270","Text":"with essentially the same mathematical idea,"},{"Start":"17:55.270 ","End":"17:56.785","Text":"but just different ways."},{"Start":"17:56.785 ","End":"17:59.410","Text":"Whether you want to draw the snake,"},{"Start":"17:59.410 ","End":"18:02.650","Text":"or whether you want to write pluses and minuses,"},{"Start":"18:02.650 ","End":"18:05.200","Text":"I\u0027m just showing you a variations."},{"Start":"18:05.200 ","End":"18:07.630","Text":"Each one will find the method suitable,"},{"Start":"18:07.630 ","End":"18:10.015","Text":"it\u0027s all the same mathematically."},{"Start":"18:10.015 ","End":"18:12.805","Text":"Now, we look at what we\u0027re collecting."},{"Start":"18:12.805 ","End":"18:17.365","Text":"We are collecting less than or equal to 0,"},{"Start":"18:17.365 ","End":"18:21.220","Text":"which means negative or 0."},{"Start":"18:21.220 ","End":"18:25.420","Text":"As for negatives, I have a negative here,"},{"Start":"18:25.420 ","End":"18:27.730","Text":"and a negative here, this is what I want."},{"Start":"18:27.730 ","End":"18:32.890","Text":"Places where it\u0027s 0 is here,"},{"Start":"18:32.890 ","End":"18:34.510","Text":"I want this also."},{"Start":"18:34.510 ","End":"18:36.865","Text":"I\u0027ve just circled the things that I want,"},{"Start":"18:36.865 ","End":"18:38.395","Text":"negatives and 0s,"},{"Start":"18:38.395 ","End":"18:40.045","Text":"not to choose this,"},{"Start":"18:40.045 ","End":"18:41.395","Text":"this is not a 0,"},{"Start":"18:41.395 ","End":"18:46.435","Text":"this is where the polynomial is not defined."},{"Start":"18:46.435 ","End":"18:49.420","Text":"Now, let\u0027s summarize what we have here."},{"Start":"18:49.420 ","End":"18:52.435","Text":"This bit together with this,"},{"Start":"18:52.435 ","End":"18:55.885","Text":"means that x is between 2.5 and 8,"},{"Start":"18:55.885 ","End":"18:58.480","Text":"including the 2.5,"},{"Start":"18:58.480 ","End":"19:05.875","Text":"so 2.5 less than or equal to x, less than 8."},{"Start":"19:05.875 ","End":"19:09.295","Text":"This is the thing to watch out for."},{"Start":"19:09.295 ","End":"19:19.270","Text":"This part here is x bigger than 8,"},{"Start":"19:19.270 ","End":"19:23.950","Text":"and that\u0027s it for this bit,"},{"Start":"19:23.950 ","End":"19:27.410","Text":"so we put them together with an or."},{"Start":"19:27.510 ","End":"19:31.840","Text":"There is one other way of doing it and I\u0027ll show you."},{"Start":"19:31.840 ","End":"19:36.160","Text":"This is certainly good enough as an answer,"},{"Start":"19:36.160 ","End":"19:41.508","Text":"but sometimes it\u0027s a bit unaesthetic to have a whole range,"},{"Start":"19:41.508 ","End":"19:44.170","Text":"and one missing point."},{"Start":"19:44.170 ","End":"19:46.720","Text":"Another way you could say it,"},{"Start":"19:46.720 ","End":"19:50.050","Text":"this is just a variation that you can ignore if you don\u0027t like it."},{"Start":"19:50.050 ","End":"19:52.990","Text":"I could say that x is from 2.5,"},{"Start":"19:52.990 ","End":"19:58.021","Text":"including 2.5 all the way to the right limitlessly."},{"Start":"19:58.021 ","End":"20:04.270","Text":"I can say x is bigger or equal to 2.5."},{"Start":"20:04.270 ","End":"20:10.195","Text":"The word but is mathematically the same as and,"},{"Start":"20:10.195 ","End":"20:12.250","Text":"x cannot be 8,"},{"Start":"20:12.250 ","End":"20:16.405","Text":"so I just say there\u0027s an exception."},{"Start":"20:16.405 ","End":"20:18.970","Text":"Everything from 2.5 onwards except 8,"},{"Start":"20:18.970 ","End":"20:21.340","Text":"it\u0027s the same as saying from 2.5 to 8,"},{"Start":"20:21.340 ","End":"20:24.025","Text":"not inclusive and from 8 onwards."},{"Start":"20:24.025 ","End":"20:27.580","Text":"This is also an answer instead of this."},{"Start":"20:27.580 ","End":"20:30.700","Text":"Maybe I\u0027ll even highlight it in a different color."},{"Start":"20:30.700 ","End":"20:35.305","Text":"Whichever one of these you like, just choose one."},{"Start":"20:35.305 ","End":"20:38.080","Text":"That does part e,"},{"Start":"20:38.080 ","End":"20:42.020","Text":"so that\u0027s the end of this exercise."}],"Thumbnail":null,"ID":5071},{"Watched":false,"Name":"Exercise 2","Duration":"22m 41s","ChapterTopicVideoID":5071,"CourseChapterTopicPlaylistID":3877,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.900","Text":"We have here 5 inequalities to solve,"},{"Start":"00:03.900 ","End":"00:07.245","Text":"and they will involve rational functions."},{"Start":"00:07.245 ","End":"00:10.650","Text":"Let\u0027s get started with the first one."},{"Start":"00:10.650 ","End":"00:17.625","Text":"The idea is to find the points which make either the numerator or the denominator 0."},{"Start":"00:17.625 ","End":"00:19.545","Text":"In the case of the numerator,"},{"Start":"00:19.545 ","End":"00:24.060","Text":"the points are x=4 minus 2,"},{"Start":"00:24.060 ","End":"00:28.080","Text":"and then the denominator x=1 will make it 0."},{"Start":"00:28.080 ","End":"00:34.340","Text":"All these 3 points have to be put on the number line in the sketch method."},{"Start":"00:34.340 ","End":"00:38.465","Text":"Here\u0027s the number line. Let\u0027s write the 3 of them in the correct order."},{"Start":"00:38.465 ","End":"00:40.610","Text":"Minus 2 comes first,"},{"Start":"00:40.610 ","End":"00:43.745","Text":"then 1, and then 4."},{"Start":"00:43.745 ","End":"00:47.705","Text":"Now in the case of 4 and minus 2,"},{"Start":"00:47.705 ","End":"00:53.510","Text":"the value of this rational function or rational expression,"},{"Start":"00:53.510 ","End":"01:00.740","Text":"if you haven\u0027t learned functions, this rational expression is 0 here and here."},{"Start":"01:00.740 ","End":"01:03.130","Text":"But in the case of x=1,"},{"Start":"01:03.130 ","End":"01:05.280","Text":"it\u0027s the denominator that 0,"},{"Start":"01:05.280 ","End":"01:07.340","Text":"so this becomes a forbidden value,"},{"Start":"01:07.340 ","End":"01:09.630","Text":"and so I\u0027ll just erase a bit,"},{"Start":"01:09.630 ","End":"01:16.069","Text":"make a little hole there and emphasize it with a gap here that we cannot let x=1,"},{"Start":"01:16.069 ","End":"01:18.515","Text":"it just doesn\u0027t make sense."},{"Start":"01:18.515 ","End":"01:21.220","Text":"Now we proceed as usual."},{"Start":"01:21.220 ","End":"01:23.790","Text":"We have 4 different intervals here,"},{"Start":"01:23.790 ","End":"01:29.240","Text":"and here and we want to pick some values representative."},{"Start":"01:29.240 ","End":"01:31.130","Text":"We can do this in our heads."},{"Start":"01:31.130 ","End":"01:35.800","Text":"For example, here I can choose x equals minus 3."},{"Start":"01:35.800 ","End":"01:38.715","Text":"We straightaway see this is negative."},{"Start":"01:38.715 ","End":"01:41.054","Text":"Minus 3 plus 2 is still negative,"},{"Start":"01:41.054 ","End":"01:45.968","Text":"3 negatives make a negative,"},{"Start":"01:45.968 ","End":"01:49.805","Text":"so we can put some point to represent it."},{"Start":"01:49.805 ","End":"01:53.645","Text":"Important thing is it should be below the line."},{"Start":"01:53.645 ","End":"01:55.850","Text":"Something in this interval,"},{"Start":"01:55.850 ","End":"01:58.785","Text":"let\u0027s say 0 is a good number."},{"Start":"01:58.785 ","End":"02:02.385","Text":"Put in 0, we\u0027ve got minus 4 plus 2 minus 1,"},{"Start":"02:02.385 ","End":"02:05.620","Text":"2 negatives and a positive, so positive."},{"Start":"02:05.620 ","End":"02:10.570","Text":"We indicate this with a point above the line."},{"Start":"02:10.570 ","End":"02:12.150","Text":"Then between 1 and 4,"},{"Start":"02:12.150 ","End":"02:13.605","Text":"let\u0027s choose 2,"},{"Start":"02:13.605 ","End":"02:15.780","Text":"2 minus 4 is negative,"},{"Start":"02:15.780 ","End":"02:17.700","Text":"2 plus 2 is positive,"},{"Start":"02:17.700 ","End":"02:20.385","Text":"2 minus 1 is positive."},{"Start":"02:20.385 ","End":"02:24.900","Text":"2 positives and a negative, so negative."},{"Start":"02:25.690 ","End":"02:28.160","Text":"Something to the right of 4,"},{"Start":"02:28.160 ","End":"02:29.390","Text":"let\u0027s say 5,"},{"Start":"02:29.390 ","End":"02:31.910","Text":"5 minus 4 is positive."},{"Start":"02:31.910 ","End":"02:37.595","Text":"Well, they\u0027re all positive and this comes out to be positive."},{"Start":"02:37.595 ","End":"02:44.420","Text":"Then if we want to use the snake method form of sketch,"},{"Start":"02:44.420 ","End":"02:45.500","Text":"we can do that."},{"Start":"02:45.500 ","End":"02:49.745","Text":"We can just put something that goes through all these,"},{"Start":"02:49.745 ","End":"02:52.290","Text":"but not through here."},{"Start":"02:55.120 ","End":"02:57.950","Text":"I like to put little arrows at the end."},{"Start":"02:57.950 ","End":"02:59.270","Text":"If say it goes on forever,"},{"Start":"02:59.270 ","End":"03:01.770","Text":"I\u0027ve just run out of space here."},{"Start":"03:02.060 ","End":"03:08.385","Text":"The 1 doesn\u0027t go through because we can\u0027t let x=1."},{"Start":"03:08.385 ","End":"03:11.990","Text":"Now we look at which type of inequality it is,"},{"Start":"03:11.990 ","End":"03:14.285","Text":"less than 0, negative,"},{"Start":"03:14.285 ","End":"03:17.030","Text":"and negative means below the line."},{"Start":"03:17.030 ","End":"03:19.340","Text":"If we use the shading method,"},{"Start":"03:19.340 ","End":"03:27.245","Text":"then I\u0027ll just shade the bits that are below the axis and strictly below,"},{"Start":"03:27.245 ","End":"03:32.690","Text":"not including the axis itself and so we get are 2 bits,"},{"Start":"03:32.690 ","End":"03:34.505","Text":"this bit and this bit."},{"Start":"03:34.505 ","End":"03:40.860","Text":"This one corresponds to x less than minus 2 and then definitely,"},{"Start":"03:40.860 ","End":"03:43.830","Text":"no limit on this side."},{"Start":"03:43.830 ","End":"03:49.135","Text":"Here we have 1 less than x, less than 4."},{"Start":"03:49.135 ","End":"03:51.710","Text":"In both cases, we\u0027re not including"},{"Start":"03:51.710 ","End":"03:57.480","Text":"this point or this point because we want a strict inequality."},{"Start":"03:58.340 ","End":"04:01.140","Text":"This is the answer."},{"Start":"04:01.140 ","End":"04:03.480","Text":"I forgot to write the or,"},{"Start":"04:03.480 ","End":"04:05.220","Text":"that\u0027s how we express it."},{"Start":"04:05.220 ","End":"04:08.740","Text":"We have 2 separate intervals."},{"Start":"04:08.870 ","End":"04:11.595","Text":"Onto the next one."},{"Start":"04:11.595 ","End":"04:13.620","Text":"What we need to do here, first of all,"},{"Start":"04:13.620 ","End":"04:15.969","Text":"is factorize, nothing to do with the numerator,"},{"Start":"04:15.969 ","End":"04:22.330","Text":"but denominator might factorize and it does factorized."},{"Start":"04:22.330 ","End":"04:24.130","Text":"I\u0027ll do the work for you."},{"Start":"04:24.130 ","End":"04:31.760","Text":"This comes out to be x minus 6, x minus 2."},{"Start":"04:32.420 ","End":"04:34.935","Text":"I\u0027ll just verify it."},{"Start":"04:34.935 ","End":"04:42.750","Text":"x times x is x^2 minus 6x minus 2x is minus 8x and minus 6 times minus 2 is 12,"},{"Start":"04:42.750 ","End":"04:44.780","Text":"so we\u0027re okay there."},{"Start":"04:44.780 ","End":"04:48.965","Text":"Now we need the points that make the numerator or denominator 0."},{"Start":"04:48.965 ","End":"04:51.935","Text":"For this term it\u0027s a 6,"},{"Start":"04:51.935 ","End":"04:53.540","Text":"and here it\u0027s a 2."},{"Start":"04:53.540 ","End":"04:58.140","Text":"These are the numbers we want to put on the number line."},{"Start":"04:58.180 ","End":"05:01.370","Text":"Make sure to put them in the right order,"},{"Start":"05:01.370 ","End":"05:02.585","Text":"2 on the left,"},{"Start":"05:02.585 ","End":"05:04.415","Text":"6 on the right."},{"Start":"05:04.415 ","End":"05:07.820","Text":"But since these make the denominator 0,"},{"Start":"05:07.820 ","End":"05:14.765","Text":"we have to indicate that somehow and I indicated by putting a little hole here and here,"},{"Start":"05:14.765 ","End":"05:18.210","Text":"so that x cannot be 2 or 6."},{"Start":"05:18.340 ","End":"05:24.830","Text":"My style is to represent it with a hollow circle to show that there\u0027s a piece missing."},{"Start":"05:24.830 ","End":"05:27.740","Text":"These are illegal values for x."},{"Start":"05:27.740 ","End":"05:30.650","Text":"Now we have 3 intervals."},{"Start":"05:30.650 ","End":"05:34.010","Text":"We pick a representative in each interval."},{"Start":"05:34.010 ","End":"05:35.540","Text":"We can do everything in our heads,"},{"Start":"05:35.540 ","End":"05:37.535","Text":"but sometimes I like to just say what I\u0027m doing."},{"Start":"05:37.535 ","End":"05:39.440","Text":"I\u0027ll choose 0 here,"},{"Start":"05:39.440 ","End":"05:42.485","Text":"I\u0027ll choose 4 here,"},{"Start":"05:42.485 ","End":"05:48.060","Text":"and I\u0027ll choose 7 here and let\u0027s see what we get."},{"Start":"05:48.060 ","End":"05:52.290","Text":"For 0, I get 1 over minus 6 times minus 2,"},{"Start":"05:52.290 ","End":"05:54.079","Text":"so altogether there\u0027s 2 negatives,"},{"Start":"05:54.079 ","End":"05:56.050","Text":"so it\u0027s a positive."},{"Start":"05:56.050 ","End":"06:00.585","Text":"Somewhere here, for x equals 4,"},{"Start":"06:00.585 ","End":"06:01.920","Text":"4 minus 6 is negative,"},{"Start":"06:01.920 ","End":"06:03.330","Text":"but 4 minus 2 is positive,"},{"Start":"06:03.330 ","End":"06:05.460","Text":"so we end up negative."},{"Start":"06:05.460 ","End":"06:08.215","Text":"Then 7, 7 minus 6, 7 minus 2,"},{"Start":"06:08.215 ","End":"06:13.350","Text":"both positive, so we are positive."},{"Start":"06:14.270 ","End":"06:17.490","Text":"Just a quick sketch, it\u0027s schematic,"},{"Start":"06:17.490 ","End":"06:21.480","Text":"it\u0027s not meant to be the graph of the function or anything."},{"Start":"06:23.590 ","End":"06:29.220","Text":"Note, doesn\u0027t go through the illegal bad points,"},{"Start":"06:29.220 ","End":"06:32.405","Text":"it goes on forever in both directions."},{"Start":"06:32.405 ","End":"06:34.880","Text":"Now let\u0027s see what are we collecting today?"},{"Start":"06:34.880 ","End":"06:37.670","Text":"We\u0027re collecting positives and 0s."},{"Start":"06:37.670 ","End":"06:39.755","Text":"Bigger or equal to 0,"},{"Start":"06:39.755 ","End":"06:42.510","Text":"positive or 0,"},{"Start":"06:43.490 ","End":"06:49.605","Text":"above the axis here and here."},{"Start":"06:49.605 ","End":"06:52.820","Text":"Let\u0027s see, are there any places where it\u0027s 0?"},{"Start":"06:52.820 ","End":"06:57.030","Text":"There just aren\u0027t any because this and this are"},{"Start":"06:57.030 ","End":"07:01.760","Text":"not places where the rational function is 0."},{"Start":"07:01.760 ","End":"07:03.470","Text":"It\u0027s places where it\u0027s undefined,"},{"Start":"07:03.470 ","End":"07:05.735","Text":"so we can\u0027t take those."},{"Start":"07:05.735 ","End":"07:07.985","Text":"What we\u0027re left with is,"},{"Start":"07:07.985 ","End":"07:13.935","Text":"from this bit x less than 2, there\u0027s an hole,"},{"Start":"07:13.935 ","End":"07:22.725","Text":"and then this bit x bigger than 6 and that\u0027s our answer to b,"},{"Start":"07:22.725 ","End":"07:30.470","Text":"and so on with c. In Part c,"},{"Start":"07:30.470 ","End":"07:32.930","Text":"we want to factorize everything."},{"Start":"07:32.930 ","End":"07:37.700","Text":"The numerator is about its factors as it can be."},{"Start":"07:37.700 ","End":"07:39.220","Text":"The denominator, well,"},{"Start":"07:39.220 ","End":"07:41.230","Text":"I can take 2 out, first of all,"},{"Start":"07:41.230 ","End":"07:49.960","Text":"that gives us x^2 squared minus 5x plus 6, that\u0027s bigger than 0."},{"Start":"07:50.060 ","End":"07:53.780","Text":"I\u0027ll save time by factoring this bit for you,"},{"Start":"07:53.780 ","End":"07:55.040","Text":"although you have 2 methods,"},{"Start":"07:55.040 ","End":"08:00.560","Text":"one with finding the roots and one using trinomial decomposition."},{"Start":"08:00.560 ","End":"08:02.620","Text":"Whatever it is,"},{"Start":"08:02.620 ","End":"08:06.630","Text":"what we get is x minus 3,"},{"Start":"08:06.630 ","End":"08:12.375","Text":"x minus 2 and I will check by multiplying out"},{"Start":"08:12.375 ","End":"08:20.055","Text":"x times x is x^2 minus 3x minus 2x is minus 5x and minus 3 times minus 2 is 6."},{"Start":"08:20.055 ","End":"08:23.060","Text":"Well, this is bigger than 0. What we"},{"Start":"08:23.060 ","End":"08:26.030","Text":"need are the places where the numerator or denominator are"},{"Start":"08:26.030 ","End":"08:33.170","Text":"0 and we\u0027ll do this for each factor."},{"Start":"08:33.170 ","End":"08:37.100","Text":"Here we have 3 makes it 0,"},{"Start":"08:37.100 ","End":"08:41.030","Text":"and here we also have 3 that makes it 0."},{"Start":"08:41.030 ","End":"08:42.920","Text":"We\u0027ll get back to that in a moment."},{"Start":"08:42.920 ","End":"08:45.050","Text":"Here we have 2."},{"Start":"08:45.050 ","End":"08:54.375","Text":"Now, what this peculiarity means with the x minus 3 appearing in both places is that,"},{"Start":"08:54.375 ","End":"08:56.580","Text":"first of all, it\u0027s in the denominator,"},{"Start":"08:56.580 ","End":"09:00.030","Text":"so it\u0027s illegal to substitute x=3,"},{"Start":"09:00.030 ","End":"09:02.330","Text":"so 3 is ruled out."},{"Start":"09:02.330 ","End":"09:05.955","Text":"But everywhere else except 3,"},{"Start":"09:05.955 ","End":"09:09.460","Text":"we could cancel the x minus 3\u0027s."},{"Start":"09:09.460 ","End":"09:13.490","Text":"But like I said, we still have the condition that x cannot be 3."},{"Start":"09:13.490 ","End":"09:15.680","Text":"Otherwise we\u0027ll get 0 over 0,"},{"Start":"09:15.680 ","End":"09:17.810","Text":"which doesn\u0027t make sense."},{"Start":"09:17.810 ","End":"09:22.010","Text":"Let me write the simplified form for when x is not 3,"},{"Start":"09:22.010 ","End":"09:24.080","Text":"what we have is"},{"Start":"09:24.080 ","End":"09:29.490","Text":"just 1 over"},{"Start":"09:37.130 ","End":"09:40.010","Text":"2(X-2)."},{"Start":"09:40.010 ","End":"09:42.335","Text":"That\u0027s what we have after we take out this factor."},{"Start":"09:42.335 ","End":"09:46.880","Text":"But we still have the x not equal to 3,"},{"Start":"09:46.880 ","End":"09:49.055","Text":"this is still bigger than 0."},{"Start":"09:49.055 ","End":"09:51.340","Text":"Now let\u0027s sketch."},{"Start":"09:51.340 ","End":"09:56.370","Text":"I will place both the 2 and the 3 here."},{"Start":"09:56.370 ","End":"09:59.820","Text":"The 3 won\u0027t be involved with the calculations,"},{"Start":"09:59.820 ","End":"10:09.140","Text":"it will just be as a reminder that there\u0027s a missing point here that x cannot be 3,"},{"Start":"10:09.140 ","End":"10:11.660","Text":"there is a gap, a hole,"},{"Start":"10:11.660 ","End":"10:15.330","Text":"I\u0027ll just put a little hollow circle instead of it."},{"Start":"10:20.780 ","End":"10:23.835","Text":"Now 2 is also in the denominator,"},{"Start":"10:23.835 ","End":"10:26.475","Text":"we can\u0027t be 2 either,"},{"Start":"10:26.475 ","End":"10:29.700","Text":"also a hollow circle here."},{"Start":"10:29.700 ","End":"10:36.705","Text":"But as for the calculations of positive and negative,"},{"Start":"10:36.705 ","End":"10:39.890","Text":"we just have to look at the 2."},{"Start":"10:40.470 ","End":"10:46.555","Text":"If x is less than 2,"},{"Start":"10:46.555 ","End":"10:50.682","Text":"let\u0027s say x is 0,"},{"Start":"10:50.682 ","End":"10:53.280","Text":"then when I plug it in,"},{"Start":"10:53.280 ","End":"10:57.635","Text":"I get 1 over 2 times minus 2, so it\u0027s negative."},{"Start":"10:57.635 ","End":"11:00.860","Text":"I\u0027ll put some point below the line here,"},{"Start":"11:00.860 ","End":"11:03.755","Text":"corresponding to the 0."},{"Start":"11:03.755 ","End":"11:06.560","Text":"Now any point to the right of 2,"},{"Start":"11:06.560 ","End":"11:08.364","Text":"for example, 4,"},{"Start":"11:08.364 ","End":"11:15.420","Text":"then I\u0027ll get 1 over 2 times 4 minus 2 is still 1 over positive,"},{"Start":"11:15.420 ","End":"11:17.920","Text":"positive, it\u0027s positive."},{"Start":"11:19.800 ","End":"11:25.120","Text":"I know it\u0027s going to be positive here also because everywhere to the right of 2,"},{"Start":"11:25.120 ","End":"11:26.875","Text":"It\u0027s the same sign."},{"Start":"11:26.875 ","End":"11:31.240","Text":"But I actually put it in an extra positive point here also."},{"Start":"11:31.240 ","End":"11:32.830","Text":"I don\u0027t need to plug it in,"},{"Start":"11:32.830 ","End":"11:35.485","Text":"but I know that it doesn\u0027t change sign."},{"Start":"11:35.485 ","End":"11:37.030","Text":"But because I have a missing point,"},{"Start":"11:37.030 ","End":"11:40.630","Text":"I\u0027d like to duplicate this because when I do the drawing,"},{"Start":"11:40.630 ","End":"11:42.370","Text":"I can\u0027t get from here to here,"},{"Start":"11:42.370 ","End":"11:44.630","Text":"I can\u0027t go through here."},{"Start":"11:44.760 ","End":"11:50.920","Text":"What happens is that I draw something like this, stop here,"},{"Start":"11:50.920 ","End":"11:55.810","Text":"go to here and then I don\u0027t know what happens in the middle,"},{"Start":"11:55.810 ","End":"11:57.565","Text":"but another is a gap."},{"Start":"11:57.565 ","End":"11:59.305","Text":"As far as I know,"},{"Start":"11:59.305 ","End":"12:04.120","Text":"it might even continue different way."},{"Start":"12:04.120 ","End":"12:09.130","Text":"Now it\u0027s not changing sign because x is still bigger than 2 in both of these."},{"Start":"12:09.130 ","End":"12:11.095","Text":"But there\u0027s something missing,"},{"Start":"12:11.095 ","End":"12:12.580","Text":"some kind of gap. I don\u0027t know."},{"Start":"12:12.580 ","End":"12:15.700","Text":"Perhaps I\u0027ll put a dotted line to indicate that there\u0027s"},{"Start":"12:15.700 ","End":"12:21.320","Text":"a gap because of an illegal value."},{"Start":"12:22.260 ","End":"12:25.435","Text":"Take 2 on the last bit."},{"Start":"12:25.435 ","End":"12:28.030","Text":"I don\u0027t know exactly how it continues here,"},{"Start":"12:28.030 ","End":"12:29.624","Text":"but however it continues,"},{"Start":"12:29.624 ","End":"12:34.630","Text":"there also has to be that gap in the function because it\u0027s"},{"Start":"12:34.630 ","End":"12:39.820","Text":"not defined at x=3."},{"Start":"12:39.820 ","End":"12:41.950","Text":"We get something like this."},{"Start":"12:41.950 ","End":"12:47.272","Text":"Now, we just look at what it is that we\u0027re collecting."},{"Start":"12:47.272 ","End":"12:48.755","Text":"Oh, wrong 1, sorry."},{"Start":"12:48.755 ","End":"12:52.090","Text":"We\u0027re collecting greater than 0."},{"Start":"12:52.090 ","End":"12:58.670","Text":"Greater than 0 is everything strictly above the axis."},{"Start":"12:59.910 ","End":"13:02.800","Text":"What we have is this bit."},{"Start":"13:02.800 ","End":"13:05.365","Text":"If I shade it, the greater than 0,"},{"Start":"13:05.365 ","End":"13:07.960","Text":"it\u0027s all this stuff."},{"Start":"13:07.960 ","End":"13:14.530","Text":"However, I can\u0027t just say x is bigger than 2."},{"Start":"13:14.530 ","End":"13:16.990","Text":"I could say that as a start,"},{"Start":"13:16.990 ","End":"13:19.390","Text":"I could say x is bigger than 2."},{"Start":"13:19.390 ","End":"13:22.160","Text":"But I have to add,"},{"Start":"13:23.280 ","End":"13:25.690","Text":"in English I would say but,"},{"Start":"13:25.690 ","End":"13:28.180","Text":"in mathematics it\u0027s and,"},{"Start":"13:28.180 ","End":"13:32.215","Text":"x not equal to 3."},{"Start":"13:32.215 ","End":"13:35.750","Text":"It\u0027s everything here except the O,"},{"Start":"13:35.750 ","End":"13:40.600","Text":"because the 3 is impossible to substitute,"},{"Start":"13:40.600 ","End":"13:42.340","Text":"we get 0 in the denominator."},{"Start":"13:42.340 ","End":"13:46.900","Text":"Actually 0 in the numerator 2 is 0 over 0 is not defined."},{"Start":"13:46.900 ","End":"13:50.845","Text":"That\u0027s the answer and it\u0027ll highlight it."},{"Start":"13:50.845 ","End":"13:54.385","Text":"Just want to give an alternative form of this."},{"Start":"13:54.385 ","End":"14:00.970","Text":"You might not want to say that this is from 2 onwards excluding 3."},{"Start":"14:00.970 ","End":"14:06.055","Text":"Another way of looking at it is to say from 2 to 3 and then from 3 onwards."},{"Start":"14:06.055 ","End":"14:10.000","Text":"You could also say x is"},{"Start":"14:10.000 ","End":"14:18.310","Text":"between 2 and 3 or x bigger than 3."},{"Start":"14:18.310 ","End":"14:24.190","Text":"Something like that and that will be an alternative."},{"Start":"14:24.190 ","End":"14:26.890","Text":"Just write that 1 it\u0027s just an alternative."},{"Start":"14:26.890 ","End":"14:31.435","Text":"Whichever 1 you think is easier to explain."},{"Start":"14:31.435 ","End":"14:34.170","Text":"The number is greater than 2 except 3."},{"Start":"14:34.170 ","End":"14:36.100","Text":"I think this is simpler,"},{"Start":"14:36.100 ","End":"14:40.255","Text":"but you can also say between 2 and 3 or from 3 onwards."},{"Start":"14:40.255 ","End":"14:45.175","Text":"That\u0027s part c. Now"},{"Start":"14:45.175 ","End":"14:51.775","Text":"onto part d. We have d and e still to go."},{"Start":"14:51.775 ","End":"14:55.675","Text":"Let\u0027s look where everything is 0."},{"Start":"14:55.675 ","End":"14:58.960","Text":"Here, it happens when x is 5, here,"},{"Start":"14:58.960 ","End":"15:00.190","Text":"when x is, let\u0027s see,"},{"Start":"15:00.190 ","End":"15:02.560","Text":"minus 1/3 that\u0027s so,"},{"Start":"15:02.560 ","End":"15:06.355","Text":"when this is 0, this is 0 when x is 2,"},{"Start":"15:06.355 ","End":"15:09.820","Text":"and this is 0 when x is minus 7."},{"Start":"15:09.820 ","End":"15:11.860","Text":"Because these 2 are different than these 2,"},{"Start":"15:11.860 ","End":"15:13.720","Text":"because these make denominator 0,"},{"Start":"15:13.720 ","End":"15:17.800","Text":"so these are invalid illegal points,"},{"Start":"15:17.800 ","End":"15:24.790","Text":"and these are points where the rational expression is 0."},{"Start":"15:24.790 ","End":"15:26.860","Text":"Let\u0027s sketch that."},{"Start":"15:26.860 ","End":"15:31.735","Text":"The full points in order of increasing minus 7,"},{"Start":"15:31.735 ","End":"15:35.950","Text":"next minus a 1/3,"},{"Start":"15:35.950 ","End":"15:40.810","Text":"next 2, and then 5."},{"Start":"15:40.810 ","End":"15:45.080","Text":"These 2 are the ones where were undefined."},{"Start":"15:45.600 ","End":"15:50.620","Text":"My little o is here and here."},{"Start":"15:50.620 ","End":"15:52.420","Text":"These 2 are okay."},{"Start":"15:52.420 ","End":"15:59.335","Text":"This is where the numerators 0 and the whole thing is 0 here and here."},{"Start":"15:59.335 ","End":"16:02.230","Text":"Now we just have to figure out what sample points,"},{"Start":"16:02.230 ","End":"16:04.735","Text":"what are the signs."},{"Start":"16:04.735 ","End":"16:09.740","Text":"Let\u0027s start with an easy 1. I\u0027ll start with 0."},{"Start":"16:09.960 ","End":"16:13.960","Text":"When x is 0, we have minus,"},{"Start":"16:13.960 ","End":"16:21.841","Text":"plus, also plus because it\u0027s plus 2 and plus 7."},{"Start":"16:21.841 ","End":"16:24.265","Text":"Let\u0027s see."},{"Start":"16:24.265 ","End":"16:28.643","Text":"That was a minus and everything else was plus."},{"Start":"16:28.643 ","End":"16:31.934","Text":"It\u0027s minus when x is 0."},{"Start":"16:31.934 ","End":"16:37.850","Text":"Let\u0027s just use the minus plus system instead of doing the snake."},{"Start":"16:37.890 ","End":"16:42.295","Text":"Let\u0027s take say minus 8,"},{"Start":"16:42.295 ","End":"16:45.610","Text":"so that would make that negative."},{"Start":"16:45.610 ","End":"16:48.025","Text":"That would be negative."},{"Start":"16:48.025 ","End":"16:50.635","Text":"2 minus, minus x is positive,"},{"Start":"16:50.635 ","End":"16:53.560","Text":"minus x plus 7, still negative."},{"Start":"16:53.560 ","End":"16:55.825","Text":"We have 1 positive, 3 negatives."},{"Start":"16:55.825 ","End":"16:59.620","Text":"That makes it negative."},{"Start":"16:59.620 ","End":"17:01.600","Text":"What do we have in this range?"},{"Start":"17:01.600 ","End":"17:03.175","Text":"Say minus 1."},{"Start":"17:03.175 ","End":"17:06.085","Text":"Minus 1 will make this negative."},{"Start":"17:06.085 ","End":"17:08.845","Text":"Minus 3 plus 1, that\u0027s also negative."},{"Start":"17:08.845 ","End":"17:11.320","Text":"2 minus minus 1, positive,"},{"Start":"17:11.320 ","End":"17:12.670","Text":"minus 1 plus 7,"},{"Start":"17:12.670 ","End":"17:14.680","Text":"positive, negative, negative, positive,"},{"Start":"17:14.680 ","End":"17:21.190","Text":"positive, so positive because an even number of negatives between 2 and 5, let\u0027s say 3."},{"Start":"17:21.190 ","End":"17:25.330","Text":"For x, 3 minus 5 is negative."},{"Start":"17:25.330 ","End":"17:28.210","Text":"9 plus 1, positive."},{"Start":"17:28.210 ","End":"17:33.130","Text":"2 minus 3 is negative"},{"Start":"17:33.130 ","End":"17:39.295","Text":"and 3 plus 7 positive."},{"Start":"17:39.295 ","End":"17:42.250","Text":"Altogether we have, let\u0027s see,"},{"Start":"17:42.250 ","End":"17:45.140","Text":"how many negatives did we have?"},{"Start":"17:45.330 ","End":"17:47.830","Text":"We had negative, negative, positive,"},{"Start":"17:47.830 ","End":"17:50.980","Text":"positive, anyway, so it\u0027s positive."},{"Start":"17:50.980 ","End":"17:53.680","Text":"Then bigger than 5, let\u0027s say 6."},{"Start":"17:53.680 ","End":"17:57.160","Text":"We have positive, positive, 2 minus,"},{"Start":"17:57.160 ","End":"18:02.638","Text":"negative, 6 plus 7 is positive."},{"Start":"18:02.638 ","End":"18:06.790","Text":"Only this one\u0027s negative, so it\u0027s minus."},{"Start":"18:06.790 ","End":"18:11.050","Text":"I did this as the alternative to the snake."},{"Start":"18:11.050 ","End":"18:14.170","Text":"Now we look at what we were collecting."},{"Start":"18:14.170 ","End":"18:16.700","Text":"We\u0027re collecting negatives."},{"Start":"18:17.610 ","End":"18:21.370","Text":"We would like this bit. It\u0027s a negative."},{"Start":"18:21.370 ","End":"18:23.364","Text":"This bit is negative."},{"Start":"18:23.364 ","End":"18:25.390","Text":"This bit here is a negative."},{"Start":"18:25.390 ","End":"18:28.060","Text":"In terms of intervals with x,"},{"Start":"18:28.060 ","End":"18:33.490","Text":"this gives us x less than minus 7."},{"Start":"18:33.490 ","End":"18:39.235","Text":"They\u0027re all connected by O or minus 1/3 less than x,"},{"Start":"18:39.235 ","End":"18:41.050","Text":"less than 2,"},{"Start":"18:41.050 ","End":"18:45.280","Text":"or 5 less than x or x bigger than 5,"},{"Start":"18:45.280 ","End":"18:47.860","Text":"whichever way you want to write it."},{"Start":"18:47.860 ","End":"18:50.860","Text":"This will be the answer."},{"Start":"18:50.860 ","End":"18:53.365","Text":"I will highlight it."},{"Start":"18:53.365 ","End":"18:57.970","Text":"Finally the last 1. Let\u0027s see."},{"Start":"18:57.970 ","End":"19:00.850","Text":"It\u0027s all factorized already."},{"Start":"19:00.850 ","End":"19:04.765","Text":"The values were collecting were something is 0."},{"Start":"19:04.765 ","End":"19:11.080","Text":"Here, x is 6 here minus 1 and here x=2 makes this 0."},{"Start":"19:11.080 ","End":"19:13.030","Text":"The 2 is the 1 on the denominator,"},{"Start":"19:13.030 ","End":"19:16.630","Text":"and this is the 1 that we have to rule out."},{"Start":"19:16.630 ","End":"19:18.640","Text":"In the right order."},{"Start":"19:18.640 ","End":"19:21.325","Text":"First we have minus 1,"},{"Start":"19:21.325 ","End":"19:28.555","Text":"then we have the 2 and then we have the 6."},{"Start":"19:28.555 ","End":"19:31.540","Text":"The 2 is the 1 which has the gap."},{"Start":"19:31.540 ","End":"19:34.555","Text":"Just erase a bit here."},{"Start":"19:34.555 ","End":"19:38.965","Text":"I\u0027ll just emphasize the gap with a hollow circle."},{"Start":"19:38.965 ","End":"19:40.615","Text":"At minus 1 and 6,"},{"Start":"19:40.615 ","End":"19:44.870","Text":"we are 0 because those are from the numerator."},{"Start":"19:46.320 ","End":"19:49.030","Text":"Here it says something about the square here."},{"Start":"19:49.030 ","End":"19:54.235","Text":"Of course, when x minus x squared is 0 is the same as x minus 6 itself is 0."},{"Start":"19:54.235 ","End":"19:56.875","Text":"I should have just mentioned that there."},{"Start":"19:56.875 ","End":"20:00.505","Text":"Let\u0027s go again with this plus-minus method."},{"Start":"20:00.505 ","End":"20:04.480","Text":"I\u0027ve put the foot snake for this exercise."},{"Start":"20:04.480 ","End":"20:10.060","Text":"I will put in substitute some sample values,"},{"Start":"20:10.060 ","End":"20:12.745","Text":"let\u0027s say minus 2."},{"Start":"20:12.745 ","End":"20:17.350","Text":"If we are minus 2 for x,"},{"Start":"20:17.350 ","End":"20:20.260","Text":"then this thing is always positive because it\u0027s squared."},{"Start":"20:20.260 ","End":"20:24.085","Text":"Minus 2 plus 1 is negative,"},{"Start":"20:24.085 ","End":"20:26.650","Text":"minus 2, minus 2 is negative."},{"Start":"20:26.650 ","End":"20:28.690","Text":"We have 2 negatives."},{"Start":"20:28.690 ","End":"20:31.360","Text":"This part is positive."},{"Start":"20:31.360 ","End":"20:36.220","Text":"In this interval we have 0. This is positive."},{"Start":"20:36.220 ","End":"20:37.840","Text":"0 plus 1 is positive,"},{"Start":"20:37.840 ","End":"20:41.365","Text":"0 minus 2 is negative. That\u0027s negative."},{"Start":"20:41.365 ","End":"20:46.615","Text":"Between 2 and 6, let\u0027s take 3 plus 1 positive,"},{"Start":"20:46.615 ","End":"20:51.850","Text":"3 minus 2 positive, everything is positive."},{"Start":"20:51.850 ","End":"20:55.630","Text":"If we take on the other side 7,"},{"Start":"20:55.630 ","End":"20:58.657","Text":"let\u0027s say, this is squared, so positive."},{"Start":"20:58.657 ","End":"20:59.680","Text":"7 plus 1 is positive,"},{"Start":"20:59.680 ","End":"21:00.895","Text":"7 minus 2 positive."},{"Start":"21:00.895 ","End":"21:03.220","Text":"Once again, positive."},{"Start":"21:03.220 ","End":"21:05.020","Text":"What do we want?"},{"Start":"21:05.020 ","End":"21:09.170","Text":"We want positives without 0."},{"Start":"21:09.240 ","End":"21:11.740","Text":"What is good for us?"},{"Start":"21:11.740 ","End":"21:13.345","Text":"This is good for us."},{"Start":"21:13.345 ","End":"21:16.105","Text":"This is good for us and this is good for us."},{"Start":"21:16.105 ","End":"21:19.510","Text":"These are all the places where it\u0027s positive and so we can write that"},{"Start":"21:19.510 ","End":"21:24.610","Text":"down as x less than minus 1,"},{"Start":"21:24.610 ","End":"21:29.980","Text":"or x between 2 and"},{"Start":"21:29.980 ","End":"21:37.540","Text":"6 or x bigger than 6, 6 less than x."},{"Start":"21:37.540 ","End":"21:40.375","Text":"Either way, this is the answer."},{"Start":"21:40.375 ","End":"21:42.820","Text":"There I\u0027ve highlighted it."},{"Start":"21:42.820 ","End":"21:44.410","Text":"But that was previously."},{"Start":"21:44.410 ","End":"21:47.455","Text":"There is another way of describing it."},{"Start":"21:47.455 ","End":"21:50.245","Text":"Equally valid depends on your taste."},{"Start":"21:50.245 ","End":"22:00.620","Text":"To say that this person is merged together and just to say that x is bigger than 2,"},{"Start":"22:00.950 ","End":"22:05.670","Text":"except for this point."},{"Start":"22:05.670 ","End":"22:09.760","Text":"Because here it\u0027s not positive, it\u0027s 0."},{"Start":"22:10.700 ","End":"22:14.140","Text":"This or this,"},{"Start":"22:15.120 ","End":"22:25.675","Text":"but we\u0027ll have to put it in brackets and x not equal to 6."},{"Start":"22:25.675 ","End":"22:29.050","Text":"I\u0027m thinking this looks better."},{"Start":"22:29.050 ","End":"22:32.774","Text":"Let\u0027s just leave it at that."},{"Start":"22:32.774 ","End":"22:39.110","Text":"But possible. Of course,"},{"Start":"22:39.110 ","End":"22:41.610","Text":"that\u0027s the end of the exercise."}],"Thumbnail":null,"ID":5072},{"Watched":false,"Name":"Exercise 3","Duration":"31m 37s","ChapterTopicVideoID":5081,"CourseChapterTopicPlaylistID":3877,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.889","Text":"In this exercise we have five inequalities."},{"Start":"00:04.889 ","End":"00:10.349","Text":"The thing is that we would like to have 0 on the right."},{"Start":"00:10.349 ","End":"00:12.030","Text":"We want to compare something to 0."},{"Start":"00:12.030 ","End":"00:13.860","Text":"In this case it\u0027s not 0."},{"Start":"00:13.860 ","End":"00:17.890","Text":"All it means is that you have to do a bit of extra algebra,"},{"Start":"00:17.890 ","End":"00:23.234","Text":"do some manipulation in order to get the comparison to be to 0,"},{"Start":"00:23.234 ","End":"00:25.920","Text":"bigger or equal to or less than 0."},{"Start":"00:25.920 ","End":"00:28.575","Text":"So just a bit of extra work."},{"Start":"00:28.575 ","End":"00:31.409","Text":"Let\u0027s get started on number a,"},{"Start":"00:31.409 ","End":"00:34.840","Text":"every written below with a bit more space."},{"Start":"00:35.780 ","End":"00:40.359","Text":"In this one, I will bring the 3 over to the other side."},{"Start":"00:40.359 ","End":"00:46.858","Text":"We\u0027ll get x-1 over"},{"Start":"00:46.858 ","End":"00:56.330","Text":"3x+2+3 is bigger or equal to 0."},{"Start":"00:56.330 ","End":"01:01.504","Text":"I deliberately left a space here because I want to save a line by"},{"Start":"01:01.504 ","End":"01:09.369","Text":"writing this times 3x+2 over 3x+2."},{"Start":"01:09.369 ","End":"01:13.670","Text":"I\u0027m starting to work on a common denominator of 3x+2."},{"Start":"01:13.670 ","End":"01:17.380","Text":"I can multiply top and bottom by the same thing."},{"Start":"01:17.380 ","End":"01:26.895","Text":"Then if I put it all over 3x+2,"},{"Start":"01:26.895 ","End":"01:34.589","Text":"then I get x-1+3 times the numerator."},{"Start":"01:34.589 ","End":"01:39.250","Text":"That\u0027s 9x+6."},{"Start":"01:39.250 ","End":"01:41.465","Text":"That\u0027s bigger or equal to 0."},{"Start":"01:41.465 ","End":"01:43.564","Text":"Continue over here."},{"Start":"01:43.564 ","End":"01:51.739","Text":"X+9x is 10x-1+6 is"},{"Start":"01:51.739 ","End":"02:01.030","Text":"plus 5 over 3x+2 is bigger or equal to 0."},{"Start":"02:01.460 ","End":"02:10.276","Text":"I can take the 5 out and say 5 times 2x+1"},{"Start":"02:10.276 ","End":"02:19.305","Text":"over 3x+2 is bigger or equal to 0."},{"Start":"02:19.305 ","End":"02:21.634","Text":"Now I have something compared to 0."},{"Start":"02:21.634 ","End":"02:25.040","Text":"I could actually divide both sides by 5,"},{"Start":"02:25.040 ","End":"02:27.289","Text":"which is a positive number."},{"Start":"02:27.289 ","End":"02:29.825","Text":"In fact, why don\u0027t I do that."},{"Start":"02:29.825 ","End":"02:35.314","Text":"I\u0027ll just divide both sides by 5 and cross the 5 out."},{"Start":"02:35.314 ","End":"02:37.940","Text":"Not that it really hurts me to keep it there,"},{"Start":"02:37.940 ","End":"02:40.120","Text":"but might as well get rid of it."},{"Start":"02:40.120 ","End":"02:44.314","Text":"On the right-hand side, 0 over 5, which is 0."},{"Start":"02:44.314 ","End":"02:47.690","Text":"Now we\u0027re in the form that we\u0027re more familiar with."},{"Start":"02:47.690 ","End":"02:54.739","Text":"We want to put the values of x which make the numerator or denominator 0."},{"Start":"02:54.739 ","End":"02:57.350","Text":"Here it would be if 2x+1 is 0,"},{"Start":"02:57.350 ","End":"03:00.020","Text":"x is minus 1 over 2."},{"Start":"03:00.020 ","End":"03:01.640","Text":"Here to get this 0,"},{"Start":"03:01.640 ","End":"03:07.125","Text":"I need x to be minus 2 over 3."},{"Start":"03:07.125 ","End":"03:10.940","Text":"If I now sketch a number line, let\u0027s see,"},{"Start":"03:10.940 ","End":"03:12.800","Text":"it\u0027s not immediately obvious,"},{"Start":"03:12.800 ","End":"03:14.780","Text":"which is to the left."},{"Start":"03:14.780 ","End":"03:17.419","Text":"2 over 3 is bigger than 1 over 2,"},{"Start":"03:17.419 ","End":"03:24.135","Text":"so minus 2 over 3 is less than minus 1 over 2."},{"Start":"03:24.135 ","End":"03:26.340","Text":"This is the right order."},{"Start":"03:26.340 ","End":"03:32.194","Text":"The difference is that the minus 2 over 3 makes the denominator 0."},{"Start":"03:32.194 ","End":"03:34.490","Text":"The expression is not defined there,"},{"Start":"03:34.490 ","End":"03:38.255","Text":"so I erase a little bit and make a little hole there and"},{"Start":"03:38.255 ","End":"03:42.889","Text":"just fill it in with a hollow circle to point out the whole,"},{"Start":"03:42.889 ","End":"03:46.090","Text":"x cannot be minus 2 over 3."},{"Start":"03:46.090 ","End":"03:50.640","Text":"Right from the start, you can\u0027t have 3x+2 being 0."},{"Start":"03:50.640 ","End":"03:56.940","Text":"Minus 1 over 2 is a different story."},{"Start":"03:57.780 ","End":"04:01.450","Text":"That makes the whole expression equal to 0,"},{"Start":"04:01.450 ","End":"04:06.024","Text":"so this indicate with a solid circle."},{"Start":"04:06.024 ","End":"04:08.980","Text":"We have 3 intervals now and we want to know on"},{"Start":"04:08.980 ","End":"04:13.305","Text":"each interval whether we\u0027re positive or negative."},{"Start":"04:13.305 ","End":"04:16.315","Text":"Let\u0027s pick some sample values."},{"Start":"04:16.315 ","End":"04:21.860","Text":"Let\u0027s say here I\u0027ll take minus 1."},{"Start":"04:22.800 ","End":"04:26.725","Text":"What\u0027s the number between 2/3 and 1/2?"},{"Start":"04:26.725 ","End":"04:31.619","Text":"Between 0.5 and 0.666 and so on?"},{"Start":"04:31.619 ","End":"04:34.719","Text":"You could take minus 0.6."},{"Start":"04:34.719 ","End":"04:39.490","Text":"Let\u0027s take it as then minus 3 over 5."},{"Start":"04:39.490 ","End":"04:44.405","Text":"That should work. Here I\u0027ve got an easy one that\u0027s 0."},{"Start":"04:44.405 ","End":"04:48.290","Text":"Each of these three I\u0027ll plug in to this,"},{"Start":"04:48.290 ","End":"04:51.305","Text":"at least just to find out if it\u0027s positive or negative,"},{"Start":"04:51.305 ","End":"04:54.440","Text":"I put minus 1, then I\u0027ve got,"},{"Start":"04:54.440 ","End":"04:58.559","Text":"this is negative and minus 1 times 3,"},{"Start":"04:58.559 ","End":"05:03.070","Text":"also negative, so positive."},{"Start":"05:05.210 ","End":"05:08.694","Text":"Anywhere just above the line."},{"Start":"05:08.694 ","End":"05:11.735","Text":"For minus 3/5."},{"Start":"05:11.735 ","End":"05:14.240","Text":"Let\u0027s see what I get."},{"Start":"05:14.240 ","End":"05:21.280","Text":"Times 2 is minus 6/5 minus 1/5 plus 1 is still negative."},{"Start":"05:21.280 ","End":"05:29.180","Text":"Here 3 times minus 3/5 is minus 9/5 minus 1 and 4/5."},{"Start":"05:29.180 ","End":"05:32.180","Text":"The two brings it over the 0."},{"Start":"05:32.180 ","End":"05:37.615","Text":"Minus over plus is minus."},{"Start":"05:37.615 ","End":"05:41.164","Text":"Below here I\u0027ve got a point somewhere."},{"Start":"05:41.164 ","End":"05:43.939","Text":"For 0, that\u0027s easy."},{"Start":"05:43.939 ","End":"05:46.220","Text":"It\u0027s just 1 over 2,"},{"Start":"05:46.220 ","End":"05:51.145","Text":"positive over positive is positive."},{"Start":"05:51.145 ","End":"05:54.050","Text":"Somewhere, say here."},{"Start":"05:54.050 ","End":"06:00.090","Text":"Now let\u0027s say I\u0027ll use the snake method this time just to illustrate."},{"Start":"06:00.130 ","End":"06:04.530","Text":"I don\u0027t go through the point."},{"Start":"06:06.870 ","End":"06:14.720","Text":"Something like this goes on forever here and it goes on forever there."},{"Start":"06:14.760 ","End":"06:16.989","Text":"Now we\u0027ve got the sketch."},{"Start":"06:16.989 ","End":"06:18.340","Text":"Now let\u0027s see what we\u0027re collecting."},{"Start":"06:18.340 ","End":"06:22.910","Text":"We\u0027re collecting positives and zeros."},{"Start":"06:22.910 ","End":"06:28.390","Text":"Positive being above the axis and zero being on the axis."},{"Start":"06:28.390 ","End":"06:31.525","Text":"This is good for us."},{"Start":"06:31.525 ","End":"06:38.484","Text":"This bit and this bit of shading I\u0027ll use highlighting."},{"Start":"06:38.484 ","End":"06:48.859","Text":"A positive bit could be here but not quite reaching positive here,"},{"Start":"06:48.859 ","End":"06:54.129","Text":"but also on going for zeros so up to here."},{"Start":"06:54.560 ","End":"06:57.760","Text":"If I write that in terms of x, let\u0027s see,"},{"Start":"06:57.760 ","End":"07:03.519","Text":"this bit is x less than minus 2 over 3."},{"Start":"07:03.519 ","End":"07:06.680","Text":"It can\u0027t include the minus 2 over 3,"},{"Start":"07:06.750 ","End":"07:11.139","Text":"the expression, sorry is not defined there."},{"Start":"07:11.139 ","End":"07:18.099","Text":"Here. From minus 1 over 2 and further to the right as far as we can go."},{"Start":"07:18.099 ","End":"07:29.109","Text":"That would be or x the bigger or equal to minus 1.5."},{"Start":"07:29.109 ","End":"07:32.659","Text":"This is our answer."},{"Start":"07:33.480 ","End":"07:36.504","Text":"Onto the next one."},{"Start":"07:36.504 ","End":"07:39.580","Text":"Here we have to once again get a 0 on the right."},{"Start":"07:39.580 ","End":"07:48.180","Text":"I bring the 2 over to the other side, x-3 over x-2-2."},{"Start":"07:48.180 ","End":"07:49.925","Text":"But instead of just minus 2,"},{"Start":"07:49.925 ","End":"07:58.315","Text":"I can already multiply it by 1 and then multiply it by x-2 over x-2."},{"Start":"07:58.315 ","End":"08:01.300","Text":"Which is okay because this is 1."},{"Start":"08:01.300 ","End":"08:05.359","Text":"Then less than 0."},{"Start":"08:06.240 ","End":"08:13.600","Text":"Then I can put it all over a common denominator, x-2."},{"Start":"08:13.600 ","End":"08:16.144","Text":"Well, let\u0027s see if we can do it in our heads."},{"Start":"08:16.144 ","End":"08:23.625","Text":"X-3-2x+4."},{"Start":"08:23.625 ","End":"08:32.055","Text":"Let\u0027s see the x-2x is minus x and the minus 3 plus the 4 is plus 1,"},{"Start":"08:32.055 ","End":"08:37.090","Text":"and then less than 0."},{"Start":"08:37.090 ","End":"08:42.330","Text":"Let\u0027s see what makes the denominator 0 and the numerator."},{"Start":"08:42.770 ","End":"08:45.690","Text":"Here, it\u0027s x=2,"},{"Start":"08:45.690 ","End":"08:47.205","Text":"that will make it 0."},{"Start":"08:47.205 ","End":"08:50.930","Text":"Therefore 2 is not in the domain of this expression."},{"Start":"08:50.930 ","End":"08:54.830","Text":"We can\u0027t substitute x=2,"},{"Start":"08:54.830 ","End":"09:01.160","Text":"here minus x+1 is 0 when x= 1-1+1."},{"Start":"09:01.460 ","End":"09:05.660","Text":"These are the two numbers we have to put on the number line."},{"Start":"09:05.660 ","End":"09:07.640","Text":"Here\u0027s a number line,"},{"Start":"09:07.640 ","End":"09:11.775","Text":"here\u0027s 1, and here\u0027s 2."},{"Start":"09:11.775 ","End":"09:13.470","Text":"The thing is with 2,"},{"Start":"09:13.470 ","End":"09:15.854","Text":"the 2 is forbidden as a value."},{"Start":"09:15.854 ","End":"09:24.979","Text":"I like to make a little hole like so and fill it in with a hollow circle."},{"Start":"09:24.979 ","End":"09:28.400","Text":"The 1 belongs to the numerator."},{"Start":"09:28.400 ","End":"09:33.465","Text":"That\u0027s okay, that\u0027s where the expression gets a value of 0."},{"Start":"09:33.465 ","End":"09:38.875","Text":"Now we want to worry about the positives and the negatives, three separate intervals."},{"Start":"09:38.875 ","End":"09:41.719","Text":"Let\u0027s take a sample from each."},{"Start":"09:41.880 ","End":"09:48.717","Text":"Let\u0027s do this with the plus-minus system for a change."},{"Start":"09:48.717 ","End":"09:50.440","Text":"Its gotten a bit messy here,"},{"Start":"09:50.440 ","End":"09:54.549","Text":"let me just emphasize what it is that we\u0027re substituting into."},{"Start":"09:54.549 ","End":"09:58.330","Text":"This is the thing that we\u0027re substituting in."},{"Start":"09:58.330 ","End":"10:01.000","Text":"When x is 0,"},{"Start":"10:01.000 ","End":"10:03.630","Text":"we\u0027ve got 1 over minus 2,"},{"Start":"10:03.630 ","End":"10:10.305","Text":"which is negative and then if x is say,"},{"Start":"10:10.305 ","End":"10:15.789","Text":"1-1.5, that\u0027s negative."},{"Start":"10:15.789 ","End":"10:18.610","Text":"1.5-2 is also negative,"},{"Start":"10:18.610 ","End":"10:21.310","Text":"so that makes it positive here."},{"Start":"10:21.310 ","End":"10:25.599","Text":"Let\u0027s say if x is 3 and this interval,"},{"Start":"10:25.599 ","End":"10:28.669","Text":"1-3, that\u0027s negative,"},{"Start":"10:28.669 ","End":"10:32.424","Text":"3-2 is positive so here, negative."},{"Start":"10:32.424 ","End":"10:35.274","Text":"Now let\u0027s see what we\u0027re collecting."},{"Start":"10:35.274 ","End":"10:39.040","Text":"We want only negatives,"},{"Start":"10:39.040 ","End":"10:41.034","Text":"no zeros, no positives,"},{"Start":"10:41.034 ","End":"10:49.524","Text":"just negatives, negatives I have here and we have here."},{"Start":"10:49.524 ","End":"10:52.000","Text":"We want to take these two bits."},{"Start":"10:52.000 ","End":"10:54.819","Text":"Regardless of which circle they are,"},{"Start":"10:54.819 ","End":"10:58.180","Text":"we don\u0027t care about what\u0027s on the axis anyway."},{"Start":"10:58.180 ","End":"11:00.930","Text":"We only care about negative."},{"Start":"11:00.930 ","End":"11:06.105","Text":"This one gives us that x is less than 1."},{"Start":"11:06.105 ","End":"11:12.675","Text":"From here we get x is bigger than 2 and there\u0027s an or in-between."},{"Start":"11:12.675 ","End":"11:18.760","Text":"This is our answer so onto part C."},{"Start":"11:20.280 ","End":"11:22.509","Text":"In this one, once again,"},{"Start":"11:22.509 ","End":"11:24.370","Text":"we don\u0027t have a 0 on the right,"},{"Start":"11:24.370 ","End":"11:26.305","Text":"so we have to do some algebra."},{"Start":"11:26.305 ","End":"11:34.450","Text":"What we do is take this expression I\u0027ll just copy it first, x^2-3x+2."},{"Start":"11:34.450 ","End":"11:38.305","Text":"Then say -3,"},{"Start":"11:38.305 ","End":"11:41.545","Text":"then less than 0,"},{"Start":"11:41.545 ","End":"11:44.589","Text":"I\u0027ve left a gap here because I\u0027m"},{"Start":"11:44.589 ","End":"11:48.100","Text":"allowed to multiply numerator and denominator by the same thing."},{"Start":"11:48.100 ","End":"11:52.130","Text":"Let\u0027s take this and multiply it."},{"Start":"11:52.560 ","End":"11:57.789","Text":"X squared minus 3x plus 2 and on"},{"Start":"11:57.789 ","End":"12:04.540","Text":"the denominator also x squared minus 3x plus 2."},{"Start":"12:04.540 ","End":"12:11.394","Text":"Then we can write all this over the same denominator."},{"Start":"12:11.394 ","End":"12:13.674","Text":"Let me just continue over here."},{"Start":"12:13.674 ","End":"12:19.780","Text":"That denominator is going to be x squared minus 3x plus 2."},{"Start":"12:19.780 ","End":"12:27.174","Text":"We have x minus 3 minus 3 times"},{"Start":"12:27.174 ","End":"12:30.940","Text":"x squared minus 3x plus 2."},{"Start":"12:30.940 ","End":"12:35.184","Text":"All this less than 0."},{"Start":"12:35.184 ","End":"12:38.110","Text":"Let\u0027s see what we get in the numerator."},{"Start":"12:38.110 ","End":"12:41.259","Text":"Let\u0027s collect together the like terms."},{"Start":"12:41.259 ","End":"12:43.255","Text":"First of all, collect x squared."},{"Start":"12:43.255 ","End":"12:47.140","Text":"We have minus 3x squared from here."},{"Start":"12:47.140 ","End":"12:49.254","Text":"We have 1x from here,"},{"Start":"12:49.254 ","End":"12:52.645","Text":"and we have another plus 9x from here."},{"Start":"12:52.645 ","End":"12:58.045","Text":"That gives us 10x for numbers we have minus 3."},{"Start":"12:58.045 ","End":"13:04.749","Text":"Then we have minus 6 from this times this minus 3 minus 6 is minus"},{"Start":"13:04.749 ","End":"13:10.240","Text":"9 over x squared"},{"Start":"13:10.240 ","End":"13:17.209","Text":"minus 3x plus 2 is less than 0."},{"Start":"13:18.960 ","End":"13:23.020","Text":"Now, we\u0027re going to try and factorize this."},{"Start":"13:23.020 ","End":"13:26.770","Text":"The easier bit is the denominator."},{"Start":"13:26.770 ","End":"13:28.720","Text":"I\u0027ll just tell you what the answer is."},{"Start":"13:28.720 ","End":"13:32.079","Text":"So you too have two techniques that you can factorize with."},{"Start":"13:32.079 ","End":"13:34.735","Text":"The answer is x minus 1,"},{"Start":"13:34.735 ","End":"13:36.220","Text":"x minus 2,"},{"Start":"13:36.220 ","End":"13:37.614","Text":"or the other way round."},{"Start":"13:37.614 ","End":"13:43.165","Text":"Let\u0027s just check x squared minus x minus 2x minus 3x plus 2."},{"Start":"13:43.165 ","End":"13:47.409","Text":"The numerator has no roots."},{"Start":"13:47.409 ","End":"13:51.220","Text":"If you check the b squared minus 4ac,"},{"Start":"13:51.220 ","End":"13:55.315","Text":"it comes out to be 100 minus 4 times 3 times 9,"},{"Start":"13:55.315 ","End":"13:58.960","Text":"which is 108, it comes out minus 8."},{"Start":"13:58.960 ","End":"14:00.954","Text":"There are no roots."},{"Start":"14:00.954 ","End":"14:03.234","Text":"If there are no roots,"},{"Start":"14:03.234 ","End":"14:06.399","Text":"then polynomial or quadratic expression"},{"Start":"14:06.399 ","End":"14:09.595","Text":"has to be either always positive or always negative."},{"Start":"14:09.595 ","End":"14:13.375","Text":"How can I tell which it is if this is always positive or always negative?"},{"Start":"14:13.375 ","End":"14:18.639","Text":"Answer, simple, substitute any value you want. I\u0027ll choose 0."},{"Start":"14:18.639 ","End":"14:21.130","Text":"I plug in x equals 0,"},{"Start":"14:21.130 ","End":"14:24.490","Text":"then I\u0027ve got 00 and minus 9,"},{"Start":"14:24.490 ","End":"14:26.605","Text":"so it\u0027s always negative."},{"Start":"14:26.605 ","End":"14:33.804","Text":"I\u0027ll write the word negative just to emphasize it rather than just a minus sign."},{"Start":"14:33.804 ","End":"14:36.489","Text":"I\u0027ll just repeat. This is a quadratic."},{"Start":"14:36.489 ","End":"14:37.674","Text":"You try to factorize it,"},{"Start":"14:37.674 ","End":"14:41.980","Text":"you fail whatever method you use because there are no factors,"},{"Start":"14:41.980 ","End":"14:44.559","Text":"you can\u0027t factor it and it has no roots,"},{"Start":"14:44.559 ","End":"14:46.510","Text":"has no roots means it\u0027s never 0,"},{"Start":"14:46.510 ","End":"14:48.715","Text":"means it\u0027s always positive or always negative."},{"Start":"14:48.715 ","End":"14:50.769","Text":"Substitute the value 0,"},{"Start":"14:50.769 ","End":"14:52.855","Text":"get minus 9, which is negative."},{"Start":"14:52.855 ","End":"14:57.115","Text":"This is what we have. Now, less than 0."},{"Start":"14:57.115 ","End":"15:01.000","Text":"Now, we make a table, another table,"},{"Start":"15:01.000 ","End":"15:08.649","Text":"sketch with the number line and we place 1 and 2 on it."},{"Start":"15:08.649 ","End":"15:11.709","Text":"These are the places where the denominator is 0."},{"Start":"15:11.709 ","End":"15:15.859","Text":"For one thing, these are the illegal values."},{"Start":"15:16.410 ","End":"15:21.580","Text":"Let\u0027s put one here and two here."},{"Start":"15:21.580 ","End":"15:26.109","Text":"Both of these make the denominator 0,"},{"Start":"15:26.109 ","End":"15:28.179","Text":"so they\u0027re both illegal."},{"Start":"15:28.179 ","End":"15:32.875","Text":"Make little holes and make a hole here and a hole here."},{"Start":"15:32.875 ","End":"15:39.070","Text":"Further emphasize with a hollow circle or ellipse or whatever it is."},{"Start":"15:39.070 ","End":"15:40.930","Text":"You know that these are not allowed to be"},{"Start":"15:40.930 ","End":"15:43.749","Text":"substituted then they can\u0027t be part of the answer."},{"Start":"15:43.749 ","End":"15:46.570","Text":"Now, we have to find out what happens in each of"},{"Start":"15:46.570 ","End":"15:49.555","Text":"these three intervals whether we\u0027re positive or negative."},{"Start":"15:49.555 ","End":"15:51.669","Text":"We do sample values."},{"Start":"15:51.669 ","End":"15:54.935","Text":"If x is 0,"},{"Start":"15:54.935 ","End":"15:57.330","Text":"I\u0027m just write the samples, I\u0027m going to use,"},{"Start":"15:57.330 ","End":"15:58.649","Text":"I\u0027m going to use 0 here,"},{"Start":"15:58.649 ","End":"16:00.570","Text":"going to use 1 and 1/2 here."},{"Start":"16:00.570 ","End":"16:02.039","Text":"I\u0027m going to you use 3 here?"},{"Start":"16:02.039 ","End":"16:06.849","Text":"If x is 0, then I\u0027ve got 0 minus 1 is negative,"},{"Start":"16:06.849 ","End":"16:08.290","Text":"0 minus 2 is negative."},{"Start":"16:08.290 ","End":"16:10.179","Text":"This we know is negative, negative,"},{"Start":"16:10.179 ","End":"16:12.805","Text":"negative, negative an odd number of times."},{"Start":"16:12.805 ","End":"16:14.810","Text":"So it\u0027s negative."},{"Start":"16:14.810 ","End":"16:18.600","Text":"We put a point below the axis."},{"Start":"16:18.600 ","End":"16:23.100","Text":"1 1/2 minus 1 is positive,"},{"Start":"16:23.100 ","End":"16:25.739","Text":"minus 2 is negative, negative,"},{"Start":"16:25.739 ","End":"16:27.045","Text":"negative, positive."},{"Start":"16:27.045 ","End":"16:29.715","Text":"So positive."},{"Start":"16:29.715 ","End":"16:33.549","Text":"3 minus 1 is positive,"},{"Start":"16:33.549 ","End":"16:34.839","Text":"3 minus 2 is positive,"},{"Start":"16:34.839 ","End":"16:38.725","Text":"but then as a negative, so negative here."},{"Start":"16:38.725 ","End":"16:44.070","Text":"We sketch we call the snake here."},{"Start":"16:44.070 ","End":"16:47.129","Text":"But since these are invalid or illegal values,"},{"Start":"16:47.129 ","End":"16:49.390","Text":"we don\u0027t go through them."},{"Start":"16:49.580 ","End":"16:52.950","Text":"Something like this."},{"Start":"16:52.950 ","End":"17:01.104","Text":"Continue to wherever indefinitely."},{"Start":"17:01.104 ","End":"17:02.890","Text":"Let\u0027s see what we\u0027re interested in."},{"Start":"17:02.890 ","End":"17:06.205","Text":"We\u0027re interested in negative values."},{"Start":"17:06.205 ","End":"17:09.805","Text":"We don\u0027t care whether these are hollow or solid."},{"Start":"17:09.805 ","End":"17:13.760","Text":"We just care about what\u0027s negative."},{"Start":"17:14.580 ","End":"17:17.695","Text":"Say I use the shading method."},{"Start":"17:17.695 ","End":"17:20.889","Text":"Negative is this bit here,"},{"Start":"17:20.889 ","End":"17:23.380","Text":"this bit here to forever."},{"Start":"17:23.380 ","End":"17:26.319","Text":"We\u0027ve got two parts."},{"Start":"17:26.319 ","End":"17:29.799","Text":"This part represents 1 and downwards."},{"Start":"17:29.799 ","End":"17:32.230","Text":"In other words, x less than 1."},{"Start":"17:32.230 ","End":"17:36.190","Text":"This part represents whatever\u0027s above the 2,"},{"Start":"17:36.190 ","End":"17:37.465","Text":"not including the 2."},{"Start":"17:37.465 ","End":"17:39.610","Text":"Like here we didn\u0027t include the 1."},{"Start":"17:39.610 ","End":"17:42.700","Text":"It\u0027s x bigger than 2,"},{"Start":"17:42.700 ","End":"17:47.260","Text":"or we could write 2 less than x with an or joining them."},{"Start":"17:47.260 ","End":"17:50.829","Text":"This is the answer."},{"Start":"17:50.829 ","End":"17:58.270","Text":"We move on to Part D. In Part D, it\u0027s embarrassing."},{"Start":"17:58.270 ","End":"18:00.790","Text":"I miss copied the question it should not be a 7."},{"Start":"18:00.790 ","End":"18:02.485","Text":"It should be a 1."},{"Start":"18:02.485 ","End":"18:04.944","Text":"Sorry about that. Let\u0027s continue."},{"Start":"18:04.944 ","End":"18:08.035","Text":"I\u0027m going to bring the 1 to the left."},{"Start":"18:08.035 ","End":"18:11.470","Text":"So 2x plus 1."},{"Start":"18:11.470 ","End":"18:16.644","Text":"We\u0027ve done already a couple of these so we can do it to take a bit of a shortcut."},{"Start":"18:16.644 ","End":"18:19.420","Text":"I\u0027m going to bring it over to the left so it\u0027s minus 1,"},{"Start":"18:19.420 ","End":"18:23.904","Text":"but also multiply top and bottom by this denominator."},{"Start":"18:23.904 ","End":"18:28.460","Text":"I\u0027m going to have everything over this denominator."},{"Start":"18:30.450 ","End":"18:36.399","Text":"Here this minus 1 will be minus the denominator over the denominator,"},{"Start":"18:36.399 ","End":"18:42.535","Text":"I have minus x squared minus 6x plus 8."},{"Start":"18:42.535 ","End":"18:46.915","Text":"I save this step because it\u0027s a bit cramped here."},{"Start":"18:46.915 ","End":"18:49.705","Text":"Let\u0027s see what this gives us."},{"Start":"18:49.705 ","End":"18:59.904","Text":"This gives us the numerator I have minus x squared."},{"Start":"18:59.904 ","End":"19:04.030","Text":"Let\u0027s see, minus is"},{"Start":"19:04.030 ","End":"19:11.510","Text":"plus 6x plus 2x is 8x."},{"Start":"19:11.880 ","End":"19:21.760","Text":"Then 1 minus 8 is minus 7 over same thing,"},{"Start":"19:21.760 ","End":"19:26.485","Text":"x squared minus 6x plus 8."},{"Start":"19:26.485 ","End":"19:29.155","Text":"It\u0027s getting a bit cramped."},{"Start":"19:29.155 ","End":"19:32.690","Text":"I\u0027ll just move this over here."},{"Start":"19:33.120 ","End":"19:40.374","Text":"That\u0027s better Now, it\u0027s still bigger or equal to 0."},{"Start":"19:40.374 ","End":"19:43.764","Text":"Did I not write bigger or equal to 0 here?"},{"Start":"19:43.764 ","End":"19:47.019","Text":"Now, we\u0027re going to factorize this."},{"Start":"19:47.019 ","End":"19:51.040","Text":"Tell you what though, I don\u0027t like this minus here."},{"Start":"19:51.040 ","End":"19:53.425","Text":"Multiply both sides by minus 1."},{"Start":"19:53.425 ","End":"19:55.420","Text":"It gets multiplied on the top."},{"Start":"19:55.420 ","End":"20:00.130","Text":"It becomes x squared minus 8x plus"},{"Start":"20:00.130 ","End":"20:06.675","Text":"7 over x squared minus 6x plus 8."},{"Start":"20:06.675 ","End":"20:09.374","Text":"Because I multiplied by a minus,"},{"Start":"20:09.374 ","End":"20:13.365","Text":"it becomes less than or equal to 0."},{"Start":"20:13.365 ","End":"20:19.270","Text":"Now, it\u0027s easier for me to see with x-squared having a positive coefficient."},{"Start":"20:19.270 ","End":"20:21.009","Text":"We\u0027re going to factorize."},{"Start":"20:21.009 ","End":"20:25.614","Text":"I\u0027ll let you check this on your own by"},{"Start":"20:25.614 ","End":"20:31.330","Text":"either method the trinomial decomposition or finding the two roots."},{"Start":"20:31.330 ","End":"20:38.200","Text":"In any event what we get here is x minus 7,"},{"Start":"20:38.200 ","End":"20:43.315","Text":"x minus 1 on the top, on the bottom,"},{"Start":"20:43.315 ","End":"20:46.490","Text":"x minus 4,"},{"Start":"20:46.650 ","End":"20:49.345","Text":"x minus 2,"},{"Start":"20:49.345 ","End":"20:53.000","Text":"and all this less than or equal to 0."},{"Start":"20:53.250 ","End":"21:00.819","Text":"The values of x, which makes the top 0, 7 and 1."},{"Start":"21:00.819 ","End":"21:05.079","Text":"For the bottom, we need 4 and 2,"},{"Start":"21:05.079 ","End":"21:08.109","Text":"and these are values at which the expression is"},{"Start":"21:08.109 ","End":"21:11.590","Text":"not defined because it makes a 0 on the denominator."},{"Start":"21:11.590 ","End":"21:14.904","Text":"In the event we put all these four on the number line."},{"Start":"21:14.904 ","End":"21:16.539","Text":"Here\u0027s a number line."},{"Start":"21:16.539 ","End":"21:18.459","Text":"Let\u0027s put them in the right order."},{"Start":"21:18.459 ","End":"21:20.785","Text":"The order is first the 1,"},{"Start":"21:20.785 ","End":"21:22.450","Text":"then the 2,"},{"Start":"21:22.450 ","End":"21:26.035","Text":"then the 4 and then the 7."},{"Start":"21:26.035 ","End":"21:28.000","Text":"Let\u0027s see which ones are the bad ones,"},{"Start":"21:28.000 ","End":"21:29.410","Text":"2 and 4 bad are the bad ones,"},{"Start":"21:29.410 ","End":"21:31.959","Text":"meaning the expression is not defined there."},{"Start":"21:31.959 ","End":"21:38.265","Text":"I indicate that by making a little hole here and a little hole"},{"Start":"21:38.265 ","End":"21:45.384","Text":"here and maybe further emphasizing it with a little hollow circle."},{"Start":"21:45.384 ","End":"21:47.266","Text":"At 1 and 7,"},{"Start":"21:47.266 ","End":"21:53.454","Text":"it\u0027s just the place where the expression is 0, the numerator is 0."},{"Start":"21:53.454 ","End":"21:56.379","Text":"We have no problem with this."},{"Start":"21:56.379 ","End":"22:02.229","Text":"These are allowed, and we get a 0 here for the rational expression."},{"Start":"22:02.229 ","End":"22:04.615","Text":"Now what are we collecting today?"},{"Start":"22:04.615 ","End":"22:09.235","Text":"We are collecting after we did all this transformation,"},{"Start":"22:09.235 ","End":"22:13.150","Text":"we are now going for less than or equal to 0."},{"Start":"22:13.150 ","End":"22:17.515","Text":"It means negatives and 0s."},{"Start":"22:17.515 ","End":"22:22.855","Text":"But we haven\u0027t yet said where we\u0027re positive or negative."},{"Start":"22:22.855 ","End":"22:27.100","Text":"Let\u0027s see. I\u0027ll use the method of the plus and minus."},{"Start":"22:27.100 ","End":"22:30.925","Text":"0 is in this first interval."},{"Start":"22:30.925 ","End":"22:32.529","Text":"If I put 0 in,"},{"Start":"22:32.529 ","End":"22:34.180","Text":"I\u0027ve got minus, minus,"},{"Start":"22:34.180 ","End":"22:37.960","Text":"minus, minus, altogether plus."},{"Start":"22:37.960 ","End":"22:41.380","Text":"If I try 1.5,"},{"Start":"22:41.380 ","End":"22:45.955","Text":"then this will be negative, positive,"},{"Start":"22:45.955 ","End":"22:50.320","Text":"negative, negative, 3 negatives."},{"Start":"22:50.320 ","End":"22:52.165","Text":"If I try 3,"},{"Start":"22:52.165 ","End":"22:54.730","Text":"I\u0027ve got negative,"},{"Start":"22:54.730 ","End":"22:58.269","Text":"positive, negative, positive."},{"Start":"22:58.269 ","End":"23:00.370","Text":"Two negatives is a positive."},{"Start":"23:00.370 ","End":"23:02.680","Text":"If I try x equals 5."},{"Start":"23:02.680 ","End":"23:05.050","Text":"This is 5 minus 7 is negative."},{"Start":"23:05.050 ","End":"23:08.350","Text":"5 minus 1 is positive, positive,"},{"Start":"23:08.350 ","End":"23:13.825","Text":"positive, three positives and a negative negative."},{"Start":"23:13.825 ","End":"23:19.129","Text":"The last you can check for in x equals 8, it\u0027s positive."},{"Start":"23:19.260 ","End":"23:22.299","Text":"All of them come up positive."},{"Start":"23:22.299 ","End":"23:24.395","Text":"Now, as we said,"},{"Start":"23:24.395 ","End":"23:27.655","Text":"we\u0027re collecting negative and 0."},{"Start":"23:27.655 ","End":"23:29.080","Text":"Let\u0027s see what\u0027s good for us."},{"Start":"23:29.080 ","End":"23:30.909","Text":"Negative is good for us,"},{"Start":"23:30.909 ","End":"23:32.620","Text":"negative is good for us."},{"Start":"23:32.620 ","End":"23:37.164","Text":"We\u0027re collecting those and we\u0027re also collecting 0,"},{"Start":"23:37.164 ","End":"23:42.280","Text":"so this is good and this is good."},{"Start":"23:42.280 ","End":"23:48.830","Text":"What do we get altogether from the strange symbols here?"},{"Start":"23:49.230 ","End":"23:54.040","Text":"This negative is all the way from 1 to 2."},{"Start":"23:54.040 ","End":"23:58.524","Text":"The fact that I\u0027m collecting zeros also means that 1 is in,"},{"Start":"23:58.524 ","End":"24:01.765","Text":"2 is not in because it\u0027s just not defined there."},{"Start":"24:01.765 ","End":"24:07.825","Text":"We get that x is between 1 and 2, but including,"},{"Start":"24:07.825 ","End":"24:10.405","Text":"I\u0027m stressing this, including the 1,"},{"Start":"24:10.405 ","End":"24:15.770","Text":"and as for the other bit between 4 and 7."},{"Start":"24:17.220 ","End":"24:19.810","Text":"We\u0027re also collecting zeros,"},{"Start":"24:19.810 ","End":"24:21.009","Text":"so the 7 is in,"},{"Start":"24:21.009 ","End":"24:25.585","Text":"so make sure that this is less than or equal to."},{"Start":"24:25.585 ","End":"24:33.819","Text":"There is an or in the middle and I highlight it and that\u0027s it."},{"Start":"24:33.819 ","End":"24:38.020","Text":"Now on to part E. I\u0027m going to do two steps in one,"},{"Start":"24:38.020 ","End":"24:39.490","Text":"so I\u0027m going to bring this over to"},{"Start":"24:39.490 ","End":"24:42.835","Text":"the other side and I\u0027m going to put a common denominator."},{"Start":"24:42.835 ","End":"24:48.985","Text":"What we get is 7x squared plus 1x,"},{"Start":"24:48.985 ","End":"24:55.690","Text":"which is just like x over 3x minus 4."},{"Start":"24:55.690 ","End":"25:02.860","Text":"Then I bring this over to the other side and it\u0027s minus 2x minus 10."},{"Start":"25:02.860 ","End":"25:05.605","Text":"Only I\u0027m not going to leave it as 2x minus 10."},{"Start":"25:05.605 ","End":"25:12.505","Text":"I\u0027m going to multiply top and bottom by 3x minus 4, 3x minus 4."},{"Start":"25:12.505 ","End":"25:15.468","Text":"I haven\u0027t changed anything because it\u0027s top and bottom,"},{"Start":"25:15.468 ","End":"25:18.530","Text":"and now it\u0027s bigger or equal to 0."},{"Start":"25:18.660 ","End":"25:23.120","Text":"Now I can put the common denominator,"},{"Start":"25:23.120 ","End":"25:32.679","Text":"so 7x squared plus 1x, I\u0027ll just write it as x."},{"Start":"25:32.679 ","End":"25:40.300","Text":"Now let\u0027s see minus and I\u0027m going to multiply this by this."},{"Start":"25:40.300 ","End":"25:44.110","Text":"Just put the brackets for the moment and it\u0027s all going to be over"},{"Start":"25:44.110 ","End":"25:51.009","Text":"3x minus 4 and bigger or equal to 0."},{"Start":"25:51.009 ","End":"25:54.445","Text":"What we get inside the brackets, let\u0027s see."},{"Start":"25:54.445 ","End":"25:57.294","Text":"2x times 3x is"},{"Start":"25:57.294 ","End":"26:06.684","Text":"6x squared minus 30x minus 8x is minus 38."},{"Start":"26:06.684 ","End":"26:12.470","Text":"X minus 10 times minus 4 is plus 40."},{"Start":"26:14.370 ","End":"26:17.724","Text":"Let\u0027s do one more step."},{"Start":"26:17.724 ","End":"26:24.550","Text":"We can combine the numerator collect terms."},{"Start":"26:24.550 ","End":"26:29.830","Text":"7x squared minus 6x squared is x squared."},{"Start":"26:29.830 ","End":"26:34.015","Text":"Let\u0027s see, plus x plus 30x,"},{"Start":"26:34.015 ","End":"26:39.820","Text":"that\u0027s minus minus makes it plus 39x."},{"Start":"26:39.820 ","End":"26:45.160","Text":"Then we have a minus 40"},{"Start":"26:45.160 ","End":"26:52.165","Text":"on all this over 3x minus 4,"},{"Start":"26:52.165 ","End":"26:55.309","Text":"bigger or equal to 0."},{"Start":"26:55.500 ","End":"27:00.310","Text":"Now I[\u0027ll factorize this for you to save time."},{"Start":"27:00.310 ","End":"27:08.380","Text":"What we get here is x plus 40 times x minus 1."},{"Start":"27:08.380 ","End":"27:09.729","Text":"You can check it,"},{"Start":"27:09.729 ","End":"27:16.150","Text":"x squared plus 4tx minus x is 39x and minus 40."},{"Start":"27:16.150 ","End":"27:25.960","Text":"On the denominator, we have 3x minus 4 and bigger or equal to 0."},{"Start":"27:25.960 ","End":"27:31.375","Text":"What we need are the roots for the numerator and for the denominator."},{"Start":"27:31.375 ","End":"27:41.455","Text":"I\u0027m going to rewrite the denominator as 3 times x minus 4 over 3."},{"Start":"27:41.455 ","End":"27:44.155","Text":"You can see here, if I take the 3 out."},{"Start":"27:44.155 ","End":"27:52.420","Text":"Then we can easiest see the roots of the numerator minus"},{"Start":"27:52.420 ","End":"28:01.014","Text":"40 and 1 and the roots of the denominator,"},{"Start":"28:01.014 ","End":"28:04.195","Text":"that\u0027s just one of them and it\u0027s 4 over 3."},{"Start":"28:04.195 ","End":"28:09.100","Text":"These are the points we have to put on the number line."},{"Start":"28:09.100 ","End":"28:13.810","Text":"Let\u0027s see. Somewhere here we have minus 40."},{"Start":"28:13.810 ","End":"28:16.225","Text":"Here we have 1,"},{"Start":"28:16.225 ","End":"28:17.980","Text":"and here we have 4/3,"},{"Start":"28:17.980 ","End":"28:19.824","Text":"which is 1 and 4/3."},{"Start":"28:19.824 ","End":"28:23.120","Text":"Let\u0027s just call it 4/3."},{"Start":"28:23.280 ","End":"28:29.410","Text":"What happens is that here and here,"},{"Start":"28:29.410 ","End":"28:32.244","Text":"the expression is 0."},{"Start":"28:32.244 ","End":"28:34.390","Text":"But here it\u0027s undefined,"},{"Start":"28:34.390 ","End":"28:37.480","Text":"so it\u0027s a hollow circle."},{"Start":"28:37.480 ","End":"28:40.435","Text":"Then we need sample points."},{"Start":"28:40.435 ","End":"28:44.470","Text":"Let\u0027s take minus 41."},{"Start":"28:44.470 ","End":"28:46.737","Text":"Here I can take 0."},{"Start":"28:46.737 ","End":"28:50.635","Text":"Here I can take 2,"},{"Start":"28:50.635 ","End":"28:54.520","Text":"and if we substitute minus 41,"},{"Start":"28:54.520 ","End":"28:59.287","Text":"then everything here is negative,"},{"Start":"28:59.287 ","End":"29:02.620","Text":"negative, negative, negative means negative."},{"Start":"29:02.620 ","End":"29:08.599","Text":"Below the line, below 0."},{"Start":"29:08.599 ","End":"29:12.369","Text":"We have positive, negative,"},{"Start":"29:12.369 ","End":"29:18.349","Text":"negative, so all together positive, above the line."},{"Start":"29:20.790 ","End":"29:32.599","Text":"Let\u0027s see. I think I missed out a sample point between 1 and 4 over 3."},{"Start":"29:33.840 ","End":"29:42.340","Text":"I don\u0027t know. Let\u0027s say 1.2 or something because this is 1.333."},{"Start":"29:42.340 ","End":"29:46.524","Text":"Anyway, if I put in 1.2,"},{"Start":"29:46.524 ","End":"29:49.210","Text":"then this is positive, this is positive,"},{"Start":"29:49.210 ","End":"29:52.225","Text":"but this is negative."},{"Start":"29:52.225 ","End":"29:58.720","Text":"It\u0027s below the line somewhere here."},{"Start":"29:58.720 ","End":"30:00.910","Text":"When we put in 2,"},{"Start":"30:00.910 ","End":"30:03.530","Text":"everything here is positive,"},{"Start":"30:03.530 ","End":"30:05.740","Text":"so we\u0027re above the line."},{"Start":"30:05.740 ","End":"30:12.925","Text":"If we use the snake method and a snake look something like this, positive,"},{"Start":"30:12.925 ","End":"30:17.649","Text":"zero, negative, undefined here,"},{"Start":"30:17.649 ","End":"30:22.465","Text":"positive and continues to be positive forever."},{"Start":"30:22.465 ","End":"30:25.550","Text":"Here, negative forever."},{"Start":"30:26.220 ","End":"30:28.869","Text":"Let\u0027s see now. What do we want?"},{"Start":"30:28.869 ","End":"30:30.820","Text":"We want bigger or equal to 0,"},{"Start":"30:30.820 ","End":"30:35.139","Text":"which means we want positives and we want 0."},{"Start":"30:35.139 ","End":"30:37.765","Text":"I\u0027ll use the highlighting method."},{"Start":"30:37.765 ","End":"30:40.914","Text":"For positive or 0, so this is in,"},{"Start":"30:40.914 ","End":"30:42.860","Text":"and all this is positive,"},{"Start":"30:42.860 ","End":"30:46.420","Text":"so this is including this point."},{"Start":"30:46.420 ","End":"30:51.919","Text":"Here, this is not included, so this bit."},{"Start":"30:52.140 ","End":"30:55.480","Text":"If we look what this means on the x-axis,"},{"Start":"30:55.480 ","End":"31:00.489","Text":"it means that we go from minus 40 inclusive"},{"Start":"31:00.489 ","End":"31:07.180","Text":"up to 1 inclusive and everything in-between."},{"Start":"31:07.180 ","End":"31:10.330","Text":"Then from 4 over 3,"},{"Start":"31:10.330 ","End":"31:15.415","Text":"but not inclusive, and on indefinitely."},{"Start":"31:15.415 ","End":"31:19.270","Text":"If we write that down in terms of x,"},{"Start":"31:19.270 ","End":"31:24.069","Text":"it means that minus 40 less than or equal to x,"},{"Start":"31:24.069 ","End":"31:26.155","Text":"less than or equal to 1,"},{"Start":"31:26.155 ","End":"31:36.700","Text":"or x bigger than 4 over 3 and that\u0027s it."}],"Thumbnail":null,"ID":5073},{"Watched":false,"Name":"Exercise 4","Duration":"22m 22s","ChapterTopicVideoID":5072,"CourseChapterTopicPlaylistID":3877,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.360","Text":"Here we have a couple of exercises with rational expressions or almost."},{"Start":"00:06.360 ","End":"00:08.940","Text":"We just have to do a little bit of work bringing"},{"Start":"00:08.940 ","End":"00:12.030","Text":"everything to one side and putting a common denominator."},{"Start":"00:12.030 ","End":"00:15.930","Text":"Let\u0027s start with the first one, part A."},{"Start":"00:15.930 ","End":"00:18.390","Text":"First thing I\u0027ll bring everything over to the left"},{"Start":"00:18.390 ","End":"00:21.644","Text":"because the idea is to compare something to 0."},{"Start":"00:21.644 ","End":"00:27.120","Text":"We\u0027ll do 25 over x^2 minus x plus 7"},{"Start":"00:27.120 ","End":"00:32.595","Text":"over x minus 3 over x plus 2,"},{"Start":"00:32.595 ","End":"00:35.440","Text":"bigger or equal to 0."},{"Start":"00:35.650 ","End":"00:39.980","Text":"Clearly, the common denominator we want is x^2,"},{"Start":"00:39.980 ","End":"00:43.685","Text":"so let\u0027s bring everything over x^2."},{"Start":"00:43.685 ","End":"00:46.880","Text":"I\u0027m going to write a dividing line here with x^2"},{"Start":"00:46.880 ","End":"00:49.903","Text":"on the bottom and bigger or equal to 0 here."},{"Start":"00:49.903 ","End":"00:52.145","Text":"Let\u0027s just see what each one of them is."},{"Start":"00:52.145 ","End":"00:55.505","Text":"Here we have just 25 over x^2."},{"Start":"00:55.505 ","End":"00:57.515","Text":"For this one and for this one,"},{"Start":"00:57.515 ","End":"01:03.165","Text":"we\u0027ll multiply top and bottom by x. I have minus,"},{"Start":"01:03.165 ","End":"01:06.680","Text":"and the minus is at the side here in case it wasn\u0027t clear,"},{"Start":"01:06.680 ","End":"01:12.150","Text":"so it\u0027s minus x times x plus 7."},{"Start":"01:12.740 ","End":"01:18.660","Text":"Here we have minus 3x multiplying top and bottom"},{"Start":"01:18.660 ","End":"01:24.670","Text":"by x and here we need to multiply by x^2."},{"Start":"01:24.670 ","End":"01:30.830","Text":"This is 2x^2 over x^2 is 2, bigger or equal to 0."},{"Start":"01:30.830 ","End":"01:34.460","Text":"Now we just want to simplify the numerator here."},{"Start":"01:34.460 ","End":"01:37.615","Text":"Let\u0027s see if we can do it in one go in our heads."},{"Start":"01:37.615 ","End":"01:40.080","Text":"I\u0027ll collect the x^2 first."},{"Start":"01:40.080 ","End":"01:42.110","Text":"We have 2x^2 from here,"},{"Start":"01:42.110 ","End":"01:44.600","Text":"but minus x^2 from here,"},{"Start":"01:44.600 ","End":"01:47.850","Text":"so that\u0027s 1x^2. Let\u0027s see."},{"Start":"01:47.850 ","End":"01:52.875","Text":"For xs, we have minus 7x from this times this,"},{"Start":"01:52.875 ","End":"01:57.310","Text":"and minus 3x here, so minus 10x."},{"Start":"01:57.650 ","End":"02:01.740","Text":"As for numbers, just the 25."},{"Start":"02:01.740 ","End":"02:07.955","Text":"Plus 25, all this over x^2,"},{"Start":"02:07.955 ","End":"02:10.640","Text":"bigger or equal to 0."},{"Start":"02:10.640 ","End":"02:15.590","Text":"Now we want to factorize a rational expression."},{"Start":"02:15.590 ","End":"02:18.838","Text":"Denominator is already factorized as x times x,"},{"Start":"02:18.838 ","End":"02:23.945","Text":"and here it\u0027s one of those special binomial products."},{"Start":"02:23.945 ","End":"02:25.670","Text":"To just remind you which one I\u0027m talking about."},{"Start":"02:25.670 ","End":"02:29.620","Text":"I\u0027m talking about the a minus b squared,"},{"Start":"02:29.620 ","End":"02:35.178","Text":"is a^2 minus 2ab plus b^2,"},{"Start":"02:35.178 ","End":"02:38.660","Text":"only this time I\u0027m going from right to left on this one."},{"Start":"02:38.660 ","End":"02:41.945","Text":"If I put a is x and b is 5,"},{"Start":"02:41.945 ","End":"02:46.110","Text":"you\u0027ll see that 2ab is really 10x."},{"Start":"02:46.110 ","End":"02:50.630","Text":"Not sure about it, just think about it for a moment or expand."},{"Start":"02:50.630 ","End":"02:59.070","Text":"This is x minus 5 squared over x^2 is bigger or equal to 0."},{"Start":"02:59.830 ","End":"03:05.690","Text":"Now as usual, we see what makes the denominator 0,"},{"Start":"03:05.690 ","End":"03:08.045","Text":"what makes the numerator 0."},{"Start":"03:08.045 ","End":"03:10.730","Text":"Denominator is 0 when x is 0."},{"Start":"03:10.730 ","End":"03:15.020","Text":"Numerator is 0 when x minus 5 is 0,"},{"Start":"03:15.020 ","End":"03:17.330","Text":"because the square is 0 when the thing itself is 0."},{"Start":"03:17.330 ","End":"03:20.885","Text":"In other words, x is 5."},{"Start":"03:20.885 ","End":"03:23.750","Text":"Sketch this on a number line."},{"Start":"03:23.750 ","End":"03:26.460","Text":"Here\u0027s a number line."},{"Start":"03:26.480 ","End":"03:29.250","Text":"We need 0 and 5 on it,"},{"Start":"03:29.250 ","End":"03:32.450","Text":"0 here, 5 here."},{"Start":"03:32.450 ","End":"03:39.875","Text":"The difference between them is that 5 is the value which makes the expression 0,"},{"Start":"03:39.875 ","End":"03:44.300","Text":"but 0 is the value which makes the expression undefined."},{"Start":"03:44.300 ","End":"03:47.075","Text":"It\u0027s illegal, invalid,"},{"Start":"03:47.075 ","End":"03:51.520","Text":"not good input, not in the domain, whatever."},{"Start":"03:51.520 ","End":"03:59.885","Text":"I indicate that by taking a little piece out of the x-axis here."},{"Start":"03:59.885 ","End":"04:01.970","Text":"Something like this."},{"Start":"04:01.970 ","End":"04:06.580","Text":"Just so you see it, I like to put a bit of a hollow circle here."},{"Start":"04:06.580 ","End":"04:10.570","Text":"In other words, this is not allowed for x at all in this expression."},{"Start":"04:10.570 ","End":"04:12.310","Text":"The 5 is a different story."},{"Start":"04:12.310 ","End":"04:16.370","Text":"That\u0027s the place where the expression is 0."},{"Start":"04:17.830 ","End":"04:21.465","Text":"Indicate it like that."},{"Start":"04:21.465 ","End":"04:26.409","Text":"All we have left is to find out where it\u0027s positive and negative."},{"Start":"04:26.409 ","End":"04:29.910","Text":"We know we\u0027re at 0, we\u0027re at positive and negative,"},{"Start":"04:29.910 ","End":"04:33.085","Text":"so we take a sample value in each interval."},{"Start":"04:33.085 ","End":"04:37.915","Text":"For example, I could choose minus 1 for this interval."},{"Start":"04:37.915 ","End":"04:40.719","Text":"I could choose 1 for this interval,"},{"Start":"04:40.719 ","End":"04:43.270","Text":"and I could choose 6 for this interval."},{"Start":"04:43.270 ","End":"04:45.600","Text":"Then if I plugged it in here,"},{"Start":"04:45.600 ","End":"04:47.340","Text":"each one separately,"},{"Start":"04:47.340 ","End":"04:49.580","Text":"for the minus 1,"},{"Start":"04:49.580 ","End":"04:53.330","Text":"we get something not 0."},{"Start":"04:53.330 ","End":"04:57.415","Text":"Squared is positive and x^2 is also positive."},{"Start":"04:57.415 ","End":"05:00.170","Text":"I think you can see that we\u0027ll always get something"},{"Start":"05:00.170 ","End":"05:02.585","Text":"positive because of the squared and squared."},{"Start":"05:02.585 ","End":"05:07.010","Text":"Here we\u0027ll get a positive number and x is 1,"},{"Start":"05:07.010 ","End":"05:08.840","Text":"1 minus 5 squared is positive,"},{"Start":"05:08.840 ","End":"05:09.910","Text":"1^2 is positive,"},{"Start":"05:09.910 ","End":"05:12.905","Text":"we\u0027ll also get some other positive number."},{"Start":"05:12.905 ","End":"05:17.615","Text":"At 6, each of these is not 0,"},{"Start":"05:17.615 ","End":"05:19.280","Text":"and when your square is positive,"},{"Start":"05:19.280 ","End":"05:20.690","Text":"so another positive number,"},{"Start":"05:20.690 ","End":"05:22.355","Text":"I don\u0027t know, say here."},{"Start":"05:22.355 ","End":"05:23.960","Text":"It\u0027s just schematic here,"},{"Start":"05:23.960 ","End":"05:25.250","Text":"it\u0027s nothing to scale,"},{"Start":"05:25.250 ","End":"05:27.355","Text":"it\u0027s not a graph or anything."},{"Start":"05:27.355 ","End":"05:31.339","Text":"Let\u0027s join these up in what we call the snake."},{"Start":"05:31.339 ","End":"05:34.205","Text":"It goes on indefinitely,"},{"Start":"05:34.205 ","End":"05:36.830","Text":"and then down here,"},{"Start":"05:36.830 ","End":"05:40.050","Text":"but doesn\u0027t go through this point,"},{"Start":"05:40.050 ","End":"05:42.570","Text":"this point is illegal."},{"Start":"05:42.570 ","End":"05:44.730","Text":"Something like this,"},{"Start":"05:44.730 ","End":"05:48.575","Text":"maybe down here and up here,"},{"Start":"05:48.575 ","End":"05:50.450","Text":"because it\u0027s going to stay positive."},{"Start":"05:50.450 ","End":"05:53.000","Text":"It\u0027s positive everywhere except"},{"Start":"05:53.000 ","End":"05:56.495","Text":"for this point where it\u0027s undefined and this point where it\u0027s 0."},{"Start":"05:56.495 ","End":"05:58.910","Text":"Sometimes I\u0027ll put an arrow to say, okay,"},{"Start":"05:58.910 ","End":"06:01.430","Text":"the graph stopped just because it ran out of space,"},{"Start":"06:01.430 ","End":"06:02.980","Text":"but it doesn\u0027t really stop,"},{"Start":"06:02.980 ","End":"06:04.445","Text":"it goes on forever."},{"Start":"06:04.445 ","End":"06:06.455","Text":"That\u0027s it. What are we collecting?"},{"Start":"06:06.455 ","End":"06:09.725","Text":"We\u0027re collecting bigger or equal to 0,"},{"Start":"06:09.725 ","End":"06:14.070","Text":"so positive is good and 0 is good."},{"Start":"06:14.080 ","End":"06:19.250","Text":"Positive we have, instead of the shading method,"},{"Start":"06:19.250 ","End":"06:20.600","Text":"I\u0027ll use the highlighting method."},{"Start":"06:20.600 ","End":"06:27.105","Text":"Positive is here up to not including the 0, which is illegal."},{"Start":"06:27.105 ","End":"06:31.223","Text":"Here we have a positive bit,"},{"Start":"06:31.223 ","End":"06:34.900","Text":"and here we have a positive bit,"},{"Start":"06:35.720 ","End":"06:40.245","Text":"0 is also good for us so this is also good."},{"Start":"06:40.245 ","End":"06:43.645","Text":"It means that the whole thing here is good."},{"Start":"06:43.645 ","End":"06:45.945","Text":"Now if we look at it,"},{"Start":"06:45.945 ","End":"06:51.230","Text":"what we\u0027ve covered is everything except for the 1 hole here."},{"Start":"06:51.230 ","End":"07:01.390","Text":"The answer is going to be x not equal to 0 and you could leave it at that."},{"Start":"07:01.390 ","End":"07:08.270","Text":"However, some people would look at it differently and you could write that this bit is"},{"Start":"07:08.270 ","End":"07:15.645","Text":"x less than 0 and this bit is x bigger than 0."},{"Start":"07:15.645 ","End":"07:20.150","Text":"You could say x is less than 0 or x is bigger than 0,"},{"Start":"07:20.150 ","End":"07:22.340","Text":"which is logically the same as this,"},{"Start":"07:22.340 ","End":"07:26.585","Text":"but this is an alternative answer that\u0027s perfectly acceptable too."},{"Start":"07:26.585 ","End":"07:29.765","Text":"Anyway, we\u0027re done."},{"Start":"07:29.765 ","End":"07:34.205","Text":"But I\u0027d just like to note that we could\u0027ve taken a shortcut."},{"Start":"07:34.205 ","End":"07:37.420","Text":"Once we got to this point,"},{"Start":"07:37.420 ","End":"07:40.985","Text":"we could see here that both on top and bottom,"},{"Start":"07:40.985 ","End":"07:42.640","Text":"we have a square,"},{"Start":"07:42.640 ","End":"07:45.575","Text":"so it has to be positive or 0,"},{"Start":"07:45.575 ","End":"07:49.160","Text":"something squared over something squared can never be negative."},{"Start":"07:49.160 ","End":"07:53.210","Text":"Basically, every x will work to give you bigger or equal to"},{"Start":"07:53.210 ","End":"07:57.200","Text":"0 except for illegal values of x,"},{"Start":"07:57.200 ","End":"08:00.170","Text":"and that\u0027s where we rule out the 0"},{"Start":"08:00.170 ","End":"08:03.680","Text":"because just the denominator can\u0027t be 0 and other than that,"},{"Start":"08:03.680 ","End":"08:05.990","Text":"every x will make this bigger or equal to 0."},{"Start":"08:05.990 ","End":"08:09.480","Text":"We could have gone from here straight to here."},{"Start":"08:09.980 ","End":"08:12.810","Text":"I\u0027m just saying that. I did it the full way."},{"Start":"08:12.810 ","End":"08:17.410","Text":"Even though, sometimes there\u0027s shortcuts if you want to take them, good."},{"Start":"08:17.410 ","End":"08:21.235","Text":"That was part A."},{"Start":"08:21.235 ","End":"08:24.530","Text":"Let\u0027s go on to part B."},{"Start":"08:25.050 ","End":"08:27.789","Text":"Same general idea."},{"Start":"08:27.789 ","End":"08:29.980","Text":"We want to bring everything to the left with"},{"Start":"08:29.980 ","End":"08:34.930","Text":"a common denominator and write it as a rational expression compared to 0,"},{"Start":"08:34.930 ","End":"08:36.640","Text":"in this case less than 0 say."},{"Start":"08:36.640 ","End":"08:45.280","Text":"Let\u0027s bring it over and we have x minus 10"},{"Start":"08:45.320 ","End":"08:51.235","Text":"over x plus 2 over 3"},{"Start":"08:51.235 ","End":"08:58.830","Text":"minus x minus 6"},{"Start":"08:58.830 ","End":"09:03.925","Text":"over x minus 2, less than 0."},{"Start":"09:03.925 ","End":"09:11.540","Text":"Now, I want to write the left-hand side with a common denominator just using fractions."},{"Start":"09:11.540 ","End":"09:16.918","Text":"I\u0027m going to put it all over the same denominator,"},{"Start":"09:16.918 ","End":"09:19.745","Text":"and what will that common denominator be?"},{"Start":"09:19.745 ","End":"09:24.590","Text":"Well, there\u0027s nothing really to save but take the product, there\u0027s nothing."},{"Start":"09:24.590 ","End":"09:26.960","Text":"I don\u0027t see anything, less that will do."},{"Start":"09:26.960 ","End":"09:28.910","Text":"Let\u0027s take this times this times this,"},{"Start":"09:28.910 ","End":"09:30.050","Text":"I\u0027ll just choose the order."},{"Start":"09:30.050 ","End":"09:31.940","Text":"I prefer to write the 3 first,"},{"Start":"09:31.940 ","End":"09:36.340","Text":"then the x, and then the x minus 2."},{"Start":"09:36.340 ","End":"09:39.890","Text":"Let me just use the method where we put"},{"Start":"09:39.890 ","End":"09:45.230","Text":"a little separator on each of them and then we see what factor we\u0027re missing."},{"Start":"09:45.230 ","End":"09:47.450","Text":"For x, to go into this,"},{"Start":"09:47.450 ","End":"09:51.060","Text":"we\u0027re missing the 3 and the x minus 2,"},{"Start":"09:51.060 ","End":"09:54.330","Text":"so we write here 3, x minus 2."},{"Start":"09:54.330 ","End":"09:56.670","Text":"In case of the 3, we need x,"},{"Start":"09:56.670 ","End":"10:02.955","Text":"x minus 2 and in case of the x minus 2, we need 3x."},{"Start":"10:02.955 ","End":"10:06.650","Text":"Then we just multiply this by this and so on."},{"Start":"10:06.650 ","End":"10:10.155","Text":"We get 3,"},{"Start":"10:10.155 ","End":"10:13.290","Text":"x minus 2,"},{"Start":"10:13.290 ","End":"10:16.485","Text":"x minus 10 for the first bit,"},{"Start":"10:16.485 ","End":"10:19.710","Text":"then plus 2x,"},{"Start":"10:19.710 ","End":"10:23.260","Text":"x minus 2 and"},{"Start":"10:25.100 ","End":"10:32.635","Text":"then minus 3x times x minus 6."},{"Start":"10:32.635 ","End":"10:36.800","Text":"Notice that I need brackets like here and here."},{"Start":"10:36.800 ","End":"10:41.750","Text":"In general, in rules of order of operations and all that,"},{"Start":"10:41.750 ","End":"10:45.690","Text":"a dividing line is considered to be a bracket."},{"Start":"10:46.530 ","End":"10:50.650","Text":"There\u0027s a bracket here and a bracket here is like x minus 10 was in bracket."},{"Start":"10:50.650 ","End":"10:54.415","Text":"I\u0027m just pointing these things out from time to time in case you forget."},{"Start":"10:54.415 ","End":"10:58.885","Text":"This is what we have on the left and we\u0027re still less than 0."},{"Start":"10:58.885 ","End":"11:02.095","Text":"I didn\u0027t multiply both sides of the equation by anything."},{"Start":"11:02.095 ","End":"11:03.534","Text":"You don\u0027t do that with inequalities."},{"Start":"11:03.534 ","End":"11:06.430","Text":"I just rewrote it with a common denominator."},{"Start":"11:06.430 ","End":"11:09.115","Text":"Now we want to simplify the numerator."},{"Start":"11:09.115 ","End":"11:14.890","Text":"The denominator already is in simplified form like factorized it here."},{"Start":"11:14.890 ","End":"11:16.960","Text":"Let\u0027s see what we get."},{"Start":"11:16.960 ","End":"11:21.655","Text":"I\u0027ll expand each of these 3 separately."},{"Start":"11:21.655 ","End":"11:26.140","Text":"We have here, I\u0027ll do the product of this and"},{"Start":"11:26.140 ","End":"11:30.490","Text":"each term I\u0027ll multiply by 3 also like this by this,"},{"Start":"11:30.490 ","End":"11:32.590","Text":"this by this, this like this, this times this."},{"Start":"11:32.590 ","End":"11:36.010","Text":"But each time we\u0027ll multiply by 3 also, save a step."},{"Start":"11:36.010 ","End":"11:38.260","Text":"X times x is x squared,"},{"Start":"11:38.260 ","End":"11:44.950","Text":"so 3x squared minus 2x minus 10x is minus 12x."},{"Start":"11:44.950 ","End":"11:49.870","Text":"I\u0027ll write minus 36x squared following the 3."},{"Start":"11:49.870 ","End":"11:51.310","Text":"Combining with the 3,"},{"Start":"11:51.310 ","End":"11:59.080","Text":"minus 2 times minus 10 is plus 20 so plus 60. That\u0027s the first bit."},{"Start":"11:59.080 ","End":"12:03.610","Text":"Now here, 2 times x times x minus 2,"},{"Start":"12:03.610 ","End":"12:12.599","Text":"what we get is 2x squared minus 4x,"},{"Start":"12:12.599 ","End":"12:17.200","Text":"and here minus 3x squared plus"},{"Start":"12:17.200 ","End":"12:24.790","Text":"18x dividing line 3x x"},{"Start":"12:24.790 ","End":"12:27.595","Text":"minus 2 less than 0."},{"Start":"12:27.595 ","End":"12:30.910","Text":"Now let\u0027s see, collect this is quadratic,"},{"Start":"12:30.910 ","End":"12:34.743","Text":"collect x-squared, then collect x\u0027s and then numbers."},{"Start":"12:34.743 ","End":"12:36.655","Text":"For x squared, I\u0027ve got,"},{"Start":"12:36.655 ","End":"12:38.695","Text":"let\u0027s see,3 of them here,"},{"Start":"12:38.695 ","End":"12:42.970","Text":"another 2 here, that\u0027s 5 minus 3."},{"Start":"12:42.970 ","End":"12:46.315","Text":"That gives me 2x squared."},{"Start":"12:46.315 ","End":"12:49.630","Text":"Next, x\u0027s, what do I have?"},{"Start":"12:49.630 ","End":"12:55.450","Text":"Minus 36 minus 4 is minus 40 plus"},{"Start":"12:55.450 ","End":"13:03.609","Text":"18 makes it minus 22x."},{"Start":"13:11.130 ","End":"13:16.585","Text":"Numbers are just the 60,"},{"Start":"13:16.585 ","End":"13:24.955","Text":"nothing else over 3xx"},{"Start":"13:24.955 ","End":"13:29.690","Text":"minus 2, less than 0."},{"Start":"13:30.990 ","End":"13:36.070","Text":"What I\u0027d like to do is I like to reduce."},{"Start":"13:36.070 ","End":"13:39.160","Text":"Let\u0027s take 2 outside the brackets,"},{"Start":"13:39.160 ","End":"13:40.733","Text":"work with smaller numbers,"},{"Start":"13:40.733 ","End":"13:44.395","Text":"one more step twice."},{"Start":"13:44.395 ","End":"13:49.735","Text":"It\u0027s optional I just recommend taking numbers out whenever you can."},{"Start":"13:49.735 ","End":"13:57.280","Text":"X squared minus 11x plus 30"},{"Start":"13:57.280 ","End":"14:05.260","Text":"over 3x x minus 2 less than"},{"Start":"14:05.260 ","End":"14:15.860","Text":"0 then there\u0027s one more thing I want to do and that is to factorize these brackets."},{"Start":"14:17.100 ","End":"14:22.795","Text":"As usual, I don\u0027t want to spend a little time factorizing the techniques."},{"Start":"14:22.795 ","End":"14:28.090","Text":"I\u0027ll just give you the answer that this is x minus 5,"},{"Start":"14:28.090 ","End":"14:31.240","Text":"x minus 6 or x minus 6,"},{"Start":"14:31.240 ","End":"14:39.220","Text":"x minus 5 over 3x x minus 2."},{"Start":"14:39.220 ","End":"14:41.020","Text":"Now although I didn\u0027t show you how I got this,"},{"Start":"14:41.020 ","End":"14:47.485","Text":"the least I can do is to show you that it is the right factorization by multiplying out."},{"Start":"14:47.485 ","End":"14:58.075","Text":"X times x is x squared minus 5x minus 6x minus 11x minus 5 times minus 6 is 30."},{"Start":"14:58.075 ","End":"15:01.600","Text":"This is right, and you know how I did it."},{"Start":"15:01.600 ","End":"15:05.420","Text":"One of the methods it doesn\u0027t matter."},{"Start":"15:05.420 ","End":"15:08.410","Text":"I\u0027ll leave it for you to do it."},{"Start":"15:09.320 ","End":"15:14.580","Text":"The methods of finding the roots or the trinomial reduction method,"},{"Start":"15:14.580 ","End":"15:17.405","Text":"so less than 0."},{"Start":"15:17.405 ","End":"15:20.320","Text":"Now we\u0027re at the good point where we want to be."},{"Start":"15:20.320 ","End":"15:23.800","Text":"We have rational expression or factorized."},{"Start":"15:23.800 ","End":"15:26.620","Text":"We look for the places which make it 0,"},{"Start":"15:26.620 ","End":"15:30.580","Text":"at least separately for the numerator 0 and the denominator 0."},{"Start":"15:30.580 ","End":"15:36.430","Text":"The numbers here don\u0027t make any difference but when x is 5,"},{"Start":"15:36.430 ","End":"15:38.425","Text":"that would make this factor 0."},{"Start":"15:38.425 ","End":"15:39.835","Text":"When x is 6,"},{"Start":"15:39.835 ","End":"15:41.845","Text":"it would make this factor 0,"},{"Start":"15:41.845 ","End":"15:48.235","Text":"when x is 2 that would make the denominator 0."},{"Start":"15:48.235 ","End":"15:50.170","Text":"Now, these get a different treatment."},{"Start":"15:50.170 ","End":"15:52.550","Text":"I\u0027ll put the mores on the number line."},{"Start":"15:52.550 ","End":"15:55.620","Text":"Here\u0027s a number line and then let\u0027s see,"},{"Start":"15:55.620 ","End":"15:57.720","Text":"we have to put them in the right order though."},{"Start":"15:57.720 ","End":"16:00.225","Text":"Of course, it\u0027s 0, 2, 5, 6."},{"Start":"16:00.225 ","End":"16:01.830","Text":"Scale is not important."},{"Start":"16:01.830 ","End":"16:08.060","Text":"Let\u0027s say 0, 2, 5, and 6."},{"Start":"16:08.060 ","End":"16:11.170","Text":"Then there is a difference between the ones on"},{"Start":"16:11.170 ","End":"16:15.130","Text":"the denominator because these are ones where x is illegal,"},{"Start":"16:15.130 ","End":"16:17.965","Text":"mustn\u0027t be substituted not in the domain."},{"Start":"16:17.965 ","End":"16:24.355","Text":"My way of indicating that is by making a little hole here,"},{"Start":"16:24.355 ","End":"16:29.500","Text":"and we see it better."},{"Start":"16:29.500 ","End":"16:36.820","Text":"I emphasize it by putting a little circle or oval here and here that x cannot be 0 or 2,"},{"Start":"16:36.820 ","End":"16:38.395","Text":"at least not in this expression."},{"Start":"16:38.395 ","End":"16:40.930","Text":"You can see it right from the beginning that if x is"},{"Start":"16:40.930 ","End":"16:44.380","Text":"0 we have a denominator 0 here and if x is 2,"},{"Start":"16:44.380 ","End":"16:47.853","Text":"we just can\u0027t have 0 in the denominator."},{"Start":"16:47.853 ","End":"16:50.365","Text":"Alpha 5 and 6 that\u0027s something else."},{"Start":"16:50.365 ","End":"16:54.910","Text":"They used to make the numerator 0 and therefore the whole expression is 0."},{"Start":"16:54.910 ","End":"16:58.210","Text":"I indicate this by putting a solid circle"},{"Start":"16:58.210 ","End":"17:02.050","Text":"here to say that the expression is 0 here and here."},{"Start":"17:02.050 ","End":"17:04.675","Text":"Undefined 0."},{"Start":"17:04.675 ","End":"17:08.890","Text":"What remains is to find out in each of the intervals here,"},{"Start":"17:08.890 ","End":"17:11.200","Text":"how many do we have 1, 2, 3, 4, 5,"},{"Start":"17:11.200 ","End":"17:16.240","Text":"in each of the 5 of them whether the expression here,"},{"Start":"17:16.240 ","End":"17:21.445","Text":"it\u0027s the same as here is positive or negative."},{"Start":"17:21.445 ","End":"17:25.959","Text":"The usual technique is to take a sample value."},{"Start":"17:25.959 ","End":"17:28.330","Text":"Let\u0027s use the sample value."},{"Start":"17:28.330 ","End":"17:30.280","Text":"Sometimes we do it all in our heads."},{"Start":"17:30.280 ","End":"17:34.810","Text":"I\u0027d like to just write what my samples are between 0 and 2 I\u0027ll take 1."},{"Start":"17:34.810 ","End":"17:37.255","Text":"I like to take whole numbers where possible."},{"Start":"17:37.255 ","End":"17:42.640","Text":"They like to choose 0 but I can\u0027t choose 0 because is no good."},{"Start":"17:42.640 ","End":"17:45.925","Text":"But here I can take say,"},{"Start":"17:45.925 ","End":"17:48.670","Text":"negative 1, between 2 and 5,"},{"Start":"17:48.670 ","End":"17:50.153","Text":"I\u0027ll take I don\u0027t know 3 or 4,"},{"Start":"17:50.153 ","End":"17:54.970","Text":"let\u0027s go for 3 between 5 and 6 I have to choose a fraction."},{"Start":"17:54.970 ","End":"17:59.755","Text":"I will go with 5 and 1/2 and beyond 6, I\u0027ll choose 7."},{"Start":"17:59.755 ","End":"18:02.620","Text":"For each of these 1, 2, 3, 4, 5,"},{"Start":"18:02.620 ","End":"18:05.320","Text":"I have to substitute in here but I don\u0027t"},{"Start":"18:05.320 ","End":"18:08.575","Text":"need the answer just whether it\u0027s positive or negative."},{"Start":"18:08.575 ","End":"18:10.690","Text":"For minus 1,"},{"Start":"18:10.690 ","End":"18:18.625","Text":"we can ignore the 2 and the 3 because they are positive and they don\u0027t affect anything."},{"Start":"18:18.625 ","End":"18:27.910","Text":"Minus 1, minus 5 is minus 6 is negative."},{"Start":"18:27.910 ","End":"18:31.225","Text":"Minus 1 minus 6 negative."},{"Start":"18:31.225 ","End":"18:33.820","Text":"Minus 1 is negative,"},{"Start":"18:33.820 ","End":"18:38.020","Text":"minus 1 minus 2, negative 4 negatives."},{"Start":"18:38.020 ","End":"18:43.075","Text":"It\u0027s a positive. For a change,"},{"Start":"18:43.075 ","End":"18:45.100","Text":"I\u0027m not going to use a snake method."},{"Start":"18:45.100 ","End":"18:47.560","Text":"We\u0027ll use the method where we just write the plus"},{"Start":"18:47.560 ","End":"18:50.350","Text":"or minus to say whether they\u0027re positive or negative."},{"Start":"18:50.350 ","End":"18:56.740","Text":"Minus 1 we are plus because accounted the minuses is an even number of them."},{"Start":"18:56.740 ","End":"19:00.350","Text":"Now, over here, what do we get?"},{"Start":"19:01.680 ","End":"19:04.825","Text":"When x is 1,"},{"Start":"19:04.825 ","End":"19:08.575","Text":"1 minus 5 is negative,"},{"Start":"19:08.575 ","End":"19:10.090","Text":"1 minus 6,"},{"Start":"19:10.090 ","End":"19:16.940","Text":"negative 1 itself, positive 1 minus 2, negative."},{"Start":"19:17.400 ","End":"19:23.840","Text":"I got a bit lazy. I really should have written something like minus times minus,"},{"Start":"19:24.330 ","End":"19:34.810","Text":"over, plus times minus instead of this abbreviation."},{"Start":"19:34.810 ","End":"19:36.820","Text":"But you know what I mean?"},{"Start":"19:36.820 ","End":"19:41.080","Text":"Together we get 3 minuses, an odd number."},{"Start":"19:41.080 ","End":"19:46.010","Text":"That gives us a minus."},{"Start":"19:47.100 ","End":"19:52.820","Text":"Maybe a bit messy but I think it\u0027s perfectly clear what I\u0027m doing."},{"Start":"19:55.560 ","End":"19:59.680","Text":"I\u0027ll write the pluses and minuses in the bottom, that\u0027s better."},{"Start":"19:59.680 ","End":"20:01.510","Text":"For our 3,"},{"Start":"20:01.510 ","End":"20:03.700","Text":"we get, if we put it in here,"},{"Start":"20:03.700 ","End":"20:05.305","Text":"3 minus 5,"},{"Start":"20:05.305 ","End":"20:07.810","Text":"negative 3 minus 6,"},{"Start":"20:07.810 ","End":"20:11.305","Text":"negative 3, positive,"},{"Start":"20:11.305 ","End":"20:13.990","Text":"3 minus 2 positive."},{"Start":"20:13.990 ","End":"20:20.330","Text":"That makes this a"},{"Start":"20:20.330 ","End":"20:25.090","Text":"plus and for 5 and 1/2, what do I get?"},{"Start":"20:25.090 ","End":"20:27.100","Text":"5 and 1/2 minus 5,"},{"Start":"20:27.100 ","End":"20:29.410","Text":"positive 5 and 1/2 Minus 6,"},{"Start":"20:29.410 ","End":"20:35.380","Text":"negative 5 and 1/2 positive 5 and 1/2 Minus 2 positive."},{"Start":"20:35.380 ","End":"20:38.860","Text":"A minus at 5 and 1/2."},{"Start":"20:38.860 ","End":"20:40.945","Text":"Finally 7."},{"Start":"20:40.945 ","End":"20:42.910","Text":"7 minus 5 is positive,"},{"Start":"20:42.910 ","End":"20:44.560","Text":"7 minus 6 is a positive,"},{"Start":"20:44.560 ","End":"20:51.265","Text":"positive plus, plus and so plus."},{"Start":"20:51.265 ","End":"20:54.445","Text":"Let\u0027s say these are just sample points for the whole interval,"},{"Start":"20:54.445 ","End":"20:56.103","Text":"all the way up to here, a plus,"},{"Start":"20:56.103 ","End":"20:58.900","Text":"and then from here to here, we have minus and so on."},{"Start":"20:58.900 ","End":"21:01.670","Text":"Now, what are we collecting?"},{"Start":"21:01.670 ","End":"21:05.985","Text":"By which I mean we\u0027re looking for positive or negative or 0 or what?"},{"Start":"21:05.985 ","End":"21:10.830","Text":"We\u0027re looking for negative less than 0 means I want the places where it\u0027s negative,"},{"Start":"21:10.830 ","End":"21:13.230","Text":"strictly negative, 0 not included."},{"Start":"21:13.230 ","End":"21:16.390","Text":"Strictly negative would be here."},{"Start":"21:16.390 ","End":"21:18.939","Text":"We have a negative and here I have a negative."},{"Start":"21:18.939 ","End":"21:23.874","Text":"What this refers to is not just the sample point, the whole interval."},{"Start":"21:23.874 ","End":"21:26.200","Text":"Since it\u0027s strictly negative,"},{"Start":"21:26.200 ","End":"21:27.880","Text":"we don\u0027t include the endpoints,"},{"Start":"21:27.880 ","End":"21:30.625","Text":"but we wouldn\u0027t include them anyway because it\u0027s not defined there."},{"Start":"21:30.625 ","End":"21:34.510","Text":"But for example here also we wouldn\u0027t include the 5 and the 6."},{"Start":"21:34.510 ","End":"21:44.935","Text":"What we get basically is this bit here which is 0 less than x, less than 2."},{"Start":"21:44.935 ","End":"21:54.205","Text":"This bit here also we get which is 5 less than x, less than 6."},{"Start":"21:54.205 ","End":"21:59.660","Text":"But the connecting operator is all."},{"Start":"21:59.850 ","End":"22:03.130","Text":"Because each x it\u0027s allowed,"},{"Start":"22:03.130 ","End":"22:04.390","Text":"it\u0027s either here or here,"},{"Start":"22:04.390 ","End":"22:05.800","Text":"even though we take them both."},{"Start":"22:05.800 ","End":"22:10.910","Text":"But there is always confusion about the or and or is the correct thing."},{"Start":"22:10.910 ","End":"22:15.400","Text":"That\u0027s the answer which I would like to highlight."},{"Start":"22:15.400 ","End":"22:21.510","Text":"Then declare that we are done because there was only 2 parts to this exercise."}],"Thumbnail":null,"ID":5074},{"Watched":false,"Name":"Exercise 5 part a","Duration":"13m 37s","ChapterTopicVideoID":5073,"CourseChapterTopicPlaylistID":3877,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.570","Text":"In this exercise, there are 2 parts separate."},{"Start":"00:03.570 ","End":"00:07.170","Text":"In each 1, we have an inequality,"},{"Start":"00:07.170 ","End":"00:14.445","Text":"which we are going to get as a rational expression compared to 0."},{"Start":"00:14.445 ","End":"00:15.970","Text":"We have to do a bit of algebra,"},{"Start":"00:15.970 ","End":"00:19.350","Text":"put everything on the left and get it into rational expression"},{"Start":"00:19.350 ","End":"00:23.340","Text":"and factorize in each of the cases and then we\u0027re on familiar territory."},{"Start":"00:23.340 ","End":"00:26.025","Text":"Let\u0027s begin with the first 1."},{"Start":"00:26.025 ","End":"00:31.140","Text":"First thing to do is just move everything to the left-hand side,"},{"Start":"00:31.140 ","End":"00:38.939","Text":"so 2x plus 7 over 5 minus x and then minus"},{"Start":"00:38.939 ","End":"00:47.185","Text":"7 and then plus 6x over x plus 3,"},{"Start":"00:47.185 ","End":"00:50.120","Text":"this will be less than 0."},{"Start":"00:50.120 ","End":"00:53.690","Text":"I want to rewrite the left-hand side,"},{"Start":"00:53.690 ","End":"00:57.110","Text":"like we do with fractions in terms of a common denominator."},{"Start":"00:57.110 ","End":"00:59.600","Text":"I\u0027m not multiplying both sides by anything."},{"Start":"00:59.600 ","End":"01:03.635","Text":"I\u0027m just rewriting the left-hand side with a common denominator."},{"Start":"01:03.635 ","End":"01:06.920","Text":"Now what is that common denominator going to be?"},{"Start":"01:06.920 ","End":"01:09.980","Text":"It\u0027s going to be just the product of this times this."},{"Start":"01:09.980 ","End":"01:12.770","Text":"There\u0027s nothing better that I can see,"},{"Start":"01:12.770 ","End":"01:17.800","Text":"so 5 minus x,"},{"Start":"01:17.800 ","End":"01:25.025","Text":"x plus 3, less than 0."},{"Start":"01:25.025 ","End":"01:28.910","Text":"Now I just want to write everything over this."},{"Start":"01:28.910 ","End":"01:30.500","Text":"Now, 7, if it helps,"},{"Start":"01:30.500 ","End":"01:35.515","Text":"you can think of as 7 over 1 and usually not necessary."},{"Start":"01:35.515 ","End":"01:41.825","Text":"Then we can use the method of saying what is the missing factor or factors?"},{"Start":"01:41.825 ","End":"01:47.020","Text":"Like 5 minus x goes into this x plus 3 times."},{"Start":"01:47.020 ","End":"01:49.900","Text":"What I usually do is put a separator."},{"Start":"01:49.900 ","End":"01:56.455","Text":"Well, put 1 here and here also while I\u0027m at it and then put the x plus 3,"},{"Start":"01:56.455 ","End":"02:00.140","Text":"what it means is that afterwards we have to multiply this by this."},{"Start":"02:00.140 ","End":"02:04.050","Text":"Then for the 7, there\u0027s a 1 here,"},{"Start":"02:04.050 ","End":"02:09.640","Text":"so I\u0027m missing both of them so I have to multiply by the whole 5 minus x,"},{"Start":"02:09.640 ","End":"02:16.195","Text":"x plus 3 and here the missing factor is 5 minus x."},{"Start":"02:16.195 ","End":"02:18.110","Text":"What I\u0027m really doing is saying,"},{"Start":"02:18.110 ","End":"02:20.465","Text":"this is what I\u0027m multiplying top and bottom by."},{"Start":"02:20.465 ","End":"02:22.295","Text":"On the bottom, I\u0027ll always get this."},{"Start":"02:22.295 ","End":"02:23.780","Text":"When it\u0027s the same denominator,"},{"Start":"02:23.780 ","End":"02:26.210","Text":"we just collect the numerators."},{"Start":"02:26.210 ","End":"02:29.960","Text":"We now have this times this,"},{"Start":"02:29.960 ","End":"02:32.170","Text":"x plus 3,"},{"Start":"02:32.170 ","End":"02:38.696","Text":"2x plus 7 minus 7 times this,"},{"Start":"02:38.696 ","End":"02:41.040","Text":"I can multiply in whichever order I want,"},{"Start":"02:41.040 ","End":"02:42.525","Text":"I took 7 first,"},{"Start":"02:42.525 ","End":"02:47.250","Text":"times 5 minus x."},{"Start":"02:47.250 ","End":"02:49.860","Text":"Am I running out of space?"},{"Start":"02:49.860 ","End":"02:58.710","Text":"I\u0027ll just move this further along and then I\u0027ll make this dividing line a bit longer."},{"Start":"03:00.280 ","End":"03:03.390","Text":"This should do it."},{"Start":"03:03.920 ","End":"03:06.930","Text":"We were just in the middle of 7 times this."},{"Start":"03:06.930 ","End":"03:08.370","Text":"We got the 5 minus x."},{"Start":"03:08.370 ","End":"03:11.700","Text":"We also need the x plus 3."},{"Start":"03:11.700 ","End":"03:19.725","Text":"The last factor would be plus 6x times 5 minus x."},{"Start":"03:19.725 ","End":"03:22.090","Text":"Yes, we just fit it in."},{"Start":"03:22.150 ","End":"03:27.035","Text":"This has to be less than 0."},{"Start":"03:27.035 ","End":"03:29.975","Text":"Now we want to expand the numerator,"},{"Start":"03:29.975 ","End":"03:31.760","Text":"just a bit of boring algebra,"},{"Start":"03:31.760 ","End":"03:33.830","Text":"but let\u0027s do it."},{"Start":"03:33.830 ","End":"03:37.940","Text":"First pair of factors here,"},{"Start":"03:37.940 ","End":"03:42.965","Text":"we get x times 2x would be"},{"Start":"03:42.965 ","End":"03:49.575","Text":"2x^2 plus 6x plus 7x would be plus 13x,"},{"Start":"03:49.575 ","End":"03:52.500","Text":"3 times 7 is 21."},{"Start":"03:52.500 ","End":"03:58.370","Text":"I\u0027ll leave the minus 7 here and just expand this times this."},{"Start":"03:58.370 ","End":"04:00.860","Text":"Let\u0027s see, 5 times x is 5x,"},{"Start":"04:00.860 ","End":"04:06.900","Text":"minus 3x is 2x,"},{"Start":"04:06.900 ","End":"04:08.894","Text":"but I want the second."},{"Start":"04:08.894 ","End":"04:14.190","Text":"The minus x^2 is what I want first and then"},{"Start":"04:14.190 ","End":"04:21.400","Text":"the 5 with the 3 is 15."},{"Start":"04:22.540 ","End":"04:26.575","Text":"Then this I can multiply out already,"},{"Start":"04:26.575 ","End":"04:31.545","Text":"plus 6x times 5 is 30x and then"},{"Start":"04:31.545 ","End":"04:38.630","Text":"minus 6x^2 and there I just copied the rest of as it didn\u0027t change."},{"Start":"04:38.630 ","End":"04:41.570","Text":"Now I want to collect like terms in the numerator,"},{"Start":"04:41.570 ","End":"04:44.195","Text":"so let\u0027s use a bit of coloring."},{"Start":"04:44.195 ","End":"04:52.700","Text":"What we\u0027ll do is we\u0027ll color the x^2 with the yellow and let\u0027s"},{"Start":"04:52.700 ","End":"04:57.050","Text":"highlight the terms containing the x in another color and"},{"Start":"04:57.050 ","End":"05:02.255","Text":"then we\u0027ll highlight the terms containing just numbers in this color."},{"Start":"05:02.255 ","End":"05:04.265","Text":"Let\u0027s see, x^2,"},{"Start":"05:04.265 ","End":"05:11.645","Text":"we have 2x^2 plus 7x^2 is 9x^2 minus 6x^2,"},{"Start":"05:11.645 ","End":"05:15.420","Text":"that makes it 3x^2."},{"Start":"05:15.420 ","End":"05:23.340","Text":"What else? X is 13 and then minus 7 times 2 is minus 14,"},{"Start":"05:23.340 ","End":"05:27.345","Text":"so minus 1x plus 30x,"},{"Start":"05:27.345 ","End":"05:31.180","Text":"all together, 29x."},{"Start":"05:31.180 ","End":"05:36.680","Text":"As for numbers, we have 21 minus 7 times 15,"},{"Start":"05:36.680 ","End":"05:43.720","Text":"21 minus 105 minus 84."},{"Start":"05:44.060 ","End":"05:49.710","Text":"Same denominator, 5 minus x,"},{"Start":"05:49.710 ","End":"05:51.540","Text":"x plus 3,"},{"Start":"05:51.540 ","End":"05:54.850","Text":"and still less than 0."},{"Start":"05:54.850 ","End":"06:00.835","Text":"The next thing to do is to factorize this numerator."},{"Start":"06:00.835 ","End":"06:06.830","Text":"As usual I will do it for you to save time since you know how to"},{"Start":"06:06.830 ","End":"06:11.900","Text":"factorize and the answer comes out to be"},{"Start":"06:11.900 ","End":"06:21.225","Text":"3x minus 7x plus 12."},{"Start":"06:21.225 ","End":"06:23.250","Text":"Let\u0027s just check that,"},{"Start":"06:23.250 ","End":"06:25.920","Text":"3x times x is 3x^2,"},{"Start":"06:25.920 ","End":"06:31.260","Text":"minus 7x plus 36x is 29x,"},{"Start":"06:31.260 ","End":"06:35.115","Text":"minus 7 times 12 is minus 84,"},{"Start":"06:35.115 ","End":"06:37.695","Text":"so that part\u0027s okay,"},{"Start":"06:37.695 ","End":"06:43.815","Text":"over same thing, 5 minus x,"},{"Start":"06:43.815 ","End":"06:48.930","Text":"x plus 3, less than 0."},{"Start":"06:48.930 ","End":"06:53.005","Text":"This is how we like the setup to be."},{"Start":"06:53.005 ","End":"06:58.480","Text":"The rational expression is completely factorized and now we can look for"},{"Start":"06:58.480 ","End":"07:05.080","Text":"places where the numerator and the denominator are 0."},{"Start":"07:05.490 ","End":"07:12.170","Text":"This factor is 0 when x is 7 over 3,"},{"Start":"07:12.170 ","End":"07:14.895","Text":"here it\u0027s going to be minus 12,"},{"Start":"07:14.895 ","End":"07:19.320","Text":"here x is going to be 5 for this to be 0,"},{"Start":"07:19.320 ","End":"07:22.455","Text":"and here x is going to be minus 3."},{"Start":"07:22.455 ","End":"07:27.235","Text":"Let\u0027s draw these on a number line."},{"Start":"07:27.235 ","End":"07:30.970","Text":"Here\u0027s a number line and we want to place these numbers on it,"},{"Start":"07:30.970 ","End":"07:32.800","Text":"but in the right order."},{"Start":"07:32.800 ","End":"07:38.750","Text":"The smallest would be minus 12,"},{"Start":"07:39.300 ","End":"07:43.570","Text":"next would be minus 3."},{"Start":"07:43.570 ","End":"07:46.256","Text":"Then from this to this is the smaller,"},{"Start":"07:46.256 ","End":"07:48.610","Text":"instead of 7 over 3,"},{"Start":"07:48.610 ","End":"07:55.395","Text":"I\u0027ll write it as 2 and 1/3 as a mixed number because 3 into 7 goes twice, remainder 1."},{"Start":"07:55.395 ","End":"07:59.780","Text":"The next 1, last 1 is 5."},{"Start":"07:59.780 ","End":"08:04.450","Text":"There is a difference between the ones in the denominator and in the numerator."},{"Start":"08:04.450 ","End":"08:10.270","Text":"In the denominator, they represent illegal values for substitution."},{"Start":"08:10.270 ","End":"08:13.060","Text":"They\u0027re not in the domain of this expression,"},{"Start":"08:13.060 ","End":"08:22.405","Text":"so I like to indicate that with a missing bit."},{"Start":"08:22.405 ","End":"08:28.115","Text":"Just erase that and erase a little bit."},{"Start":"08:28.115 ","End":"08:32.970","Text":"My eraser is not working right. There it is."},{"Start":"08:34.030 ","End":"08:43.425","Text":"I\u0027ll further emphasize that there\u0027s a hole here by putting a hollow circle."},{"Start":"08:43.425 ","End":"08:46.710","Text":"This is in contrast to the numerator,"},{"Start":"08:46.710 ","End":"08:49.595","Text":"where the numerator is 0, that\u0027s fine."},{"Start":"08:49.595 ","End":"08:52.880","Text":"That just means that the expression as a whole is 0,"},{"Start":"08:52.880 ","End":"08:56.120","Text":"which I indicate with a solid dot."},{"Start":"08:56.120 ","End":"09:00.425","Text":"That would be a 2 and 1/3 and minus 12."},{"Start":"09:00.425 ","End":"09:04.460","Text":"Now what we want to do is find out in which of"},{"Start":"09:04.460 ","End":"09:09.320","Text":"these intervals we are positive and where we\u0027re negative."},{"Start":"09:09.320 ","End":"09:11.955","Text":"After all, we want the negative."},{"Start":"09:11.955 ","End":"09:14.240","Text":"Actually these numbers are not important to"},{"Start":"09:14.240 ","End":"09:16.790","Text":"us where it\u0027s 0 because we\u0027re not collecting 0s,"},{"Start":"09:16.790 ","End":"09:18.530","Text":"only negatives. Let\u0027s see."},{"Start":"09:18.530 ","End":"09:23.320","Text":"We take sample values and substitute."},{"Start":"09:23.320 ","End":"09:27.120","Text":"I\u0027ll just show you what sample values I could take, I need 1 here,"},{"Start":"09:27.120 ","End":"09:28.575","Text":"I need 1 here, I need 1 here,"},{"Start":"09:28.575 ","End":"09:30.470","Text":"I need 1 here, and here and here."},{"Start":"09:30.470 ","End":"09:35.345","Text":"This one I could take as, say, minus 13."},{"Start":"09:35.345 ","End":"09:37.085","Text":"This one I could take,"},{"Start":"09:37.085 ","End":"09:39.400","Text":"I don\u0027t know, minus 4."},{"Start":"09:39.400 ","End":"09:43.635","Text":"Here I could choose 0 between 2 and 1/3 and 5."},{"Start":"09:43.635 ","End":"09:49.555","Text":"Let me go with 4 and here I\u0027ll go with 6."},{"Start":"09:49.555 ","End":"09:56.195","Text":"Now I\u0027m going to substitute each 1 in this expression and see what we get."},{"Start":"09:56.195 ","End":"09:59.585","Text":"Of course, I just care about the sign."},{"Start":"09:59.585 ","End":"10:05.115","Text":"If we put x is minus 13,"},{"Start":"10:05.115 ","End":"10:07.440","Text":"it\u0027s minus 39 minus 7,"},{"Start":"10:07.440 ","End":"10:10.745","Text":"that bit\u0027s negative, I\u0027ll just put a little minus there."},{"Start":"10:10.745 ","End":"10:18.800","Text":"Minus 13 plus 12 is going to be negative, it\u0027s minus 1."},{"Start":"10:18.800 ","End":"10:24.830","Text":"If I put in 5 minus minus 13,"},{"Start":"10:24.830 ","End":"10:29.160","Text":"that would be positive because that would be 5 plus 13,"},{"Start":"10:29.160 ","End":"10:35.910","Text":"18 is positive and minus 13 plus 3 would be negative."},{"Start":"10:35.910 ","End":"10:40.065","Text":"I\u0027m just counting, I have a negative negative over positive negative,"},{"Start":"10:40.065 ","End":"10:43.075","Text":"that would be a negative."},{"Start":"10:43.075 ","End":"10:46.630","Text":"I won\u0027t use the snake diagram method."},{"Start":"10:46.630 ","End":"10:48.980","Text":"I\u0027ll just use the minuses and pluses method,"},{"Start":"10:48.980 ","End":"10:53.510","Text":"where here in this interval or this sample point,"},{"Start":"10:53.510 ","End":"10:55.945","Text":"which is for the whole interval, it\u0027s a minus."},{"Start":"10:55.945 ","End":"11:01.910","Text":"Then if I put in minus 4 to represent this interval,"},{"Start":"11:01.910 ","End":"11:07.855","Text":"easy to see that here I\u0027ll get negative minus 4 plus 12."},{"Start":"11:07.855 ","End":"11:10.290","Text":"That would be positive,"},{"Start":"11:10.290 ","End":"11:11.580","Text":"which I might as well we\u0027ve got minus,"},{"Start":"11:11.580 ","End":"11:18.810","Text":"we got plus, 5 minus minus 4 is positive."},{"Start":"11:18.810 ","End":"11:23.480","Text":"Here, minus 4 plus 3 is negative."},{"Start":"11:23.480 ","End":"11:26.400","Text":"Let\u0027s see, there\u0027s 2 negatives."},{"Start":"11:26.440 ","End":"11:29.630","Text":"That\u0027s a positive."},{"Start":"11:29.630 ","End":"11:31.910","Text":"I\u0027ll just save time,"},{"Start":"11:31.910 ","End":"11:33.680","Text":"I\u0027ll let you continue on your own."},{"Start":"11:33.680 ","End":"11:37.670","Text":"In this case, it does happen to alternate as it often does, but not always."},{"Start":"11:37.670 ","End":"11:40.530","Text":"That\u0027s going to be a minus and that\u0027s going to be"},{"Start":"11:40.530 ","End":"11:44.430","Text":"a plus and that\u0027s going to be a minus or maybe we\u0027ll check 1 more."},{"Start":"11:44.430 ","End":"11:46.020","Text":"The 0 is easy to check."},{"Start":"11:46.020 ","End":"11:50.190","Text":"Minus 7 plus 12 plus 5 plus"},{"Start":"11:50.190 ","End":"11:56.360","Text":"3 and there\u0027s 1 minus and 3 pluses,"},{"Start":"11:56.360 ","End":"11:58.950","Text":"so it\u0027s a minus and so on."},{"Start":"11:59.200 ","End":"12:03.320","Text":"What we\u0027re interested in, like I mentioned,"},{"Start":"12:03.320 ","End":"12:07.865","Text":"is just the less than 0 strictly negative."},{"Start":"12:07.865 ","End":"12:09.800","Text":"This is interesting."},{"Start":"12:09.800 ","End":"12:13.410","Text":"This is what I\u0027m collecting"},{"Start":"12:15.440 ","End":"12:23.285","Text":"and this one also and each 1 of these corresponds to the interval."},{"Start":"12:23.285 ","End":"12:30.990","Text":"This interval, I\u0027ll even emphasize it because there\u0027s different kinds of points,"},{"Start":"12:30.990 ","End":"12:33.440","Text":"some are samples and some really are interval borders."},{"Start":"12:33.440 ","End":"12:36.720","Text":"These are the intervals."},{"Start":"12:38.800 ","End":"12:44.929","Text":"This one is the interval x less than minus"},{"Start":"12:44.929 ","End":"12:54.545","Text":"12 and this interval is between minus 3 and 2/3,"},{"Start":"12:54.545 ","End":"12:58.655","Text":"so I\u0027ll write it as minus 3 less than x less than,"},{"Start":"12:58.655 ","End":"13:02.170","Text":"not 2/3, 2 and 1/3."},{"Start":"13:02.170 ","End":"13:09.525","Text":"This last interval is x bigger than 5."},{"Start":"13:09.525 ","End":"13:14.555","Text":"If I combine these with or"},{"Start":"13:14.555 ","End":"13:19.820","Text":"this or this or this,"},{"Start":"13:19.820 ","End":"13:22.040","Text":"that gives me my answer."},{"Start":"13:22.040 ","End":"13:26.400","Text":"I\u0027ll highlight it. It came out a bit crooked,"},{"Start":"13:26.400 ","End":"13:28.065","Text":"it does not matter."},{"Start":"13:28.065 ","End":"13:34.350","Text":"X less than 12 or x in this interval or x in this interval."},{"Start":"13:34.350 ","End":"13:37.870","Text":"That\u0027s the end of part a."}],"Thumbnail":null,"ID":5075},{"Watched":false,"Name":"Exercise 5 partb","Duration":"14m 47s","ChapterTopicVideoID":5074,"CourseChapterTopicPlaylistID":3877,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.350","Text":"Here in Part B,"},{"Start":"00:01.350 ","End":"00:04.080","Text":"we started out pretty much like in Part A where we bring"},{"Start":"00:04.080 ","End":"00:07.770","Text":"everything to the left-hand side and compare it to 0."},{"Start":"00:07.770 ","End":"00:11.700","Text":"What we get is 3 over 3x minus"},{"Start":"00:11.700 ","End":"00:19.185","Text":"2 minus 1 over 4x plus 1,"},{"Start":"00:19.185 ","End":"00:24.720","Text":"then minus 3x over 9x minus"},{"Start":"00:24.720 ","End":"00:34.235","Text":"6 is greater than or equal to 0."},{"Start":"00:34.235 ","End":"00:39.630","Text":"Let\u0027s put everything over a common denominator."},{"Start":"00:39.740 ","End":"00:42.720","Text":"I just noticed something."},{"Start":"00:42.720 ","End":"00:50.765","Text":"3 will go into numerator and denominator in this last term. Just noticed it."},{"Start":"00:50.765 ","End":"00:57.065","Text":"Let\u0027s take the 3 out and I\u0027ll just work on the last term for the moment."},{"Start":"00:57.065 ","End":"01:06.205","Text":"This would be 3x over 3 times 3x minus 2."},{"Start":"01:06.205 ","End":"01:09.080","Text":"Then if I cancel the 3s,"},{"Start":"01:09.080 ","End":"01:15.965","Text":"that would just be x over 3x minus 2."},{"Start":"01:15.965 ","End":"01:20.240","Text":"As for the rest of it, I\u0027ll just copy it."},{"Start":"01:20.240 ","End":"01:29.580","Text":"Let\u0027s see here and"},{"Start":"01:29.580 ","End":"01:32.465","Text":"here just copy-paste for the rest of it."},{"Start":"01:32.465 ","End":"01:36.649","Text":"Now it\u0027s going to be easier because we only have 2 different factors,"},{"Start":"01:36.649 ","End":"01:39.395","Text":"3x minus 2 and 4x plus 1."},{"Start":"01:39.395 ","End":"01:44.430","Text":"Let\u0027s put a big dividing line there,"},{"Start":"01:44.430 ","End":"01:45.990","Text":"and on the bottom,"},{"Start":"01:45.990 ","End":"01:52.690","Text":"I want to put 3x minus 2 times 4x plus 1."},{"Start":"01:52.690 ","End":"01:55.475","Text":"That\u0027s our common denominator."},{"Start":"01:55.475 ","End":"02:02.090","Text":"Then the first term will have to multiply numerator and denominator by 4x plus 1,"},{"Start":"02:02.090 ","End":"02:08.525","Text":"which is the missing factor here by 3x minus 2."},{"Start":"02:08.525 ","End":"02:13.500","Text":"Here again by 4x plus 1."},{"Start":"02:14.020 ","End":"02:17.810","Text":"I\u0027m just putting this on a common denominator."},{"Start":"02:17.810 ","End":"02:24.260","Text":"I\u0027m not multiplying both sides by anything or in any way altering both sides."},{"Start":"02:24.260 ","End":"02:27.870","Text":"I\u0027m just rewriting the fraction with a common denominator."},{"Start":"02:28.400 ","End":"02:32.419","Text":"What we get is we\u0027ll add the numerators."},{"Start":"02:32.419 ","End":"02:33.859","Text":"When we have a common denominator,"},{"Start":"02:33.859 ","End":"02:35.060","Text":"we add the numerators,"},{"Start":"02:35.060 ","End":"02:41.420","Text":"so we get 3 times 4x plus 1 for this."},{"Start":"02:41.420 ","End":"02:47.850","Text":"Then this product just gives me 3x minus 2."},{"Start":"02:47.850 ","End":"02:49.805","Text":"That\u0027s the minus 1 there."},{"Start":"02:49.805 ","End":"02:54.835","Text":"Minus, let\u0027s see, I\u0027ll take first of all the x,"},{"Start":"02:54.835 ","End":"02:59.275","Text":"then the 4x plus 1,"},{"Start":"02:59.275 ","End":"03:02.689","Text":"and we\u0027re still bigger or equal to 0."},{"Start":"03:02.689 ","End":"03:06.110","Text":"Let\u0027s see just do"},{"Start":"03:06.110 ","End":"03:10.190","Text":"another copy-paste here because we\u0027re going to want to work on the numerator."},{"Start":"03:10.190 ","End":"03:12.275","Text":"We\u0027ve got 3 times 4x is"},{"Start":"03:12.275 ","End":"03:18.855","Text":"12x plus 3 minus 3x"},{"Start":"03:18.855 ","End":"03:23.400","Text":"plus 2 is minus,"},{"Start":"03:23.400 ","End":"03:30.650","Text":"minus, minus x times 4x is 4x^2 and minus x."},{"Start":"03:30.650 ","End":"03:32.300","Text":"We\u0027re running out of space,"},{"Start":"03:32.300 ","End":"03:34.500","Text":"scroll down a bit."},{"Start":"03:35.210 ","End":"03:38.520","Text":"Time to collect like terms."},{"Start":"03:38.520 ","End":"03:40.800","Text":"The x^2 is only 1 of them."},{"Start":"03:40.800 ","End":"03:43.920","Text":"That\u0027s minus 4x^2."},{"Start":"03:43.920 ","End":"03:46.770","Text":"Xs are having 3 places,"},{"Start":"03:46.770 ","End":"03:51.975","Text":"it\u0027s 12 minus 3 minus 1 is 8x"},{"Start":"03:51.975 ","End":"03:59.970","Text":"and numbers 3 plus 2 is plus 5,"},{"Start":"03:59.970 ","End":"04:02.750","Text":"and it\u0027s still bigger or equal to 0."},{"Start":"04:02.750 ","End":"04:08.370","Text":"Now what we want to do is factorize the numerator."},{"Start":"04:08.530 ","End":"04:14.950","Text":"Continuing over here, I already copied the denominator again."},{"Start":"04:14.950 ","End":"04:21.470","Text":"I prefer to work with quadratics with a positive a in front of the x^2 squared."},{"Start":"04:21.470 ","End":"04:25.580","Text":"So let me take a minus outside the brackets and write it"},{"Start":"04:25.580 ","End":"04:32.675","Text":"as 4x^2 minus 8x minus 5,"},{"Start":"04:32.675 ","End":"04:35.190","Text":"that suits me better."},{"Start":"04:35.950 ","End":"04:40.400","Text":"Now as usual, I do the work for you to save time"},{"Start":"04:40.400 ","End":"04:44.105","Text":"because you know how to factorize trinomials,"},{"Start":"04:44.105 ","End":"04:45.775","Text":"we have 2 main methods."},{"Start":"04:45.775 ","End":"04:49.640","Text":"Anyway, I\u0027ll just give you the answer"},{"Start":"04:49.700 ","End":"04:57.235","Text":"2x minus 1 times 2x plus 5."},{"Start":"04:57.235 ","End":"05:01.175","Text":"However, I will verify it with you."},{"Start":"05:01.175 ","End":"05:07.670","Text":"2x times 2x is 4x^2 minus 2x plus"},{"Start":"05:07.670 ","End":"05:16.310","Text":"10x,10 minus 2 is 8."},{"Start":"05:16.310 ","End":"05:19.470","Text":"This is a plus."},{"Start":"05:20.590 ","End":"05:27.390","Text":"Take 2 on the last bit."},{"Start":"05:27.520 ","End":"05:36.505","Text":"2x plus 1, 2x minus 5."},{"Start":"05:36.505 ","End":"05:40.055","Text":"At least I can verify it for you."},{"Start":"05:40.055 ","End":"05:44.015","Text":"2x times 2x is 4x^2,"},{"Start":"05:44.015 ","End":"05:48.290","Text":"plus 2x minus 10x is minus"},{"Start":"05:48.290 ","End":"05:54.750","Text":"8x and 1 times minus 5 is minus 5."},{"Start":"05:54.750 ","End":"05:57.740","Text":"This is now the situation we want."},{"Start":"05:57.740 ","End":"06:03.185","Text":"We have a completely factorized rational expression."},{"Start":"06:03.185 ","End":"06:05.660","Text":"It\u0027s possible to get rid of the minus here."},{"Start":"06:05.660 ","End":"06:10.460","Text":"You could multiply both sides by minus and then you\u0027d get rid of it."},{"Start":"06:10.460 ","End":"06:12.110","Text":"But then you\u0027d have to change the sign,"},{"Start":"06:12.110 ","End":"06:14.960","Text":"make it less than or equal to possible if you want to do it that way,"},{"Start":"06:14.960 ","End":"06:17.000","Text":"I\u0027ll just leave the minus here."},{"Start":"06:17.000 ","End":"06:21.180","Text":"We really have 5 factors here."},{"Start":"06:21.180 ","End":"06:22.860","Text":"We have 1, 2, 3, 4, 5."},{"Start":"06:22.860 ","End":"06:24.450","Text":"It will be 5 plus or minuses,"},{"Start":"06:24.450 ","End":"06:26.150","Text":"but 1 of them will always be minus."},{"Start":"06:26.150 ","End":"06:28.190","Text":"Anyway, getting ahead of myself,"},{"Start":"06:28.190 ","End":"06:30.530","Text":"let\u0027s first of all say what"},{"Start":"06:30.530 ","End":"06:36.350","Text":"the roots are in the numerator and what makes the denominator 0?"},{"Start":"06:36.350 ","End":"06:39.822","Text":"For this factor, it\u0027s minus 1/2."},{"Start":"06:39.822 ","End":"06:41.855","Text":"Does it make it 0?"},{"Start":"06:41.855 ","End":"06:45.275","Text":"But here it\u0027s 5 over 2,"},{"Start":"06:45.275 ","End":"06:47.055","Text":"or 2 and 1/2,"},{"Start":"06:47.055 ","End":"06:50.085","Text":"5 over 2 whatever."},{"Start":"06:50.085 ","End":"06:53.390","Text":"Here we need 2 on the other side divided by 3,"},{"Start":"06:53.390 ","End":"06:58.310","Text":"it\u0027s 2/3 and here we get that if this has to be 0,"},{"Start":"06:58.310 ","End":"07:02.040","Text":"x should be minus 1/4."},{"Start":"07:02.040 ","End":"07:07.555","Text":"Now we put these 4 values on the number line."},{"Start":"07:07.555 ","End":"07:10.660","Text":"First, the number line and now let\u0027s put the values on,"},{"Start":"07:10.660 ","End":"07:13.780","Text":"but it\u0027s important to put them in the right order,"},{"Start":"07:13.780 ","End":"07:16.885","Text":"in order of increasing magnitude from left to right."},{"Start":"07:16.885 ","End":"07:25.105","Text":"Let\u0027s see, the very smallest would be the minus 1/2 then we get the minus 1/4."},{"Start":"07:25.105 ","End":"07:29.275","Text":"The scale doesn\u0027t matter just the order matters."},{"Start":"07:29.275 ","End":"07:32.290","Text":"Then that\u0027s the negatives and the positives,"},{"Start":"07:32.290 ","End":"07:37.675","Text":"we have a 2/3 and we have a 2.5."},{"Start":"07:37.675 ","End":"07:45.654","Text":"That breaks our line into 5 intervals."},{"Start":"07:45.654 ","End":"07:49.855","Text":"Sometimes I like to emphasize this with dotted lines."},{"Start":"07:49.855 ","End":"07:55.959","Text":"This breaks it up into 5 pieces,"},{"Start":"07:55.959 ","End":"08:01.960","Text":"1 2 3 4 5 and now we want to know whether it\u0027s positive or negative in each piece."},{"Start":"08:01.960 ","End":"08:06.850","Text":"We choose sample values that\u0027s the usual technique."},{"Start":"08:06.850 ","End":"08:11.230","Text":"Let\u0027s see, I\u0027ll choose minus 1 for"},{"Start":"08:11.230 ","End":"08:16.300","Text":"the sample here and what\u0027s between minus 1/4 and minus 1/2?"},{"Start":"08:16.300 ","End":"08:22.750","Text":"How about minus 1/3 is the sample value between minus 1/4 and 2/3."},{"Start":"08:22.750 ","End":"08:25.640","Text":"Well, I\u0027ll go for 0 every time."},{"Start":"08:25.740 ","End":"08:34.165","Text":"Between this and this I\u0027ll take the value 1 and after 2.5,"},{"Start":"08:34.165 ","End":"08:42.250","Text":"let\u0027s take 3 and let\u0027s try substituting each of these and see what we get."},{"Start":"08:42.250 ","End":"08:48.460","Text":"If we put in minus 1 in this expression what we"},{"Start":"08:48.460 ","End":"08:54.145","Text":"get is first of all maybe I\u0027ll write it down here just to scratch."},{"Start":"08:54.145 ","End":"08:59.575","Text":"I have a minus already and then I put in x is minus 1,"},{"Start":"08:59.575 ","End":"09:08.800","Text":"so minus 2 plus 1 is minus and hen twice minus 1 minus 5 is also minus."},{"Start":"09:08.800 ","End":"09:13.570","Text":"In fact, everything comes out minus over minus,"},{"Start":"09:13.570 ","End":"09:17.440","Text":"minus there with the dividing line."},{"Start":"09:17.440 ","End":"09:19.090","Text":"I\u0027ve got 5 minuses,"},{"Start":"09:19.090 ","End":"09:23.710","Text":"so this is a minus and this time we\u0027ll use the snake method,"},{"Start":"09:23.710 ","End":"09:29.095","Text":"so I\u0027ll just draw a point below the axis to indicate the minus."},{"Start":"09:29.095 ","End":"09:33.955","Text":"For the minus 1/3;"},{"Start":"09:33.955 ","End":"09:36.925","Text":"let\u0027s see there\u0027s, the minus is always there."},{"Start":"09:36.925 ","End":"09:41.875","Text":"Minus 2/3 plus 1 is already positive,"},{"Start":"09:41.875 ","End":"09:46.780","Text":"but everything else will come out negative."},{"Start":"09:46.780 ","End":"09:53.170","Text":"Minus 2/3 minus 5 this will be minus 1 minus"},{"Start":"09:53.170 ","End":"09:59.230","Text":"2 and this will be minus 4 thirds plus 1 is still negative,"},{"Start":"09:59.230 ","End":"10:01.735","Text":"so we get just 1 positive."},{"Start":"10:01.735 ","End":"10:06.085","Text":"We get basically minus and then plus,"},{"Start":"10:06.085 ","End":"10:10.168","Text":"minus, minus, minus this full negatives,"},{"Start":"10:10.168 ","End":"10:12.010","Text":"so it\u0027s a positive."},{"Start":"10:12.010 ","End":"10:15.865","Text":"I indicate that with a dot above the line."},{"Start":"10:15.865 ","End":"10:19.705","Text":"For 0, well 0 is easy enough,"},{"Start":"10:19.705 ","End":"10:22.240","Text":"0 just makes the x\u0027s disappear."},{"Start":"10:22.240 ","End":"10:25.345","Text":"We get plus, minus, minus,"},{"Start":"10:25.345 ","End":"10:31.090","Text":"plus and plus, minus,"},{"Start":"10:31.090 ","End":"10:32.920","Text":"minus this is what we get,"},{"Start":"10:32.920 ","End":"10:36.430","Text":"the 3 minuses that\u0027s also a minus."},{"Start":"10:36.430 ","End":"10:42.910","Text":"Here, when we put in 1 the situation becomes;"},{"Start":"10:42.910 ","End":"10:44.680","Text":"basically everything is as it was,"},{"Start":"10:44.680 ","End":"10:47.530","Text":"but this one changes from a minus to a plus."},{"Start":"10:47.530 ","End":"10:50.170","Text":"We have a minus plus,"},{"Start":"10:50.170 ","End":"10:52.855","Text":"minus plus, plus situation."},{"Start":"10:52.855 ","End":"10:54.100","Text":"There\u0027s 2 minuses,"},{"Start":"10:54.100 ","End":"10:55.645","Text":"so that\u0027s a plus,"},{"Start":"10:55.645 ","End":"10:59.440","Text":"let\u0027s say here and for 3,"},{"Start":"10:59.440 ","End":"11:02.410","Text":"basically all these terms come out plus,"},{"Start":"11:02.410 ","End":"11:04.090","Text":"but there\u0027s still that minus."},{"Start":"11:04.090 ","End":"11:06.460","Text":"In other words minus plus, plus, plus,"},{"Start":"11:06.460 ","End":"11:11.395","Text":"plus that\u0027s also a minus,"},{"Start":"11:11.395 ","End":"11:14.095","Text":"hence below the line."},{"Start":"11:14.095 ","End":"11:16.765","Text":"The other thing that\u0027s good to do,"},{"Start":"11:16.765 ","End":"11:25.150","Text":"we should do is the actual points themselves in these things that make something 0."},{"Start":"11:25.150 ","End":"11:30.730","Text":"The ones on the numerator are actually values that make the expression 0."},{"Start":"11:30.730 ","End":"11:34.540","Text":"They\u0027re important, the minus1/2 that\u0027s where"},{"Start":"11:34.540 ","End":"11:42.940","Text":"the rational expression is 0 and also at 2.5, it\u0027s also 0."},{"Start":"11:42.940 ","End":"11:46.780","Text":"However, these 2 values,"},{"Start":"11:46.780 ","End":"11:48.955","Text":"the 2/3 and the minus 1/4,"},{"Start":"11:48.955 ","End":"11:52.900","Text":"are actually illegal values where I actually can\u0027t substitute in."},{"Start":"11:52.900 ","End":"11:55.810","Text":"They\u0027re not in the domain of definition."},{"Start":"11:55.810 ","End":"11:59.560","Text":"The way I like to do that is I erase a little bit of"},{"Start":"11:59.560 ","End":"12:06.970","Text":"the x\u0027s to make a hole there to show that it\u0027s not possible."},{"Start":"12:06.970 ","End":"12:16.370","Text":"I even emphasize it with whole hollow circle."},{"Start":"12:16.620 ","End":"12:22.345","Text":"This is not something that we can substitute."},{"Start":"12:22.345 ","End":"12:25.555","Text":"The snake has gaps in it."},{"Start":"12:25.555 ","End":"12:28.680","Text":"The snake, let me draw something like this,"},{"Start":"12:28.680 ","End":"12:33.460","Text":"but it doesn\u0027t go here and picks up"},{"Start":"12:33.460 ","End":"12:39.895","Text":"somewhere and maybe like this and then maybe something like this."},{"Start":"12:39.895 ","End":"12:44.620","Text":"But it does go through this point and this point certainly."},{"Start":"12:44.620 ","End":"12:49.930","Text":"It continues indefinitely I like to put an arrow it\u0027s not standard,"},{"Start":"12:49.930 ","End":"12:52.490","Text":"but I just do that to show that."},{"Start":"12:52.950 ","End":"13:00.280","Text":"That\u0027s a so called snake what we want to know is let\u0027s see what do we want to know."},{"Start":"13:00.280 ","End":"13:03.400","Text":"We want the bigger or equal to 0,"},{"Start":"13:03.400 ","End":"13:06.340","Text":"that means positive and 0."},{"Start":"13:06.340 ","End":"13:07.840","Text":"That\u0027s what we\u0027re collecting."},{"Start":"13:07.840 ","End":"13:14.170","Text":"So the positive bits I\u0027ll do it with highlighting."},{"Start":"13:14.170 ","End":"13:16.435","Text":"Sometimes I use shading, sometimes highlight."},{"Start":"13:16.435 ","End":"13:20.530","Text":"Remember positives we\u0027re collecting and 0s."},{"Start":"13:20.530 ","End":"13:24.985","Text":"This is a 0 that\u0027s in positive all the way from here to here,"},{"Start":"13:24.985 ","End":"13:28.240","Text":"but not including this illegal point."},{"Start":"13:28.240 ","End":"13:31.525","Text":"Let\u0027s see what else are we positive or 0 here?"},{"Start":"13:31.525 ","End":"13:36.610","Text":"Positive, positive and this point here,"},{"Start":"13:36.610 ","End":"13:41.650","Text":"0 included and that\u0027s it."},{"Start":"13:41.650 ","End":"13:47.440","Text":"Now we just have to see what this highlighted part represents."},{"Start":"13:47.440 ","End":"13:54.970","Text":"Here it\u0027s from minus 1/2 to minus a 1/4 here from 2/3 to 2.5."},{"Start":"13:54.970 ","End":"13:58.614","Text":"We just have to make sure what\u0027s included and what\u0027s not included."},{"Start":"13:58.614 ","End":"14:06.115","Text":"In this one, we certainly have x between minus 1/4 and minus 1/2,"},{"Start":"14:06.115 ","End":"14:08.200","Text":"but this one is included,"},{"Start":"14:08.200 ","End":"14:10.690","Text":"here we have also"},{"Start":"14:10.690 ","End":"14:15.515","Text":"the inequality also the equality is greater than or equal to, but not here."},{"Start":"14:15.515 ","End":"14:20.539","Text":"Similarly, over here we have 2/3, we have x,"},{"Start":"14:20.539 ","End":"14:23.555","Text":"we have 2.5 less than,"},{"Start":"14:23.555 ","End":"14:26.450","Text":"less than, but we want to include this point."},{"Start":"14:26.450 ","End":"14:29.675","Text":"So this is a less than or equal to."},{"Start":"14:29.675 ","End":"14:32.445","Text":"Didn\u0027t come out clear I\u0027ll rewrite it,"},{"Start":"14:32.445 ","End":"14:36.295","Text":"less than or equal to."},{"Start":"14:36.295 ","End":"14:41.050","Text":"Then we put an or in the middle and that\u0027s our end."},{"Start":"14:41.050 ","End":"14:47.640","Text":"I\u0027ll highlight it and then declare that we are done."}],"Thumbnail":null,"ID":5076},{"Watched":false,"Name":"Exercise 6 part a","Duration":"12m 51s","ChapterTopicVideoID":5075,"CourseChapterTopicPlaylistID":3877,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.620","Text":"What we want to do here is bring this to the left-hand side."},{"Start":"00:04.620 ","End":"00:09.465","Text":"Let\u0027s start by just copying the first bit,"},{"Start":"00:09.465 ","End":"00:12.930","Text":"x minus 4 over 5,"},{"Start":"00:12.930 ","End":"00:21.288","Text":"and then minus this thing which is x^2 minus 4x plus 4 over,"},{"Start":"00:21.288 ","End":"00:22.956","Text":"here it\u0027s 5x minus 20,"},{"Start":"00:22.956 ","End":"00:29.025","Text":"but I noticed that 5 goes into both of them almost can\u0027t help but notice."},{"Start":"00:29.025 ","End":"00:33.030","Text":"It will be good to take that 5 outside the brackets,"},{"Start":"00:33.030 ","End":"00:36.390","Text":"especially since I see I already have a 5 here,"},{"Start":"00:36.390 ","End":"00:39.585","Text":"so I won\u0027t have to take 5 twice."},{"Start":"00:39.585 ","End":"00:43.215","Text":"Anyway, it\u0027s less than or equal to 0."},{"Start":"00:43.215 ","End":"00:45.050","Text":"Now, we\u0027re going to do an exercise in"},{"Start":"00:45.050 ","End":"00:47.930","Text":"fractions where we put them all with the same denominator,"},{"Start":"00:47.930 ","End":"00:50.240","Text":"and then we can add or subtract the numerators."},{"Start":"00:50.240 ","End":"00:55.140","Text":"Let\u0027s take a denominator, something like this."},{"Start":"00:55.640 ","End":"01:02.010","Text":"On the denominator will be 5 times x times x minus 4."},{"Start":"01:02.010 ","End":"01:04.684","Text":"All 3 of them go into this nicely."},{"Start":"01:04.684 ","End":"01:06.769","Text":"Remember, the technique?"},{"Start":"01:06.769 ","End":"01:08.960","Text":"I put little, I don\u0027t know what to call them,"},{"Start":"01:08.960 ","End":"01:12.785","Text":"separators here and I write the missing factors."},{"Start":"01:12.785 ","End":"01:17.370","Text":"For x, what\u0027s missing is 5 times x minus 4."},{"Start":"01:17.900 ","End":"01:23.740","Text":"For 5, What\u0027s missing is x times x minus 4."},{"Start":"01:23.960 ","End":"01:28.214","Text":"For the last 1, what\u0027s missing is just the x,"},{"Start":"01:28.214 ","End":"01:31.580","Text":"and then what we do is we multiply this by this,"},{"Start":"01:31.580 ","End":"01:34.040","Text":"this by this, and that makes everything"},{"Start":"01:34.040 ","End":"01:36.945","Text":"over the same denominator and then we just add and subtract them."},{"Start":"01:36.945 ","End":"01:43.215","Text":"So 5 times x minus 4,"},{"Start":"01:43.215 ","End":"01:45.165","Text":"x plus 3,"},{"Start":"01:45.165 ","End":"01:46.815","Text":"that\u0027s the first one."},{"Start":"01:46.815 ","End":"01:50.010","Text":"The next one is x,"},{"Start":"01:50.010 ","End":"01:52.230","Text":"it\u0027s a plus x,"},{"Start":"01:52.230 ","End":"01:54.120","Text":"x minus 4,"},{"Start":"01:54.120 ","End":"01:56.210","Text":"and again, x minus 4,"},{"Start":"01:56.210 ","End":"01:59.220","Text":"so just make it squared,"},{"Start":"01:59.220 ","End":"02:07.350","Text":"minus x times x^2 minus 4x plus 4,"},{"Start":"02:07.350 ","End":"02:10.625","Text":"and all this is less than or equal to 0."},{"Start":"02:10.625 ","End":"02:15.470","Text":"I just did a copy-paste for the bit that\u0027s not going to change."},{"Start":"02:15.470 ","End":"02:17.935","Text":"We\u0027re just going to work on the numerator here."},{"Start":"02:17.935 ","End":"02:21.200","Text":"We won\u0027t do it all at once."},{"Start":"02:21.200 ","End":"02:23.720","Text":"We\u0027ll do it in bits, 5 I\u0027ll leave,"},{"Start":"02:23.720 ","End":"02:25.865","Text":"I\u0027ll just multiply these 2."},{"Start":"02:25.865 ","End":"02:33.440","Text":"I get x^2 minus 4x plus 3x is minus x,"},{"Start":"02:33.440 ","End":"02:36.125","Text":"and then minus 12."},{"Start":"02:36.125 ","End":"02:39.908","Text":"The next bit is x times,"},{"Start":"02:39.908 ","End":"02:46.235","Text":"now there\u0027s a formula for binomial squared special product a minus b^2,"},{"Start":"02:46.235 ","End":"02:50.885","Text":"I\u0027ll write it just in case you\u0027ve forgotten."},{"Start":"02:50.885 ","End":"02:52.490","Text":"I\u0027ll write that formula here,"},{"Start":"02:52.490 ","End":"03:01.290","Text":"a minus b^2 is a^2 minus 2ab plus b^2 that I\u0027m sure you probably all have memorized."},{"Start":"03:03.790 ","End":"03:07.020","Text":"So x^2 minus twice x times 4,"},{"Start":"03:07.020 ","End":"03:10.650","Text":"so that 4^2 is that."},{"Start":"03:10.650 ","End":"03:12.465","Text":"That\u0027s this bit,"},{"Start":"03:12.465 ","End":"03:18.960","Text":"and then here, I can already multiply out."},{"Start":"03:18.960 ","End":"03:27.660","Text":"This is minus x^3"},{"Start":"03:27.660 ","End":"03:35.020","Text":"plus 4x^2 and minus 4x."},{"Start":"03:37.850 ","End":"03:42.770","Text":"Once again, I did a copy-paste of the part that\u0027s not going to change."},{"Start":"03:42.770 ","End":"03:45.935","Text":"I just want to simplify the numerator."},{"Start":"03:45.935 ","End":"03:49.890","Text":"Let\u0027s see if we can do it in one step now."},{"Start":"03:55.010 ","End":"03:58.500","Text":"I\u0027ll write it out in full for public safest,"},{"Start":"03:58.500 ","End":"04:06.510","Text":"5x^2 minus 5x minus"},{"Start":"04:06.510 ","End":"04:10.945","Text":"60 and then plus x^3,"},{"Start":"04:10.945 ","End":"04:13.295","Text":"I\u0027m just raising the power of each by 1."},{"Start":"04:13.295 ","End":"04:15.980","Text":"This is x^3,"},{"Start":"04:15.980 ","End":"04:24.035","Text":"this is minus 8x^2 plus"},{"Start":"04:24.035 ","End":"04:26.720","Text":"16x that\u0027s this bit,"},{"Start":"04:26.720 ","End":"04:34.880","Text":"and then minus x^3 plus 4x^2 minus 4x."},{"Start":"04:35.800 ","End":"04:40.230","Text":"Now, let\u0027s collect terms in the numerator."},{"Start":"04:40.990 ","End":"04:47.674","Text":"We\u0027ll go for the x^3 first,"},{"Start":"04:47.674 ","End":"04:50.515","Text":"but I\u0027ll just copy this bit."},{"Start":"04:50.515 ","End":"04:54.440","Text":"As I was saying the x^3 luckily,"},{"Start":"04:54.440 ","End":"04:59.825","Text":"they cancel out because we don\u0027t know how to factorize cubics in general,"},{"Start":"04:59.825 ","End":"05:01.610","Text":"so we\u0027re just going to have a quadratic."},{"Start":"05:01.610 ","End":"05:07.700","Text":"The next term after that would be the x^2 term and we have, how many of those?"},{"Start":"05:07.700 ","End":"05:10.890","Text":"We have 1 here, and 1 here."},{"Start":"05:10.890 ","End":"05:18.165","Text":"Let\u0027s see, do the arithmetic 5 minus 8 plus 4,"},{"Start":"05:18.165 ","End":"05:19.540","Text":"I make that 1."},{"Start":"05:19.540 ","End":"05:21.725","Text":"If you look at the positives, yeah,"},{"Start":"05:21.725 ","End":"05:26.015","Text":"it\u0027s 1 so 1x^2 so just x^2."},{"Start":"05:26.015 ","End":"05:28.820","Text":"Now, the x\u0027s, how many of those do we have?"},{"Start":"05:28.820 ","End":"05:30.215","Text":"Three of them, here,"},{"Start":"05:30.215 ","End":"05:32.225","Text":"here, and here."},{"Start":"05:32.225 ","End":"05:35.070","Text":"Minus 5 plus 16,"},{"Start":"05:35.070 ","End":"05:39.225","Text":"minus 4, 16 minus 9 is 7,"},{"Start":"05:39.225 ","End":"05:43.118","Text":"so this would be plus 7x,"},{"Start":"05:43.118 ","End":"05:52.420","Text":"and now numbers don\u0027t seem to have any of those so this is all there is."},{"Start":"05:52.970 ","End":"05:57.040","Text":"There is a number minus 60."},{"Start":"05:57.500 ","End":"06:00.270","Text":"Now, that\u0027s it."},{"Start":"06:00.270 ","End":"06:04.490","Text":"The next thing we want to do is to factorize this."},{"Start":"06:04.490 ","End":"06:07.680","Text":"We want to factorize as much as possible. Let\u0027s see."},{"Start":"06:11.960 ","End":"06:15.635","Text":"Once again, I copied the bit that doesn\u0027t change."},{"Start":"06:15.635 ","End":"06:18.140","Text":"I\u0027ll just tell you the factorization of this."},{"Start":"06:18.140 ","End":"06:22.220","Text":"There are two techniques that you\u0027ve learned and this is time-consuming,"},{"Start":"06:22.220 ","End":"06:23.915","Text":"so I don\u0027t want to waste time with that."},{"Start":"06:23.915 ","End":"06:32.630","Text":"This comes out to be x plus 12 and x minus 5."},{"Start":"06:32.630 ","End":"06:34.430","Text":"However, if I didn\u0027t show you how I got to it,"},{"Start":"06:34.430 ","End":"06:36.355","Text":"at least I can show you that it\u0027s true,"},{"Start":"06:36.355 ","End":"06:41.100","Text":"x times x is x^2 plus 12x"},{"Start":"06:41.100 ","End":"06:47.625","Text":"minus 5x is 7x and 12 times minus 5 is minus 60 so this is fine."},{"Start":"06:47.625 ","End":"06:51.375","Text":"Now, this is completely factorized"},{"Start":"06:51.375 ","End":"06:57.600","Text":"we need to say which values of x make these factors 0,"},{"Start":"06:57.600 ","End":"07:04.005","Text":"x plus 12 is 0 when x is minus 12,"},{"Start":"07:04.005 ","End":"07:08.420","Text":"x minus 5 is 0 when x is 5, and so on."},{"Start":"07:08.420 ","End":"07:10.250","Text":"Here, we have the value 4,"},{"Start":"07:10.250 ","End":"07:12.260","Text":"here we have the value 0."},{"Start":"07:12.260 ","End":"07:14.354","Text":"The 5 we can basically just ignore,"},{"Start":"07:14.354 ","End":"07:15.800","Text":"it\u0027s a positive number actually,"},{"Start":"07:15.800 ","End":"07:19.640","Text":"you could even multiply both sides by 5 just get rid of it,"},{"Start":"07:19.640 ","End":"07:22.025","Text":"but I\u0027ll leave it there."},{"Start":"07:22.025 ","End":"07:23.870","Text":"But when we do our computations of plus,"},{"Start":"07:23.870 ","End":"07:25.010","Text":"minus, and so on,"},{"Start":"07:25.010 ","End":"07:27.040","Text":"the 5 is a plus won\u0027t change anything,"},{"Start":"07:27.040 ","End":"07:29.550","Text":"so I can ignore it."},{"Start":"07:29.960 ","End":"07:33.380","Text":"What I want to do now is using the sketch method,"},{"Start":"07:33.380 ","End":"07:38.345","Text":"we\u0027ll use the variation of the snake method as it\u0027s known."},{"Start":"07:38.345 ","End":"07:42.250","Text":"We start with the number line and we put these four numbers on."},{"Start":"07:42.250 ","End":"07:45.680","Text":"Here, we have a number line and we want to place these four numbers,"},{"Start":"07:45.680 ","End":"07:47.240","Text":"we have to do it in the right order,"},{"Start":"07:47.240 ","End":"07:49.040","Text":"but not to scale or anything."},{"Start":"07:49.040 ","End":"07:50.405","Text":"The first one,"},{"Start":"07:50.405 ","End":"07:53.420","Text":"leftmost one is minus 12,"},{"Start":"07:53.420 ","End":"07:56.845","Text":"and then we have a 0 and a 4 and 5,"},{"Start":"07:56.845 ","End":"07:58.600","Text":"0 somewhere here,"},{"Start":"07:58.600 ","End":"08:03.255","Text":"4 somewhere here, and 5 somewhere here."},{"Start":"08:03.255 ","End":"08:05.880","Text":"The thing about the 0 and the 4,"},{"Start":"08:05.880 ","End":"08:09.100","Text":"is different than these 2 because these are on the denominator."},{"Start":"08:09.100 ","End":"08:12.400","Text":"These are values which x is not allowed to be"},{"Start":"08:12.400 ","End":"08:15.670","Text":"illegal values or values not in the domain and I like"},{"Start":"08:15.670 ","End":"08:22.900","Text":"to put a couple of holes in the x-axis just to show that these are not valid."},{"Start":"08:22.900 ","End":"08:25.030","Text":"At 0 and 4,"},{"Start":"08:25.030 ","End":"08:30.775","Text":"I\u0027ll take an eraser and just erase a little bit and then erase a little bit and"},{"Start":"08:30.775 ","End":"08:37.835","Text":"just emphasize it with a hollow circle like a hole here and here."},{"Start":"08:37.835 ","End":"08:41.380","Text":"It\u0027s a different story with the minus 12 and the 5 there."},{"Start":"08:41.380 ","End":"08:44.140","Text":"The expression is perfectly defined and it\u0027s equal"},{"Start":"08:44.140 ","End":"08:47.425","Text":"to 0 as a matter of fact because the numerator is 0,"},{"Start":"08:47.425 ","End":"08:48.920","Text":"I think is 0,"},{"Start":"08:48.920 ","End":"08:50.570","Text":"so at 5 and minus 12,"},{"Start":"08:50.570 ","End":"08:54.140","Text":"I put a solid circle to show that it\u0027s 0."},{"Start":"08:54.140 ","End":"08:56.036","Text":"So this for 0,"},{"Start":"08:56.036 ","End":"08:58.175","Text":"this for undefined or illegal."},{"Start":"08:58.175 ","End":"09:02.375","Text":"Now, we have 5 intervals."},{"Start":"09:02.375 ","End":"09:04.820","Text":"Let me just emphasize them."},{"Start":"09:04.820 ","End":"09:06.650","Text":"If I put some vertical lines through,"},{"Start":"09:06.650 ","End":"09:08.720","Text":"you\u0027ll see that there are 5 intervals."},{"Start":"09:08.720 ","End":"09:10.627","Text":"There\u0027s this 1, 2,"},{"Start":"09:10.627 ","End":"09:12.845","Text":"3, 4, and 5."},{"Start":"09:12.845 ","End":"09:15.970","Text":"In each one, I know that"},{"Start":"09:15.970 ","End":"09:20.335","Text":"the expression is going to be the positive all the time or negative all the time,"},{"Start":"09:20.335 ","End":"09:25.800","Text":"it can\u0027t be 0 or it can\u0027t change in the middle."},{"Start":"09:25.800 ","End":"09:28.460","Text":"So we choose sample values."},{"Start":"09:28.460 ","End":"09:33.005","Text":"I\u0027ll just write some sample values here."},{"Start":"09:33.005 ","End":"09:38.000","Text":"Say minus 13,"},{"Start":"09:38.000 ","End":"09:43.640","Text":"I don\u0027t know, minus 1."},{"Start":"09:43.640 ","End":"09:45.020","Text":"Here, I can take a whole number,"},{"Start":"09:45.020 ","End":"09:46.630","Text":"so 4 and a-half,"},{"Start":"09:46.630 ","End":"09:48.975","Text":"and here I can choose 6."},{"Start":"09:48.975 ","End":"09:51.095","Text":"And each of these 5 values,"},{"Start":"09:51.095 ","End":"09:53.735","Text":"I\u0027m going to substitute in this expression here,"},{"Start":"09:53.735 ","End":"09:56.560","Text":"but I only need to know positive or negative,"},{"Start":"09:56.560 ","End":"09:59.555","Text":"that\u0027s why I said I can ignore the 5 which is positive."},{"Start":"09:59.555 ","End":"10:01.587","Text":"Let\u0027s see."},{"Start":"10:01.587 ","End":"10:03.980","Text":"If I put in minus 13,"},{"Start":"10:03.980 ","End":"10:08.510","Text":"then minus 13 plus 12 is minus 1."},{"Start":"10:08.510 ","End":"10:10.895","Text":"In fact, they\u0027re all going to be negative."},{"Start":"10:10.895 ","End":"10:15.320","Text":"I get full negatives so that makes a positive."},{"Start":"10:15.320 ","End":"10:18.350","Text":"I\u0027ll use a variation of"},{"Start":"10:18.350 ","End":"10:22.250","Text":"the snake method without actually drawing the snake just writing the pluses and minuses."},{"Start":"10:22.250 ","End":"10:24.950","Text":"That\u0027s a common variation on it,"},{"Start":"10:24.950 ","End":"10:29.880","Text":"which I don\u0027t particularly like snakes."},{"Start":"10:31.180 ","End":"10:33.695","Text":"For the minus 1,"},{"Start":"10:33.695 ","End":"10:39.545","Text":"we get minus 1 plus 12 is positive."},{"Start":"10:39.545 ","End":"10:41.670","Text":"But this is still negative,"},{"Start":"10:41.670 ","End":"10:44.310","Text":"this is negative, and this is negative."},{"Start":"10:44.310 ","End":"10:46.440","Text":"We have positive and 3 negatives,"},{"Start":"10:46.440 ","End":"10:48.907","Text":"3 negatives means negative,"},{"Start":"10:48.907 ","End":"10:51.530","Text":"so in this whole interval, it\u0027s negative."},{"Start":"10:51.530 ","End":"10:53.845","Text":"The expression, one,"},{"Start":"10:53.845 ","End":"10:57.140","Text":"we have 1 plus 12 is positive,"},{"Start":"10:57.140 ","End":"10:59.810","Text":"1 is positive, but these 2 are still negative,"},{"Start":"10:59.810 ","End":"11:01.925","Text":"2 negatives is a positive."},{"Start":"11:01.925 ","End":"11:05.810","Text":"It\u0027s often the case that it alternates signs and not always."},{"Start":"11:05.810 ","End":"11:07.475","Text":"Let\u0027s see, with 4 and a-half,"},{"Start":"11:07.475 ","End":"11:09.265","Text":"that\u0027s going to be positive."},{"Start":"11:09.265 ","End":"11:10.905","Text":"This is going to be negative."},{"Start":"11:10.905 ","End":"11:12.360","Text":"This is positive,"},{"Start":"11:12.360 ","End":"11:14.680","Text":"and this is positive."},{"Start":"11:14.680 ","End":"11:19.230","Text":"We only have one negative,"},{"Start":"11:19.230 ","End":"11:25.130","Text":"which is this one, and so this is a minus."},{"Start":"11:25.130 ","End":"11:28.085","Text":"Then the last one when we take x is 6,"},{"Start":"11:28.085 ","End":"11:32.660","Text":"they\u0027re all positive and this is positive here."},{"Start":"11:32.660 ","End":"11:36.320","Text":"Now, we know where the thing is, positive,"},{"Start":"11:36.320 ","End":"11:38.465","Text":"negative, 0,"},{"Start":"11:38.465 ","End":"11:40.460","Text":"or undefined wherever we want."},{"Start":"11:40.460 ","End":"11:42.500","Text":"What are we collecting?"},{"Start":"11:42.500 ","End":"11:45.830","Text":"We are collecting negatives and zeros."},{"Start":"11:45.830 ","End":"11:50.180","Text":"That\u0027s what less than or equal to 0 means either 0 or less than 0, which is negative."},{"Start":"11:50.180 ","End":"11:54.470","Text":"I\u0027m interested in this,"},{"Start":"11:54.470 ","End":"11:56.915","Text":"it\u0027s a negative, this is a negative."},{"Start":"11:56.915 ","End":"12:02.590","Text":"I\u0027m also interested in this because it\u0027s a 0 and I\u0027m interested in collecting this also."},{"Start":"12:02.590 ","End":"12:05.360","Text":"The ones I\u0027ve circled in this dark blue are"},{"Start":"12:05.360 ","End":"12:08.450","Text":"the things that I\u0027m collecting and if you look at it,"},{"Start":"12:08.450 ","End":"12:17.863","Text":"this and this together means that x is between 0,"},{"Start":"12:17.863 ","End":"12:22.820","Text":"and minus 12, important here,"},{"Start":"12:22.820 ","End":"12:24.410","Text":"including the minus 12,"},{"Start":"12:24.410 ","End":"12:27.830","Text":"but not including the 0 we can\u0027t take 0 is not defined there."},{"Start":"12:27.830 ","End":"12:35.010","Text":"This bit here gives us 4 less than x,"},{"Start":"12:35.010 ","End":"12:36.640","Text":"less than 5,"},{"Start":"12:36.640 ","End":"12:39.850","Text":"but that\u0027s not all because also x could equal 5,"},{"Start":"12:39.850 ","End":"12:42.565","Text":"so we put less than or equal to 5."},{"Start":"12:42.565 ","End":"12:45.268","Text":"Put the word or in the middle,"},{"Start":"12:45.268 ","End":"12:46.790","Text":"and that\u0027s our answer."},{"Start":"12:46.790 ","End":"12:51.210","Text":"I\u0027ll just highlight it and then we can move on to Part B."}],"Thumbnail":null,"ID":5077},{"Watched":false,"Name":"Exercise 6 part b","Duration":"6m 56s","ChapterTopicVideoID":5076,"CourseChapterTopicPlaylistID":3877,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.190","Text":"On to Part B."},{"Start":"00:06.170 ","End":"00:13.410","Text":"Once again, we want to bring everything to the left-hand side and compare with 0."},{"Start":"00:13.410 ","End":"00:17.730","Text":"What we get is x over,"},{"Start":"00:17.730 ","End":"00:21.942","Text":"let me just leave that blank for the moment,"},{"Start":"00:21.942 ","End":"00:31.590","Text":"plus 1 over x plus 2 minus 1 over x minus 2 less than 0."},{"Start":"00:31.590 ","End":"00:33.615","Text":"I\u0027m coming back here."},{"Start":"00:33.615 ","End":"00:37.470","Text":"What I want to do is use the difference of squares formula,"},{"Start":"00:37.470 ","End":"00:40.785","Text":"the a^2 minus b^2 thing,"},{"Start":"00:40.785 ","End":"00:46.915","Text":"to write this as x minus 2 times x plus 2."},{"Start":"00:46.915 ","End":"00:49.730","Text":"Obviously, this is a good thing"},{"Start":"00:49.730 ","End":"00:53.090","Text":"because I see that I already have an x minus 2 and x plus 2,"},{"Start":"00:53.090 ","End":"00:56.135","Text":"and that can really save me on the common denominator."},{"Start":"00:56.135 ","End":"00:58.145","Text":"For the common denominator,"},{"Start":"00:58.145 ","End":"01:03.150","Text":"all I really need is x minus 2,"},{"Start":"01:03.150 ","End":"01:08.510","Text":"x plus 2 because all the other factors go into this."},{"Start":"01:08.510 ","End":"01:12.425","Text":"What we\u0027re left with is,"},{"Start":"01:12.425 ","End":"01:16.070","Text":"if we put this as our common denominator and then add the numerators,"},{"Start":"01:16.070 ","End":"01:17.630","Text":"here it\u0027s just x,"},{"Start":"01:17.630 ","End":"01:22.640","Text":"here we have a missing factor of x minus 2,"},{"Start":"01:22.640 ","End":"01:26.340","Text":"and here we have a missing factor of x plus 2."},{"Start":"01:27.790 ","End":"01:32.790","Text":"This becomes plus x minus 2 times 1,"},{"Start":"01:32.790 ","End":"01:35.260","Text":"which is just x minus 2."},{"Start":"01:35.260 ","End":"01:38.105","Text":"I\u0027ll leave it like that just so you can see,"},{"Start":"01:38.105 ","End":"01:40.610","Text":"minus x plus 2 times 1."},{"Start":"01:40.610 ","End":"01:44.015","Text":"Again, I\u0027ll just leave it like that so we can see,"},{"Start":"01:44.015 ","End":"01:50.000","Text":"even leave the 1 in times 1, less than 0."},{"Start":"01:50.000 ","End":"01:53.605","Text":"Now, I want to simplify the numerator,"},{"Start":"01:53.605 ","End":"01:58.555","Text":"x plus x minus 2 minus x plus 2."},{"Start":"01:58.555 ","End":"02:03.565","Text":"For the xs, we got x plus x minus x is just x,"},{"Start":"02:03.565 ","End":"02:06.465","Text":"and for the numbers,"},{"Start":"02:06.465 ","End":"02:10.280","Text":"I have minus 2 from here,"},{"Start":"02:10.280 ","End":"02:12.695","Text":"and I also have minus 2 from here,"},{"Start":"02:12.695 ","End":"02:15.430","Text":"so it\u0027s x minus 4."},{"Start":"02:15.430 ","End":"02:22.165","Text":"It\u0027s still over the x minus 2, x plus 2."},{"Start":"02:22.165 ","End":"02:25.160","Text":"What we want is less than 0."},{"Start":"02:25.160 ","End":"02:30.775","Text":"As usual, we look for the numbers that make any one of these factors 0."},{"Start":"02:30.775 ","End":"02:34.545","Text":"This factor is 0 when x is 4,"},{"Start":"02:34.545 ","End":"02:36.630","Text":"this one when x is 2,"},{"Start":"02:36.630 ","End":"02:39.300","Text":"and this one when x is minus 2."},{"Start":"02:39.300 ","End":"02:41.240","Text":"I want to put these on the number line,"},{"Start":"02:41.240 ","End":"02:45.330","Text":"and then we\u0027ll distinguish numerator from denominator."},{"Start":"02:45.960 ","End":"02:51.280","Text":"Here\u0027s a number line, and I\u0027ll put these three in the right order on it."},{"Start":"02:51.280 ","End":"02:53.185","Text":"We have a minus 2,"},{"Start":"02:53.185 ","End":"02:57.140","Text":"we have a 2, and we have a 4."},{"Start":"02:58.130 ","End":"03:02.195","Text":"The difference is that at 4,"},{"Start":"03:02.195 ","End":"03:08.410","Text":"this rational expression is defined and its value is 0,"},{"Start":"03:08.410 ","End":"03:13.435","Text":"which I indicate by putting a dot on the number line here."},{"Start":"03:13.435 ","End":"03:18.699","Text":"For these two, the expression is not defined."},{"Start":"03:18.699 ","End":"03:21.280","Text":"You can\u0027t substitute 2 or minus 2."},{"Start":"03:21.280 ","End":"03:25.555","Text":"I indicate this by putting a little hole,"},{"Start":"03:25.555 ","End":"03:31.495","Text":"the eraser here and here to show that these not valid."},{"Start":"03:31.495 ","End":"03:37.760","Text":"I also like to emphasize it with a little hollow circle or whatever,"},{"Start":"03:37.760 ","End":"03:42.485","Text":"that these are not allowed and can\u0027t substitute them."},{"Start":"03:42.485 ","End":"03:45.030","Text":"They\u0027re outside the domain."},{"Start":"03:45.890 ","End":"03:49.700","Text":"Now, we want to see in each of these, how many intervals?"},{"Start":"03:49.700 ","End":"03:50.780","Text":"We have 1, 2, 3,"},{"Start":"03:50.780 ","End":"03:54.800","Text":"4 intervals, whether the expression is positive or negative."},{"Start":"03:54.800 ","End":"03:59.865","Text":"Within each interval, it stays the same sign. Let\u0027s see."},{"Start":"03:59.865 ","End":"04:02.085","Text":"We choose sample values."},{"Start":"04:02.085 ","End":"04:04.200","Text":"We need to choose a value here,"},{"Start":"04:04.200 ","End":"04:07.160","Text":"a value here, a value here, and a value here."},{"Start":"04:07.160 ","End":"04:10.700","Text":"We can do all this mentally."},{"Start":"04:10.700 ","End":"04:15.226","Text":"Suppose I choose minus 3 and I substitute it,"},{"Start":"04:15.226 ","End":"04:18.985","Text":"minus 3, minus 4 is negative."},{"Start":"04:18.985 ","End":"04:21.380","Text":"Minus 3, minus 2 is also negative."},{"Start":"04:21.380 ","End":"04:22.550","Text":"Minus 3 plus 2 is negative."},{"Start":"04:22.550 ","End":"04:24.215","Text":"We have minus, minus, minus,"},{"Start":"04:24.215 ","End":"04:25.849","Text":"which makes it minus."},{"Start":"04:25.849 ","End":"04:28.415","Text":"I\u0027ll use the snake method this time."},{"Start":"04:28.415 ","End":"04:29.720","Text":"I put something negative,"},{"Start":"04:29.720 ","End":"04:31.400","Text":"which means below the axis,"},{"Start":"04:31.400 ","End":"04:33.410","Text":"corresponding to this point."},{"Start":"04:33.410 ","End":"04:36.740","Text":"Then at this point, let\u0027s say I choose 0,"},{"Start":"04:36.740 ","End":"04:38.210","Text":"that would be the easiest,"},{"Start":"04:38.210 ","End":"04:40.955","Text":"0 minus 4 is negative."},{"Start":"04:40.955 ","End":"04:43.190","Text":"Here negative, here positive."},{"Start":"04:43.190 ","End":"04:46.609","Text":"2 negatives means that it\u0027s altogether positive."},{"Start":"04:46.609 ","End":"04:48.605","Text":"Above 0, I\u0027ll put positive."},{"Start":"04:48.605 ","End":"04:51.385","Text":"Let\u0027s suppose I take this to be 3."},{"Start":"04:51.385 ","End":"04:53.870","Text":"Then 3 minus 4 is negative,"},{"Start":"04:53.870 ","End":"04:55.910","Text":"but then positive, positive,"},{"Start":"04:55.910 ","End":"04:58.380","Text":"so it\u0027s a negative."},{"Start":"04:59.060 ","End":"05:02.386","Text":"Then let\u0027s say I take 5 here,"},{"Start":"05:02.386 ","End":"05:05.420","Text":"all these three factors will be positive,"},{"Start":"05:05.420 ","End":"05:09.530","Text":"so it\u0027s positive here. Something like that."},{"Start":"05:09.530 ","End":"05:16.610","Text":"Now, the snake, which is just a schematic representation,"},{"Start":"05:16.610 ","End":"05:19.970","Text":"not a graph, we just connect everything,"},{"Start":"05:19.970 ","End":"05:24.995","Text":"but we don\u0027t go through the bad illegal points."},{"Start":"05:24.995 ","End":"05:30.265","Text":"Here we just miss it and then continue."},{"Start":"05:30.265 ","End":"05:32.980","Text":"This is perfectly fine,"},{"Start":"05:32.980 ","End":"05:38.160","Text":"this 0.4 is good and so on."},{"Start":"05:38.160 ","End":"05:43.940","Text":"The graph continues indefinitely, but let\u0027s see."},{"Start":"05:43.940 ","End":"05:45.500","Text":"What are we collecting today?"},{"Start":"05:45.500 ","End":"05:47.330","Text":"We\u0027re collecting less than 0."},{"Start":"05:47.330 ","End":"05:49.670","Text":"In other words, we\u0027re interested in the negatives,"},{"Start":"05:49.670 ","End":"05:52.170","Text":"strictly negative."},{"Start":"05:53.120 ","End":"05:55.130","Text":"I\u0027ll highlight them."},{"Start":"05:55.130 ","End":"05:58.955","Text":"What we\u0027re talking about is this bit here and this bit here."},{"Start":"05:58.955 ","End":"06:03.980","Text":"What we want as the negative is this up to here,"},{"Start":"06:03.980 ","End":"06:07.492","Text":"and, of course, not including,"},{"Start":"06:07.492 ","End":"06:09.050","Text":"and then from here,"},{"Start":"06:09.050 ","End":"06:11.365","Text":"also not including that value,"},{"Start":"06:11.365 ","End":"06:16.350","Text":"and here also not including."},{"Start":"06:16.350 ","End":"06:19.850","Text":"Even though it\u0027s defined for it just happens to be 0 there,"},{"Start":"06:19.850 ","End":"06:21.635","Text":"and I\u0027m collecting negatives."},{"Start":"06:21.635 ","End":"06:23.435","Text":"I don\u0027t want the 4 either."},{"Start":"06:23.435 ","End":"06:24.920","Text":"I want just this bit and this bit,"},{"Start":"06:24.920 ","End":"06:27.664","Text":"and all of them without the endpoints."},{"Start":"06:27.664 ","End":"06:37.740","Text":"In terms of x, this bit can be described as x less than minus 2."},{"Start":"06:37.740 ","End":"06:43.980","Text":"This bit can be described as 2 less than x, less than 4."},{"Start":"06:43.980 ","End":"06:47.055","Text":"Together we just put an or in between."},{"Start":"06:47.055 ","End":"06:49.730","Text":"This is our answer."},{"Start":"06:49.730 ","End":"06:55.560","Text":"I\u0027ll highlight it. That\u0027s it for part B. We are done."}],"Thumbnail":null,"ID":5078},{"Watched":false,"Name":"Exercise 7 part a","Duration":"10m 14s","ChapterTopicVideoID":5077,"CourseChapterTopicPlaylistID":3877,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.150","Text":"Here we have another pair of inequalities."},{"Start":"00:03.150 ","End":"00:08.940","Text":"They\u0027re going to turn out to be rational expressions compared to 0."},{"Start":"00:08.940 ","End":"00:13.960","Text":"What we have to do first is bring things over to the left-hand side."},{"Start":"00:13.960 ","End":"00:23.160","Text":"We do compare with 0 and then simplify what\u0027s left as 1 quotient."},{"Start":"00:23.160 ","End":"00:28.065","Text":"Some numerator over denominator where each of them is the factorized polynomial,"},{"Start":"00:28.065 ","End":"00:31.005","Text":"in other words, the factorized rational expression."},{"Start":"00:31.005 ","End":"00:34.935","Text":"Then we could proceed as usual."},{"Start":"00:34.935 ","End":"00:37.530","Text":"That\u0027s just the general idea."},{"Start":"00:37.530 ","End":"00:41.200","Text":"Let\u0027s start with part a."},{"Start":"00:42.620 ","End":"00:46.130","Text":"Let\u0027s see if we can factorize something first because that"},{"Start":"00:46.130 ","End":"00:48.800","Text":"will make it easier to find the common denominator."},{"Start":"00:48.800 ","End":"00:52.220","Text":"I already see here I can take 2 out and here I can"},{"Start":"00:52.220 ","End":"00:56.760","Text":"take 3 out and here I see a difference of squares."},{"Start":"00:58.280 ","End":"01:07.290","Text":"Let\u0027s write it as 4 over 2x plus 2,"},{"Start":"01:07.290 ","End":"01:08.775","Text":"taking the 2 out."},{"Start":"01:08.775 ","End":"01:11.580","Text":"Here, we\u0027ll take the 3 out."},{"Start":"01:11.580 ","End":"01:18.730","Text":"It\u0027s 3 over 3 times x minus 2."},{"Start":"01:19.390 ","End":"01:22.370","Text":"I\u0027ll do 2 steps in 1."},{"Start":"01:22.370 ","End":"01:24.050","Text":"The less than this,"},{"Start":"01:24.050 ","End":"01:29.955","Text":"I\u0027ll put it as minus this and then less than 0."},{"Start":"01:29.955 ","End":"01:36.005","Text":"This expression, I\u0027ll also rewrite it instead of x^2 minus 4,"},{"Start":"01:36.005 ","End":"01:38.375","Text":"I\u0027ll write it as x minus 2,"},{"Start":"01:38.375 ","End":"01:42.185","Text":"x plus 2 using the difference of squares formula,"},{"Start":"01:42.185 ","End":"01:45.180","Text":"the one with a^2 minus b^2."},{"Start":"01:45.910 ","End":"01:51.380","Text":"At this point we can easily spot a common denominator."},{"Start":"01:51.380 ","End":"01:54.565","Text":"But small thing."},{"Start":"01:54.565 ","End":"01:59.100","Text":"Not so small, I would cancel this 3 with this 3,"},{"Start":"01:59.100 ","End":"02:01.140","Text":"so it just leaves a 1 here."},{"Start":"02:01.140 ","End":"02:03.960","Text":"I\u0027d also cancel this 2 with this 4."},{"Start":"02:03.960 ","End":"02:08.100","Text":"It goes in 2 so we\u0027ve got smaller numbers to work with."},{"Start":"02:08.100 ","End":"02:12.140","Text":"At this point, becomes fairly"},{"Start":"02:12.140 ","End":"02:15.830","Text":"clear as to what the common denominator is or rather dividing line."},{"Start":"02:15.830 ","End":"02:19.700","Text":"What we need is x minus 2 and x plus 2,"},{"Start":"02:19.700 ","End":"02:21.530","Text":"at least we needed for this,"},{"Start":"02:21.530 ","End":"02:26.050","Text":"but then these factors are already contained so we don\u0027t need them again."},{"Start":"02:26.050 ","End":"02:30.235","Text":"Our common denominator will just be x minus 2,"},{"Start":"02:30.235 ","End":"02:35.685","Text":"x plus 2 and here less than 0."},{"Start":"02:35.685 ","End":"02:40.730","Text":"I just want to rewrite this as one fraction over this denominator."},{"Start":"02:40.730 ","End":"02:45.320","Text":"What we do is we put it to the fractions as the same denominator by"},{"Start":"02:45.320 ","End":"02:50.735","Text":"multiplying top and bottom as appropriate and then adding all the numerators,"},{"Start":"02:50.735 ","End":"02:52.470","Text":"adding or subtracting,"},{"Start":"02:52.470 ","End":"02:55.460","Text":"because then we have the same denominator and we can do that."},{"Start":"02:55.460 ","End":"03:01.570","Text":"Here what we get is let\u0027s see."},{"Start":"03:01.570 ","End":"03:06.385","Text":"This x plus 2 is missing an x minus 2."},{"Start":"03:06.385 ","End":"03:10.235","Text":"If I multiply top and bottom by x minus 2,"},{"Start":"03:10.235 ","End":"03:17.255","Text":"it will be 2x minus 2 over this denominator."},{"Start":"03:17.255 ","End":"03:23.350","Text":"Here I have to multiply the 1 by the missing factor here,"},{"Start":"03:23.350 ","End":"03:31.465","Text":"which is x plus 2 and the 4 is already over this denominator so I just take it as is,"},{"Start":"03:31.465 ","End":"03:33.815","Text":"this is what I get."},{"Start":"03:33.815 ","End":"03:38.020","Text":"The next step, everything is linear should be easy to combine."},{"Start":"03:38.020 ","End":"03:43.700","Text":"Let\u0027s see, 2x plus 1x is 3x."},{"Start":"03:44.270 ","End":"03:51.570","Text":"Let\u0027s see, minus 4 plus 2 minus 4."},{"Start":"03:51.570 ","End":"03:55.180","Text":"I make that minus 6."},{"Start":"03:56.840 ","End":"04:04.590","Text":"It\u0027s still over x minus 2, x plus 2."},{"Start":"04:04.590 ","End":"04:07.500","Text":"This is less than 0."},{"Start":"04:07.500 ","End":"04:12.950","Text":"Look, the numerator also factorizes again because I can take 3 out."},{"Start":"04:12.950 ","End":"04:17.030","Text":"What I get is 3 times x minus"},{"Start":"04:17.030 ","End":"04:23.870","Text":"2 over x minus 2,"},{"Start":"04:23.870 ","End":"04:28.760","Text":"x plus 2, less than 0."},{"Start":"04:28.760 ","End":"04:30.635","Text":"Now here\u0027s a peculiar thing."},{"Start":"04:30.635 ","End":"04:33.625","Text":"We have an x minus 2 here."},{"Start":"04:33.625 ","End":"04:38.600","Text":"Just highlight it so really see the point I\u0027m trying to make."},{"Start":"04:38.600 ","End":"04:41.405","Text":"We have an x minus 2 and x minus 2 here."},{"Start":"04:41.405 ","End":"04:42.935","Text":"Now you might just say,"},{"Start":"04:42.935 ","End":"04:44.630","Text":"well, cancel them."},{"Start":"04:44.630 ","End":"04:46.565","Text":"It\u0027s not as simple as that."},{"Start":"04:46.565 ","End":"04:50.960","Text":"Because the thing is that if x=2,"},{"Start":"04:50.960 ","End":"04:57.150","Text":"this would make this thing 0 and we can\u0027t have 0 in the denominator."},{"Start":"04:57.840 ","End":"05:01.710","Text":"If I put x=2, I get 0 over 0,"},{"Start":"05:01.710 ","End":"05:03.165","Text":"which is not defined,"},{"Start":"05:03.165 ","End":"05:06.300","Text":"anything over 0 is not defined."},{"Start":"05:06.300 ","End":"05:08.930","Text":"Before I can cancel,"},{"Start":"05:08.930 ","End":"05:15.370","Text":"I have to write down that 2 is an illegal value."},{"Start":"05:15.520 ","End":"05:21.250","Text":"Let me just write that and I\u0027ll write it in a bold red, that x,"},{"Start":"05:21.250 ","End":"05:28.775","Text":"that\u0027s the wrong color, cannot be 2."},{"Start":"05:28.775 ","End":"05:32.585","Text":"If I remember that this is the case,"},{"Start":"05:32.585 ","End":"05:36.905","Text":"then I can cancel the x minus 2."},{"Start":"05:36.905 ","End":"05:44.959","Text":"This simplifies to 3 over x plus 2 less than 0."},{"Start":"05:44.959 ","End":"05:47.180","Text":"But if I didn\u0027t write this down,"},{"Start":"05:47.180 ","End":"05:49.820","Text":"in this expression certainly x could be 2."},{"Start":"05:49.820 ","End":"05:52.010","Text":"I\u0027d get 3/4."},{"Start":"05:52.010 ","End":"05:54.950","Text":"Nothing wrong with that."},{"Start":"05:54.950 ","End":"05:59.345","Text":"But I have to take this into account."},{"Start":"05:59.345 ","End":"06:05.720","Text":"Now we can use our usual method with the number line."},{"Start":"06:05.720 ","End":"06:08.210","Text":"Here\u0027s the number line and it\u0027s actually"},{"Start":"06:08.210 ","End":"06:11.210","Text":"2 interesting values that we should place on here."},{"Start":"06:11.210 ","End":"06:14.735","Text":"One of them is x=2."},{"Start":"06:14.735 ","End":"06:17.825","Text":"I\u0027ll just note that there\u0027s a 2 here somewhere."},{"Start":"06:17.825 ","End":"06:24.320","Text":"The other thing that\u0027s worth noting is when x plus 2 is 0,"},{"Start":"06:24.320 ","End":"06:28.250","Text":"that gives us x= -2."},{"Start":"06:28.250 ","End":"06:33.400","Text":"That\u0027s also a disallowed illegal value for x."},{"Start":"06:33.400 ","End":"06:35.590","Text":"Both the 2 and minus 2,"},{"Start":"06:35.590 ","End":"06:40.550","Text":"there are problems and I will indicate that the way I like to do it,"},{"Start":"06:40.550 ","End":"06:47.545","Text":"by making a little hole in the axis saying these 2 values are missing."},{"Start":"06:47.545 ","End":"06:53.530","Text":"To see it better else I like to put it in a little hollow circle ellipse whatever,"},{"Start":"06:53.530 ","End":"06:55.100","Text":"to show that these are missing."},{"Start":"06:55.100 ","End":"06:58.920","Text":"These are not legal values, throwing them out."},{"Start":"07:00.340 ","End":"07:02.855","Text":"I get 3 intervals,"},{"Start":"07:02.855 ","End":"07:05.090","Text":"this one, this one and this one."},{"Start":"07:05.090 ","End":"07:08.240","Text":"As before, we need to find out in"},{"Start":"07:08.240 ","End":"07:11.540","Text":"each interval whether this expression is positive or negative."},{"Start":"07:11.540 ","End":"07:15.380","Text":"It will be constant throughout each interval."},{"Start":"07:15.380 ","End":"07:19.070","Text":"The usual way we do that is with sample points."},{"Start":"07:19.070 ","End":"07:24.500","Text":"For example, here I would choose the sample point 0."},{"Start":"07:24.500 ","End":"07:26.930","Text":"Here I might choose 3,"},{"Start":"07:26.930 ","End":"07:29.705","Text":"and here I might choose minus 3."},{"Start":"07:29.705 ","End":"07:37.275","Text":"I want to see what happens in each of these sample points,"},{"Start":"07:37.275 ","End":"07:40.215","Text":"which then applies to the whole interval."},{"Start":"07:40.215 ","End":"07:44.530","Text":"If we take x= -3,"},{"Start":"07:45.160 ","End":"07:53.855","Text":"then we substitute and we get minus 3 plus 2 is minus 1."},{"Start":"07:53.855 ","End":"07:57.565","Text":"The whole thing comes out to be negative."},{"Start":"07:57.565 ","End":"07:59.690","Text":"If we\u0027re using the snake method,"},{"Start":"07:59.690 ","End":"08:03.275","Text":"I\u0027ll just put some negative value below this."},{"Start":"08:03.275 ","End":"08:12.380","Text":"At 0, we get 3 over 0 plus 2 is positive anyway,"},{"Start":"08:12.380 ","End":"08:15.600","Text":"positive over positive is positive."},{"Start":"08:16.180 ","End":"08:24.400","Text":"At 3, we get 3 over 5 is still positive."},{"Start":"08:26.990 ","End":"08:33.410","Text":"If we use the snake sketch as we call it,"},{"Start":"08:33.410 ","End":"08:34.760","Text":"it will look something like this."},{"Start":"08:34.760 ","End":"08:36.275","Text":"It goes through this point."},{"Start":"08:36.275 ","End":"08:39.140","Text":"It does not go through this point."},{"Start":"08:39.140 ","End":"08:41.150","Text":"It stops somewhere short,"},{"Start":"08:41.150 ","End":"08:48.260","Text":"continues, not exactly clear how it goes."},{"Start":"08:48.260 ","End":"08:53.520","Text":"It has to have a hold somewhere here."},{"Start":"08:54.840 ","End":"08:58.375","Text":"I\u0027ll indicate the gap with a circle."},{"Start":"08:58.375 ","End":"09:02.740","Text":"Then it continuous, not even clear from the same point or from a different point,"},{"Start":"09:02.740 ","End":"09:04.000","Text":"but it doesn\u0027t really matter."},{"Start":"09:04.000 ","End":"09:09.130","Text":"It continues and goes through here and so on,"},{"Start":"09:09.130 ","End":"09:11.980","Text":"it goes on forever."},{"Start":"09:11.980 ","End":"09:13.560","Text":"This is not a sketch of the graph,"},{"Start":"09:13.560 ","End":"09:16.090","Text":"it\u0027s just a schematic representation of where things are"},{"Start":"09:16.090 ","End":"09:20.980","Text":"positive and negative and undefined and 0,"},{"Start":"09:20.980 ","End":"09:24.155","Text":"which in this case it isn\u0027t anywhere."},{"Start":"09:24.155 ","End":"09:27.915","Text":"Then we take a look, what do we want to find?"},{"Start":"09:27.915 ","End":"09:33.085","Text":"Well we want to find places where it is less than 0, which means negative."},{"Start":"09:33.085 ","End":"09:36.500","Text":"I look where in this graph is it negative."},{"Start":"09:36.500 ","End":"09:40.525","Text":"Negative is, using the shading method,"},{"Start":"09:40.525 ","End":"09:45.780","Text":"here, not including this point."},{"Start":"09:45.980 ","End":"09:49.340","Text":"If we translate this in terms of x,"},{"Start":"09:49.340 ","End":"09:56.300","Text":"this just says that x is less than minus 2 and so on indefinitely."},{"Start":"09:56.300 ","End":"09:58.580","Text":"Some people write bigger than minus infinity,"},{"Start":"09:58.580 ","End":"10:01.235","Text":"but what\u0027s infinity anyway?"},{"Start":"10:01.235 ","End":"10:03.860","Text":"Just x less than minus 2."},{"Start":"10:03.860 ","End":"10:07.745","Text":"All this part is either positive or undefined and so is this."},{"Start":"10:07.745 ","End":"10:10.475","Text":"That\u0027s what we\u0027re left with and that is the answer."},{"Start":"10:10.475 ","End":"10:14.610","Text":"We\u0027ll highlight it and then move on to part B."}],"Thumbnail":null,"ID":5079},{"Watched":false,"Name":"Exercise 7 part b","Duration":"10m 50s","ChapterTopicVideoID":5078,"CourseChapterTopicPlaylistID":3877,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.680","Text":"Let\u0027s find part B."},{"Start":"00:01.680 ","End":"00:08.070","Text":"Here it is and what we have here is pretty much like we had before."},{"Start":"00:08.070 ","End":"00:09.600","Text":"We have an inequality,"},{"Start":"00:09.600 ","End":"00:14.460","Text":"but we want to bring everything to the left and compare 0."},{"Start":"00:14.460 ","End":"00:17.688","Text":"We\u0027ll bring this thing over,"},{"Start":"00:17.688 ","End":"00:21.810","Text":"but there\u0027s another thing I want to do is also to factorize this."},{"Start":"00:21.810 ","End":"00:23.805","Text":"If we factorize this,"},{"Start":"00:23.805 ","End":"00:24.990","Text":"then we might be able to get"},{"Start":"00:24.990 ","End":"00:28.320","Text":"this lower common denominator than just"},{"Start":"00:28.320 ","End":"00:32.025","Text":"multiplying them out and also we\u0027ll have to factorize it anyway."},{"Start":"00:32.025 ","End":"00:36.360","Text":"Let me just tell you what the factorization of that is."},{"Start":"00:36.360 ","End":"00:38.850","Text":"I don\u0027t want to spend time on that."},{"Start":"00:38.850 ","End":"00:43.425","Text":"X squared plus x minus 20"},{"Start":"00:43.425 ","End":"00:49.885","Text":"is equal to x plus 5 times x minus 4."},{"Start":"00:49.885 ","End":"00:52.460","Text":"Let\u0027s just check that this is indeed true."},{"Start":"00:52.460 ","End":"00:55.250","Text":"We did multiply out the right side,"},{"Start":"00:55.250 ","End":"01:01.810","Text":"5x minus 4x is plus x, so the middle term\u0027s right."},{"Start":"01:01.810 ","End":"01:07.775","Text":"The first one is x times x is x^2 of course and the last one 5 times minus 4 is minus 20."},{"Start":"01:07.775 ","End":"01:10.070","Text":"This is the correct factorization."},{"Start":"01:10.070 ","End":"01:11.525","Text":"If I go back here,"},{"Start":"01:11.525 ","End":"01:20.090","Text":"what I have is x over x plus 5 minus 1 over, instead of this,"},{"Start":"01:20.090 ","End":"01:28.610","Text":"I\u0027ll put x plus 5 times x minus 4 and this comes over to the left,"},{"Start":"01:28.610 ","End":"01:36.590","Text":"so we make it minus 1 over x minus 4 and all this, bigger than 0."},{"Start":"01:36.590 ","End":"01:39.365","Text":"I forgot a bracket here."},{"Start":"01:39.365 ","End":"01:43.745","Text":"Now it\u0027s pretty clear that the common denominator is just x plus 5,"},{"Start":"01:43.745 ","End":"01:47.345","Text":"x minus 4 because it already takes care of this and of this,"},{"Start":"01:47.345 ","End":"01:50.730","Text":"so here\u0027s a dividing line."},{"Start":"01:51.260 ","End":"01:57.115","Text":"Here we write x plus 5, x minus 4."},{"Start":"01:57.115 ","End":"02:00.410","Text":"We want to put everything, all these fractions over"},{"Start":"02:00.410 ","End":"02:05.615","Text":"this by multiplying top and bottom by the appropriate."},{"Start":"02:05.615 ","End":"02:08.525","Text":"Now this one doesn\u0027t have to be multiplied by anything."},{"Start":"02:08.525 ","End":"02:11.735","Text":"I\u0027ll just go with the middle 1 first, that\u0027s minus 1."},{"Start":"02:11.735 ","End":"02:13.475","Text":"Let\u0027s go for the first term."},{"Start":"02:13.475 ","End":"02:16.790","Text":"I have to multiply this by the missing factor,"},{"Start":"02:16.790 ","End":"02:18.110","Text":"x minus 4,"},{"Start":"02:18.110 ","End":"02:21.950","Text":"so here I have x, x minus 4."},{"Start":"02:21.950 ","End":"02:24.260","Text":"That would be this over this,"},{"Start":"02:24.260 ","End":"02:26.990","Text":"so we haven\u0027t changed the fraction and here,"},{"Start":"02:26.990 ","End":"02:30.380","Text":"multiply top and bottom by the x plus 5."},{"Start":"02:30.380 ","End":"02:36.195","Text":"We get minus, I\u0027ll leave the 1 here just so we see where it came from,"},{"Start":"02:36.195 ","End":"02:41.210","Text":"1 times x plus 5 is bigger than 0."},{"Start":"02:41.210 ","End":"02:47.360","Text":"Haven\u0027t changed both sides and I haven\u0027t multiplied or anything."},{"Start":"02:47.360 ","End":"02:49.370","Text":"I just put everything over a common denominator,"},{"Start":"02:49.370 ","End":"02:50.915","Text":"like we would with numbers."},{"Start":"02:50.915 ","End":"02:54.410","Text":"Let\u0027s simplify the numerator here."},{"Start":"02:54.410 ","End":"02:57.245","Text":"I think we can do this in our heads."},{"Start":"02:57.245 ","End":"03:02.495","Text":"What we have here is x^2 from here."},{"Start":"03:02.495 ","End":"03:04.685","Text":"Then we have minus 4x,"},{"Start":"03:04.685 ","End":"03:06.815","Text":"but also minus x,"},{"Start":"03:06.815 ","End":"03:12.620","Text":"so minus 5x and that takes care of this term."},{"Start":"03:12.620 ","End":"03:19.835","Text":"We also have a minus 1 and a minus 5 so altogether a minus"},{"Start":"03:19.835 ","End":"03:29.690","Text":"6 and this is over the same thing still x plus 5,"},{"Start":"03:29.690 ","End":"03:34.980","Text":"x minus 4 bigger than 0."},{"Start":"03:35.420 ","End":"03:38.000","Text":"1 thing is clear,"},{"Start":"03:38.000 ","End":"03:46.950","Text":"that the illegal values already we can see those that would be minus 5 and 4,"},{"Start":"03:46.950 ","End":"03:49.320","Text":"so x cannot be minus 5 or 4,"},{"Start":"03:49.320 ","End":"03:51.305","Text":"that would make the denominator 0."},{"Start":"03:51.305 ","End":"03:55.730","Text":"We\u0027d also like to factorize the numerator to see where the expression is 0."},{"Start":"03:55.730 ","End":"03:58.490","Text":"Let me also do that one at the side,"},{"Start":"03:58.490 ","End":"04:01.115","Text":"not at the side. I\u0027ll do it here."},{"Start":"04:01.115 ","End":"04:04.295","Text":"That gives me x minus 6,"},{"Start":"04:04.295 ","End":"04:06.750","Text":"x plus 1."},{"Start":"04:06.750 ","End":"04:09.025","Text":"That\u0027s the factorization."},{"Start":"04:09.025 ","End":"04:13.490","Text":"Before I continue, I\u0027ll just check that this really is the case,"},{"Start":"04:13.490 ","End":"04:16.040","Text":"x times x is x^2,"},{"Start":"04:16.040 ","End":"04:20.340","Text":"minus 6x plus 1x is minus 5x,"},{"Start":"04:20.340 ","End":"04:23.730","Text":"and minus 6 times 1 is minus 6, so that\u0027s fine."},{"Start":"04:23.730 ","End":"04:27.600","Text":"Over the same x plus 5,"},{"Start":"04:27.600 ","End":"04:35.180","Text":"x minus 4 is bigger than 0."},{"Start":"04:35.180 ","End":"04:37.970","Text":"We already found what makes the denominator 0,"},{"Start":"04:37.970 ","End":"04:39.170","Text":"it\u0027s these 2 values."},{"Start":"04:39.170 ","End":"04:41.635","Text":"What makes the numerator 0?"},{"Start":"04:41.635 ","End":"04:46.303","Text":"That would be 6 for this factor and minus 1 for this factor."},{"Start":"04:46.303 ","End":"04:51.710","Text":"I\u0027ll put all these 4 numbers on the number line."},{"Start":"04:51.710 ","End":"04:54.350","Text":"Here\u0027s a number line and let\u0027s put these,"},{"Start":"04:54.350 ","End":"04:57.908","Text":"but in the right order from decreasing to increasing."},{"Start":"04:57.908 ","End":"05:02.435","Text":"The one on the most left would be minus 5."},{"Start":"05:02.435 ","End":"05:08.080","Text":"Then we\u0027d have a minus 1 and then we have a 4 and then we have a 6."},{"Start":"05:08.080 ","End":"05:09.950","Text":"The scale is not important at all,"},{"Start":"05:09.950 ","End":"05:18.805","Text":"but the order is very important and the 2 bad ones are the minus 5 and the 4."},{"Start":"05:18.805 ","End":"05:24.073","Text":"To show that they\u0027re bad or excluded I\u0027ll just erase a little bit,"},{"Start":"05:24.073 ","End":"05:28.670","Text":"make a little hole in the axis for minus 5 and for 4,"},{"Start":"05:28.670 ","End":"05:31.265","Text":"make another little hole here."},{"Start":"05:31.265 ","End":"05:36.860","Text":"We\u0027ll even make it more prominent by putting a little hollow circle here."},{"Start":"05:36.860 ","End":"05:38.870","Text":"These are the values that have been excluded,"},{"Start":"05:38.870 ","End":"05:41.105","Text":"illegal, not in the domain."},{"Start":"05:41.105 ","End":"05:43.010","Text":"These 2, on the other hand,"},{"Start":"05:43.010 ","End":"05:47.090","Text":"are perfectly valid and they\u0027re what makes the expression 0."},{"Start":"05:47.090 ","End":"05:54.120","Text":"I\u0027ll indicate that by putting a solid circle here and here."},{"Start":"05:54.520 ","End":"05:58.760","Text":"Now I want to know in these,"},{"Start":"05:58.760 ","End":"06:00.107","Text":"how many, 1, 2, 3,"},{"Start":"06:00.107 ","End":"06:02.320","Text":"4, 5 intervals that remain,"},{"Start":"06:02.320 ","End":"06:04.956","Text":"whether the expression is positive or negative in each."},{"Start":"06:04.956 ","End":"06:07.495","Text":"It won\u0027t change. In each 1, it\u0027ll be constant."},{"Start":"06:07.495 ","End":"06:11.469","Text":"Let\u0027s see. Use the method of sample points."},{"Start":"06:11.469 ","End":"06:17.590","Text":"For example, I could take this sample point of minus 6."},{"Start":"06:17.590 ","End":"06:18.640","Text":"Here I could take,"},{"Start":"06:18.640 ","End":"06:21.400","Text":"let\u0027s say, minus 3."},{"Start":"06:21.400 ","End":"06:26.334","Text":"Here I can take 0, which is my favorite always if it\u0027s available."},{"Start":"06:26.334 ","End":"06:30.660","Text":"Here I\u0027ll take 5 and here I\u0027ll take 7."},{"Start":"06:30.660 ","End":"06:34.225","Text":"What I want to do is substitute in here for each of them,"},{"Start":"06:34.225 ","End":"06:37.311","Text":"but I only care about the sign."},{"Start":"06:37.311 ","End":"06:40.595","Text":"If I take the minus 6,"},{"Start":"06:40.595 ","End":"06:45.493","Text":"what I basically get is this comes out negative,"},{"Start":"06:45.493 ","End":"06:47.194","Text":"minus x plus 1 is negative,"},{"Start":"06:47.194 ","End":"06:52.969","Text":"they\u0027re all negative and I get a minus times minus over minus minus."},{"Start":"06:52.969 ","End":"06:58.280","Text":"Therefore, it\u0027s plus and I"},{"Start":"06:58.280 ","End":"07:04.310","Text":"indicate it as plus by writing a dot above it."},{"Start":"07:04.310 ","End":"07:06.199","Text":"Above the number line is plus,"},{"Start":"07:06.199 ","End":"07:08.525","Text":"below the number line is minus."},{"Start":"07:08.525 ","End":"07:11.450","Text":"Then for minus 3,"},{"Start":"07:11.450 ","End":"07:13.100","Text":"if I do that,"},{"Start":"07:13.100 ","End":"07:19.730","Text":"here I get negative negative,"},{"Start":"07:19.730 ","End":"07:21.039","Text":"but this one is already positive,"},{"Start":"07:21.039 ","End":"07:22.790","Text":"minus 3 plus 5 is positive,"},{"Start":"07:22.790 ","End":"07:29.345","Text":"so I have the minus minus plus minus."},{"Start":"07:29.345 ","End":"07:36.875","Text":"An odd number of minuses so this one is going to be minus."},{"Start":"07:36.875 ","End":"07:40.460","Text":"Doesn\u0027t really matter where as long it\u0027s below the axis."},{"Start":"07:40.460 ","End":"07:44.290","Text":"At 0, we get minus 6,"},{"Start":"07:44.290 ","End":"07:47.525","Text":"1, 5, minus 4,"},{"Start":"07:47.525 ","End":"07:50.140","Text":"which means minus, plus, plus, minus,"},{"Start":"07:50.140 ","End":"07:57.650","Text":"even number of minuses and so at 0, it\u0027s plus."},{"Start":"07:57.650 ","End":"08:00.350","Text":"Above, at 5,"},{"Start":"08:00.350 ","End":"08:03.905","Text":"we have this is negative,"},{"Start":"08:03.905 ","End":"08:06.250","Text":"positive, positive, and positive."},{"Start":"08:08.550 ","End":"08:12.595","Text":"The only 1 that\u0027s negative is the first 1."},{"Start":"08:12.595 ","End":"08:15.995","Text":"Minus, plus, plus, plus,"},{"Start":"08:15.995 ","End":"08:22.135","Text":"which makes here the value negative, say here."},{"Start":"08:22.135 ","End":"08:28.900","Text":"At 7, all of these are going to be positive if you check,"},{"Start":"08:28.900 ","End":"08:31.000","Text":"7 minus 6 is positive, they\u0027re all positive."},{"Start":"08:31.000 ","End":"08:34.810","Text":"Plus, plus, plus,"},{"Start":"08:34.810 ","End":"08:44.125","Text":"plus and so somewhere above the line and now we can sketch the snake as we call it,"},{"Start":"08:44.125 ","End":"08:47.290","Text":"which goes through here,"},{"Start":"08:47.290 ","End":"08:51.450","Text":"but it cannot go through the illegal point,"},{"Start":"08:51.450 ","End":"08:52.810","Text":"the excluded point,"},{"Start":"08:52.810 ","End":"08:56.590","Text":"continues somewhere through here."},{"Start":"08:56.590 ","End":"09:00.700","Text":"Definitely goes through here where it\u0027s 0."},{"Start":"09:00.700 ","End":"09:05.980","Text":"Again, gets stuck, stops somewhere,"},{"Start":"09:05.980 ","End":"09:09.895","Text":"but then resumes after the 4,"},{"Start":"09:09.895 ","End":"09:17.420","Text":"6 is fine, and it continues indefinitely in both directions."},{"Start":"09:17.910 ","End":"09:20.770","Text":"Now we have to see what we\u0027re interested in,"},{"Start":"09:20.770 ","End":"09:22.270","Text":"whether we\u0027re interested in positive,"},{"Start":"09:22.270 ","End":"09:23.950","Text":"negative, 0, or what."},{"Start":"09:23.950 ","End":"09:27.320","Text":"The bigger than 0 means we\u0027re interested in the positive,"},{"Start":"09:27.320 ","End":"09:32.095","Text":"which means above the axis and where is it above the axis?"},{"Start":"09:32.095 ","End":"09:34.950","Text":"I\u0027ll shade the bits that are above."},{"Start":"09:34.950 ","End":"09:37.084","Text":"This bit here is above,"},{"Start":"09:37.084 ","End":"09:40.865","Text":"this bit here is above, this bit here."},{"Start":"09:40.865 ","End":"09:45.395","Text":"Let\u0027s see what this means in terms of x and let\u0027s also be precise."},{"Start":"09:45.395 ","End":"09:48.020","Text":"This is not included."},{"Start":"09:48.020 ","End":"09:51.890","Text":"For 1 thing, it\u0027s not defined here."},{"Start":"09:51.890 ","End":"09:53.570","Text":"This is also not included."},{"Start":"09:53.570 ","End":"09:55.970","Text":"It\u0027s perfectly defined, but we\u0027re not collecting 0s,"},{"Start":"09:55.970 ","End":"09:57.920","Text":"we\u0027re only collecting positives."},{"Start":"09:57.920 ","End":"10:05.760","Text":"Basically, none of these intervals includes the endpoint and we have 3 of them."},{"Start":"10:06.560 ","End":"10:13.165","Text":"This part is x less than minus 5."},{"Start":"10:13.165 ","End":"10:18.395","Text":"This hump here is a snake or it\u0027s a camel, I don\u0027t know."},{"Start":"10:18.395 ","End":"10:20.015","Text":"It\u0027s got a hump anyway,"},{"Start":"10:20.015 ","End":"10:23.490","Text":"it\u0027s between minus 1 and 4."},{"Start":"10:25.220 ","End":"10:30.734","Text":"Again, not including the endpoints and then from this onwards,"},{"Start":"10:30.734 ","End":"10:35.930","Text":"which I\u0027ll write as x bigger than 6 or you could write 6 less than x."},{"Start":"10:35.930 ","End":"10:39.331","Text":"If you put the word or in-between,"},{"Start":"10:39.331 ","End":"10:43.125","Text":"then this is our final answer."},{"Start":"10:43.125 ","End":"10:46.040","Text":"I like to highlight it and say, yes,"},{"Start":"10:46.040 ","End":"10:50.130","Text":"this is it, we\u0027re done with part b and that\u0027s it."}],"Thumbnail":null,"ID":5080},{"Watched":false,"Name":"Exercise 8","Duration":"32m 2s","ChapterTopicVideoID":5079,"CourseChapterTopicPlaylistID":3877,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.720","Text":"Here, we have another pair of inequalities to solve,"},{"Start":"00:03.720 ","End":"00:09.104","Text":"which will eventually lead to a rational expression being compared to 0."},{"Start":"00:09.104 ","End":"00:11.804","Text":"Let\u0027s start with the first."},{"Start":"00:11.804 ","End":"00:16.140","Text":"We want to bring everything to the left and compare to 0,"},{"Start":"00:16.140 ","End":"00:19.319","Text":"but we\u0027d also like to do some factorization."},{"Start":"00:19.319 ","End":"00:24.360","Text":"Notice, that this x^2 minus 6x plus"},{"Start":"00:24.360 ","End":"00:30.645","Text":"9 is one of those special binomial products expansions."},{"Start":"00:30.645 ","End":"00:37.680","Text":"It\u0027s modeled on the a^2 minus 2ab plus b^2 =(a-b)^2."},{"Start":"00:37.680 ","End":"00:42.080","Text":"A formula is usually given from right to left,"},{"Start":"00:42.080 ","End":"00:43.504","Text":"but it works both ways."},{"Start":"00:43.504 ","End":"00:45.274","Text":"If we apply it here,"},{"Start":"00:45.274 ","End":"00:49.200","Text":"clearly this means that this is x minus 3^2."},{"Start":"00:49.300 ","End":"00:54.300","Text":"Also, the 3x minus 9,"},{"Start":"00:54.300 ","End":"00:56.460","Text":"I\u0027m doing side exercises."},{"Start":"00:56.460 ","End":"01:00.675","Text":"This is clearly 3 times x minus 3."},{"Start":"01:00.675 ","End":"01:01.830","Text":"Notice x minus 3,"},{"Start":"01:01.830 ","End":"01:05.439","Text":"x minus 3, so I\u0027m onto something here."},{"Start":"01:05.570 ","End":"01:08.890","Text":"Let\u0027s put everything on the left."},{"Start":"01:08.890 ","End":"01:15.050","Text":"27 over x minus 3^2"},{"Start":"01:15.050 ","End":"01:24.980","Text":"plus 10x over 3 times x minus 3,"},{"Start":"01:24.980 ","End":"01:31.920","Text":"minus 3 is less than 0."},{"Start":"01:31.920 ","End":"01:35.999","Text":"Now, we want the common denominator for the stuff on the left."},{"Start":"01:35.999 ","End":"01:38.239","Text":"Once we have the common denominator,"},{"Start":"01:38.239 ","End":"01:41.974","Text":"we can write each one in terms of that common denominator."},{"Start":"01:41.974 ","End":"01:45.050","Text":"Then, using the rules of fractions with the same denominator,"},{"Start":"01:45.050 ","End":"01:48.390","Text":"we can add or subtract the numerators."},{"Start":"01:48.410 ","End":"01:53.570","Text":"The denominator that suits all of them will be,"},{"Start":"01:53.570 ","End":"01:57.035","Text":"x minus 3^2, at least we need that."},{"Start":"01:57.035 ","End":"01:59.150","Text":"But we also need a 3 for this,"},{"Start":"01:59.150 ","End":"02:02.790","Text":"so this should cover everything."},{"Start":"02:03.110 ","End":"02:09.260","Text":"What we do is, we multiply each numerator by the missing factors on the denominator."},{"Start":"02:09.260 ","End":"02:12.709","Text":"From here into here,"},{"Start":"02:12.709 ","End":"02:14.465","Text":"we\u0027re missing a factor of 3,"},{"Start":"02:14.465 ","End":"02:17.514","Text":"so it\u0027s 3 times 27."},{"Start":"02:17.514 ","End":"02:22.590","Text":"For the 10x,"},{"Start":"02:22.590 ","End":"02:24.709","Text":"from here to here, we\u0027re missing an x minus"},{"Start":"02:24.709 ","End":"02:27.079","Text":"3 because here it\u0027s squared and here it isn\u0027t."},{"Start":"02:27.079 ","End":"02:30.024","Text":"Here, we put x minus 3."},{"Start":"02:30.024 ","End":"02:32.099","Text":"Then, the 3,"},{"Start":"02:32.099 ","End":"02:34.664","Text":"which is 3 over 1 is missing everything."},{"Start":"02:34.664 ","End":"02:38.325","Text":"I have to put minus 3 times the whole thing."},{"Start":"02:38.325 ","End":"02:40.590","Text":"There\u0027s another 3, that\u0027s this 3,"},{"Start":"02:40.590 ","End":"02:45.060","Text":"then there\u0027s this 3 times x minus 3^2."},{"Start":"02:45.060 ","End":"02:48.580","Text":"It\u0027s still less than 0."},{"Start":"02:52.850 ","End":"02:57.620","Text":"I copied the denominator using copy-paste to speed things up."},{"Start":"02:57.620 ","End":"03:04.874","Text":"3 times 27 is 81.10x times x is 10x^2."},{"Start":"03:04.874 ","End":"03:13.110","Text":"Then we have minus 30x minus 9,"},{"Start":"03:13.110 ","End":"03:16.165","Text":"that\u0027s 3 times 3, and then x minus 3^2."},{"Start":"03:16.165 ","End":"03:18.004","Text":"We\u0027re going to use this formula again,"},{"Start":"03:18.004 ","End":"03:19.879","Text":"this time, in the other direction."},{"Start":"03:19.879 ","End":"03:26.605","Text":"It\u0027s x^2 minus twice x times 3 is 6x plus 3^2 is 9,"},{"Start":"03:26.605 ","End":"03:29.770","Text":"still less than 0."},{"Start":"03:30.850 ","End":"03:35.074","Text":"I\u0027m going to expand and collect like terms all in one go."},{"Start":"03:35.074 ","End":"03:40.900","Text":"Let\u0027s look for x^2. We have x^2 here and here."},{"Start":"03:40.900 ","End":"03:43.590","Text":"I think, I\u0027ll use color."},{"Start":"03:43.590 ","End":"03:48.360","Text":"The x^2s and then xs would be here and here,"},{"Start":"03:48.360 ","End":"03:53.620","Text":"and just numbers here and this with its product."},{"Start":"03:53.900 ","End":"03:57.899","Text":"Let\u0027s see what we get. The yellow ones first."},{"Start":"03:57.899 ","End":"04:02.535","Text":"10x^2 minus 9x^2 is x^2."},{"Start":"04:02.535 ","End":"04:08.265","Text":"Then xs, we have minus 30x plus 54x,"},{"Start":"04:08.265 ","End":"04:10.589","Text":"9 times 6 is 54."},{"Start":"04:10.589 ","End":"04:16.200","Text":"Minus 30 plus 54 is 24x."},{"Start":"04:16.200 ","End":"04:17.640","Text":"Let\u0027s see in numbers."},{"Start":"04:17.640 ","End":"04:22.590","Text":"I have 81, and I have minus 81,"},{"Start":"04:22.590 ","End":"04:26.080","Text":"so that\u0027s all that\u0027s left."},{"Start":"04:26.240 ","End":"04:28.620","Text":"I just shortened the line there,"},{"Start":"04:28.620 ","End":"04:33.570","Text":"it\u0027s wasteful, still less than 0."},{"Start":"04:33.570 ","End":"04:39.185","Text":"One final simplification is to take the x out."},{"Start":"04:39.185 ","End":"04:44.509","Text":"We have x times x plus"},{"Start":"04:44.509 ","End":"04:54.035","Text":"24 over 3 times x minus 3^2,"},{"Start":"04:54.035 ","End":"04:57.965","Text":"all this is less than 0."},{"Start":"04:57.965 ","End":"05:02.239","Text":"Now, we have our rational expression completely factored,"},{"Start":"05:02.239 ","End":"05:06.905","Text":"and we want to know where each factor is 0;"},{"Start":"05:06.905 ","End":"05:09.830","Text":"x is 0 when x is 0,"},{"Start":"05:09.830 ","End":"05:11.450","Text":"sort of obvious,"},{"Start":"05:11.450 ","End":"05:16.025","Text":"x plus 24 is 0 when x is minus 24,"},{"Start":"05:16.025 ","End":"05:18.560","Text":"3 is never 0,"},{"Start":"05:18.560 ","End":"05:20.570","Text":"it\u0027s always 3,"},{"Start":"05:20.570 ","End":"05:25.259","Text":"x minus 3^2 is 0 when x minus 3 is 0,"},{"Start":"05:25.259 ","End":"05:28.125","Text":"in other words, when x is 3."},{"Start":"05:28.125 ","End":"05:30.860","Text":"These are the three important values we\u0027re going to"},{"Start":"05:30.860 ","End":"05:34.249","Text":"place on the number line when we do the sketch."},{"Start":"05:34.249 ","End":"05:36.529","Text":"Here\u0027s a number line."},{"Start":"05:36.529 ","End":"05:39.890","Text":"I want to put these three in the correct order."},{"Start":"05:39.890 ","End":"05:45.570","Text":"Leftmost would be minus 24,"},{"Start":"05:45.890 ","End":"05:50.004","Text":"and then we would have the 0,"},{"Start":"05:50.004 ","End":"05:52.885","Text":"and then we have the 3,"},{"Start":"05:52.885 ","End":"05:54.789","Text":"not necessarily to scale,"},{"Start":"05:54.789 ","End":"05:57.499","Text":"but must be in the right order."},{"Start":"05:59.550 ","End":"06:03.610","Text":"There is a difference between these"},{"Start":"06:03.610 ","End":"06:07.524","Text":"two from the numerator and this one from the denominator."},{"Start":"06:07.524 ","End":"06:13.145","Text":"The 0 and the minus 24 are places where the expression is 0."},{"Start":"06:13.145 ","End":"06:19.735","Text":"I\u0027ll indicate that by putting a solid dot on the number line to say,"},{"Start":"06:19.735 ","End":"06:22.094","Text":"the value of the expression here is 0,"},{"Start":"06:22.094 ","End":"06:24.455","Text":"and here it\u0027s 0."},{"Start":"06:24.455 ","End":"06:25.950","Text":"But, with the 3,"},{"Start":"06:25.950 ","End":"06:27.345","Text":"it\u0027s a different story."},{"Start":"06:27.345 ","End":"06:30.095","Text":"That\u0027s the place where a denominator is 0,"},{"Start":"06:30.095 ","End":"06:32.419","Text":"which means that this expression is undefined,"},{"Start":"06:32.419 ","End":"06:35.510","Text":"meaningless, 3 is illegal, whatever."},{"Start":"06:35.510 ","End":"06:40.060","Text":"I like to make a little hole in the number line,"},{"Start":"06:40.060 ","End":"06:43.949","Text":"just take out a little bite."},{"Start":"06:43.949 ","End":"06:45.570","Text":"My eraser is not working."},{"Start":"06:45.570 ","End":"06:48.885","Text":"There it is. Maybe to emphasize that,"},{"Start":"06:48.885 ","End":"06:52.800","Text":"I\u0027ll put here a hollow circle ellipse,"},{"Start":"06:52.800 ","End":"06:55.695","Text":"whatever, to say that it\u0027s missing."},{"Start":"06:55.695 ","End":"06:59.530","Text":"Always know you cannot put 3 in, illegal."},{"Start":"06:59.650 ","End":"07:05.929","Text":"After this, we see that the line is split up into 4 separate intervals here,"},{"Start":"07:05.929 ","End":"07:07.894","Text":"here, here, and here."},{"Start":"07:07.894 ","End":"07:10.610","Text":"Each one of those pieces we know,"},{"Start":"07:10.610 ","End":"07:13.370","Text":"it\u0027s positive or negative."},{"Start":"07:13.370 ","End":"07:16.234","Text":"It doesn\u0027t change in the middle of one of these intervals."},{"Start":"07:16.234 ","End":"07:19.069","Text":"How do we find out, say,"},{"Start":"07:19.069 ","End":"07:20.420","Text":"in this interval,"},{"Start":"07:20.420 ","End":"07:22.309","Text":"if it\u0027s positive or negative always?"},{"Start":"07:22.309 ","End":"07:24.620","Text":"If it\u0027s always the same, we can just substitute the value,"},{"Start":"07:24.620 ","End":"07:26.560","Text":"that\u0027s the sample value."},{"Start":"07:26.560 ","End":"07:29.624","Text":"I like to choose whole numbers."},{"Start":"07:29.624 ","End":"07:35.345","Text":"Let\u0027s say I\u0027ll choose a sample value of minus 25 for this interval,"},{"Start":"07:35.345 ","End":"07:37.650","Text":"which goes on forever."},{"Start":"07:37.650 ","End":"07:41.190","Text":"Between here and here,"},{"Start":"07:41.190 ","End":"07:43.214","Text":"I\u0027ll go for minus 1,"},{"Start":"07:43.214 ","End":"07:45.740","Text":"probably smaller numbers are easier to compute with."},{"Start":"07:45.740 ","End":"07:48.574","Text":"Between 0 and 3, I\u0027ll choose 1,"},{"Start":"07:48.574 ","End":"07:52.175","Text":"and to the right of 3, I\u0027ll choose 4."},{"Start":"07:52.175 ","End":"07:54.460","Text":"Each of these, I want to substitute here,"},{"Start":"07:54.460 ","End":"07:58.210","Text":"but only to find out the sign positive or negative."},{"Start":"07:58.210 ","End":"08:02.049","Text":"I don\u0027t care about the actual value. Let\u0027s do them in order."},{"Start":"08:02.049 ","End":"08:05.604","Text":"If I put minus 25 in here,"},{"Start":"08:05.604 ","End":"08:09.229","Text":"that\u0027s negative, minus 25 or 24 is negative."},{"Start":"08:09.229 ","End":"08:13.595","Text":"Actually, the denominator is always going to be positive."},{"Start":"08:13.595 ","End":"08:17.414","Text":"In effect, I can ignore the denominator."},{"Start":"08:17.414 ","End":"08:19.350","Text":"I\u0027m not going put in 3, that\u0027s for sure."},{"Start":"08:19.350 ","End":"08:20.970","Text":"But as for everything else,"},{"Start":"08:20.970 ","End":"08:22.840","Text":"we\u0027ll make this positive because it\u0027s squared,"},{"Start":"08:22.840 ","End":"08:24.250","Text":"and times 3 will still be positive."},{"Start":"08:24.250 ","End":"08:26.860","Text":"I really just have to look at the numerator,"},{"Start":"08:26.860 ","End":"08:29.700","Text":"that\u0027ll make work easier."},{"Start":"08:29.700 ","End":"08:35.959","Text":"We said with the minus 25, this is negative."},{"Start":"08:35.959 ","End":"08:38.410","Text":"Even plus 24 is still negative,"},{"Start":"08:38.410 ","End":"08:41.149","Text":"so negative, negative is positive."},{"Start":"08:41.149 ","End":"08:47.060","Text":"I\u0027ll put a dot above the line,"},{"Start":"08:47.060 ","End":"08:50.330","Text":"which means positive at 24 at 0."},{"Start":"08:50.330 ","End":"08:51.855","Text":"Let\u0027s see it."},{"Start":"08:51.855 ","End":"08:53.810","Text":"I\u0027m just looking at the numerator."},{"Start":"08:53.810 ","End":"08:55.824","Text":"Now, remember, I don\u0027t need that."},{"Start":"08:55.824 ","End":"08:57.729","Text":"Minus 1 is negative,"},{"Start":"08:57.729 ","End":"09:00.409","Text":"minus 1 plus 24 is positive."},{"Start":"09:00.409 ","End":"09:02.750","Text":"Negative times positive is negative, so here,"},{"Start":"09:02.750 ","End":"09:09.330","Text":"I put a dot below the line indicating negative."},{"Start":"09:09.330 ","End":"09:13.200","Text":"At x=1, 1 is positive,"},{"Start":"09:13.200 ","End":"09:21.870","Text":"1 plus 24 is positive,"},{"Start":"09:21.870 ","End":"09:26.160","Text":"and so that\u0027s positive."},{"Start":"09:26.160 ","End":"09:28.049","Text":"When x is 4 also,"},{"Start":"09:28.049 ","End":"09:29.550","Text":"positive times positive,"},{"Start":"09:29.550 ","End":"09:33.260","Text":"so we get two positives consecutive."},{"Start":"09:33.260 ","End":"09:37.024","Text":"I\u0027m glad when that happens because people are always saying,"},{"Start":"09:37.024 ","End":"09:38.899","Text":"don\u0027t they just alternate signs,"},{"Start":"09:38.899 ","End":"09:40.584","Text":"positive, negative, positive, negative?"},{"Start":"09:40.584 ","End":"09:41.744","Text":"Well, usually,"},{"Start":"09:41.744 ","End":"09:44.719","Text":"but not always, so you can\u0027t count on it."},{"Start":"09:44.719 ","End":"09:50.875","Text":"Then we draw schematically what we call the snake,"},{"Start":"09:50.875 ","End":"09:55.245","Text":"which reminds some people of a snake,"},{"Start":"09:55.245 ","End":"09:59.569","Text":"and I can see why, here and here."},{"Start":"09:59.569 ","End":"10:06.515","Text":"However, we do not go through the place where it\u0027s undefined."},{"Start":"10:06.515 ","End":"10:10.099","Text":"It\u0027s not even clear where it continues from."},{"Start":"10:10.099 ","End":"10:11.734","Text":"It doesn\u0027t matter, it\u0027s just a scheme."},{"Start":"10:11.734 ","End":"10:13.220","Text":"We stop somewhere short,"},{"Start":"10:13.220 ","End":"10:17.389","Text":"or even getting very closer, and so on."},{"Start":"10:17.389 ","End":"10:21.384","Text":"Of course, the curve continues indefinitely onward,"},{"Start":"10:21.384 ","End":"10:26.164","Text":"it\u0027s just that we don\u0027t have an infinite amount of space to draw it all,"},{"Start":"10:26.164 ","End":"10:30.085","Text":"but you get my drift."},{"Start":"10:30.085 ","End":"10:33.540","Text":"I have to see what I\u0027m collecting. I call it, collecting."},{"Start":"10:33.540 ","End":"10:34.650","Text":"Which things are good for me?"},{"Start":"10:34.650 ","End":"10:36.179","Text":"I want negatives."},{"Start":"10:36.179 ","End":"10:38.130","Text":"Less than 0 means negatives."},{"Start":"10:38.130 ","End":"10:40.729","Text":"I don\u0027t want 0s, I don\u0027t want positives,"},{"Start":"10:40.729 ","End":"10:42.155","Text":"I just want negatives,"},{"Start":"10:42.155 ","End":"10:44.465","Text":"and I also, never take undefines."},{"Start":"10:44.465 ","End":"10:46.849","Text":"You see, here is positive 0,"},{"Start":"10:46.849 ","End":"10:50.690","Text":"negative 0, positive undefined, and positive."},{"Start":"10:50.690 ","End":"10:52.804","Text":"I just want the positives."},{"Start":"10:52.804 ","End":"10:55.984","Text":"Let me just highlight the positive bits."},{"Start":"10:55.984 ","End":"10:58.444","Text":"This bit is positive."},{"Start":"10:58.444 ","End":"11:01.220","Text":"I missed here a bit, but you understand."},{"Start":"11:01.220 ","End":"11:02.690","Text":"This bit is positive,"},{"Start":"11:02.690 ","End":"11:06.595","Text":"not including the dot because that\u0027s where it\u0027s 0."},{"Start":"11:06.595 ","End":"11:08.925","Text":"Here, not including the dot,"},{"Start":"11:08.925 ","End":"11:12.519","Text":"it\u0027s undefined, but including this."},{"Start":"11:13.310 ","End":"11:16.709","Text":"These are the positive bits,"},{"Start":"11:16.709 ","End":"11:21.515","Text":"and then I interpret this in terms of what xs this covers."},{"Start":"11:21.515 ","End":"11:24.450","Text":"I can say, for this bit,"},{"Start":"11:24.450 ","End":"11:28.685","Text":"x less than minus 24."},{"Start":"11:28.685 ","End":"11:32.500","Text":"It doesn\u0027t have a limit on the boundary on the left,"},{"Start":"11:32.500 ","End":"11:34.209","Text":"it goes on indefinitely."},{"Start":"11:34.209 ","End":"11:39.050","Text":"This bit is between 0 and 3."},{"Start":"11:39.380 ","End":"11:44.295","Text":"They\u0027re all strict inequalities, no equalities here."},{"Start":"11:44.295 ","End":"11:49.170","Text":"This bit is x bigger than 3?"},{"Start":"11:49.170 ","End":"11:52.470","Text":"If we put the word or in-between,"},{"Start":"11:52.470 ","End":"11:54.914","Text":"we actually get the answer,"},{"Start":"11:54.914 ","End":"11:57.359","Text":"which I shall highlight."},{"Start":"11:57.359 ","End":"12:00.435","Text":"But, I want to point out that there\u0027s a variation of this,"},{"Start":"12:00.435 ","End":"12:06.584","Text":"that you might find simpler or more aesthetic."},{"Start":"12:06.584 ","End":"12:09.704","Text":"The last 2 bits,"},{"Start":"12:09.704 ","End":"12:13.920","Text":"this and this, which is this,"},{"Start":"12:13.920 ","End":"12:15.360","Text":"then a gap,"},{"Start":"12:15.360 ","End":"12:16.619","Text":"and then the rest of it,"},{"Start":"12:16.619 ","End":"12:19.979","Text":"you could think of it as from 0 onwards,"},{"Start":"12:19.979 ","End":"12:22.319","Text":"but with the exception of a single point."},{"Start":"12:22.319 ","End":"12:23.939","Text":"In other words, the second half,"},{"Start":"12:23.939 ","End":"12:27.525","Text":"or second 2/3 or whatever,"},{"Start":"12:27.525 ","End":"12:32.680","Text":"could be written as x bigger than 0."},{"Start":"12:35.720 ","End":"12:43.650","Text":"But in mathematics is the same as and x not equal to 3,"},{"Start":"12:43.650 ","End":"12:47.415","Text":"and I\u0027ll put these in brackets."},{"Start":"12:47.415 ","End":"12:49.169","Text":"The first bit is still the same,"},{"Start":"12:49.169 ","End":"12:54.689","Text":"x less than minus 24, or another side,"},{"Start":"12:54.689 ","End":"12:57.720","Text":"the second bit to say that x is either between 0 and 3,"},{"Start":"12:57.720 ","End":"13:03.650","Text":"or from 3 onwards is the same as saying it\u0027s from 0 onwards except 3."},{"Start":"13:03.650 ","End":"13:07.220","Text":"That\u0027s how I write the except 3 and x is not equal to 3."},{"Start":"13:07.220 ","End":"13:09.670","Text":"This is also good,"},{"Start":"13:09.670 ","End":"13:12.825","Text":"both of them are good, but choose 1."},{"Start":"13:12.825 ","End":"13:17.025","Text":"That\u0027s the answer to a,"},{"Start":"13:17.025 ","End":"13:20.200","Text":"let\u0027s move on to b,"},{"Start":"13:20.420 ","End":"13:24.669","Text":"should be somewhere down here, there it is."},{"Start":"13:24.860 ","End":"13:28.180","Text":"Now we have b,"},{"Start":"13:29.870 ","End":"13:33.180","Text":"the general strategy as usual, first of all,"},{"Start":"13:33.180 ","End":"13:39.675","Text":"will be to put this thing on the left and then we have something bigger or equal to 0."},{"Start":"13:39.675 ","End":"13:45.150","Text":"Then, we will find a common denominator and put everything over that denominator, etc."},{"Start":"13:45.150 ","End":"13:49.860","Text":"Then, we\u0027ll also be doing some factorizing where we can."},{"Start":"13:49.860 ","End":"13:53.430","Text":"This time I\u0027m going to show you how to factorize this."},{"Start":"13:53.430 ","End":"13:57.330","Text":"Usually, I leave it for you as homework,"},{"Start":"13:57.330 ","End":"13:59.309","Text":"but this time I will do it,"},{"Start":"13:59.309 ","End":"14:06.265","Text":"will do the x^2 plus 6x minus 7."},{"Start":"14:06.265 ","End":"14:08.540","Text":"There are 2 main ways of doing it."},{"Start":"14:08.540 ","End":"14:12.740","Text":"I\u0027m going to use the trinomial decomposition method or some similar name,"},{"Start":"14:12.740 ","End":"14:14.300","Text":"and also put a 1 here,"},{"Start":"14:14.300 ","End":"14:16.100","Text":"because I want to know that this is a,"},{"Start":"14:16.100 ","End":"14:17.270","Text":"this is b,"},{"Start":"14:17.270 ","End":"14:22.740","Text":"and the minus 7 is c. What this method says,"},{"Start":"14:22.740 ","End":"14:31.485","Text":"is that we look for 2 numbers such that when they\u0027re added together,"},{"Start":"14:31.485 ","End":"14:34.080","Text":"they give us b."},{"Start":"14:34.080 ","End":"14:36.255","Text":"It\u0027s always b, in this case 6."},{"Start":"14:36.255 ","End":"14:38.595","Text":"Let\u0027s say, that in general this is b,"},{"Start":"14:38.595 ","End":"14:44.399","Text":"and the same 2 numbers that are here and here would be multiplied together."},{"Start":"14:44.399 ","End":"14:46.995","Text":"That\u0027s what I would like to give,"},{"Start":"14:46.995 ","End":"14:49.935","Text":"it\u0027s always ac,"},{"Start":"14:49.935 ","End":"14:51.045","Text":"and in this case,"},{"Start":"14:51.045 ","End":"14:56.950","Text":"it\u0027s 1 times minus 7 is minus 7."},{"Start":"14:57.620 ","End":"15:05.054","Text":"Now, when you see a negative in the product bit,"},{"Start":"15:05.054 ","End":"15:08.160","Text":"in the numbers have opposite signs,"},{"Start":"15:08.160 ","End":"15:12.539","Text":"you should think of it as a subtraction or a difference problem,"},{"Start":"15:12.539 ","End":"15:15.360","Text":"like 2 numbers multiplied together to give 7,"},{"Start":"15:15.360 ","End":"15:17.220","Text":"ignore the signs for the moment,"},{"Start":"15:17.220 ","End":"15:20.085","Text":"and the difference is 6."},{"Start":"15:20.085 ","End":"15:23.460","Text":"There\u0027s not too many things that multiply together to give 7,"},{"Start":"15:23.460 ","End":"15:25.260","Text":"it\u0027s 7 and1,1 and 7,"},{"Start":"15:25.260 ","End":"15:26.925","Text":"so that\u0027s really going to have to do,"},{"Start":"15:26.925 ","End":"15:30.240","Text":"and the question is, is the difference 6? Yes it is."},{"Start":"15:30.240 ","End":"15:36.495","Text":"Forgetting signs, it\u0027s going to be 7 minus 1 is 6,"},{"Start":"15:36.495 ","End":"15:39.405","Text":"and 7 times 1 is minus 7,"},{"Start":"15:39.405 ","End":"15:41.790","Text":"but I only temporarily ignore the signs,"},{"Start":"15:41.790 ","End":"15:43.710","Text":"I can\u0027t really do that."},{"Start":"15:43.710 ","End":"15:49.920","Text":"But the minus is going to go here because it\u0027s 7 minus 1 is 6,"},{"Start":"15:49.920 ","End":"15:52.080","Text":"and so that has also has to be minus,"},{"Start":"15:52.080 ","End":"15:55.665","Text":"and indeed 7 times minus 1 is minus 7."},{"Start":"15:55.665 ","End":"16:00.560","Text":"These are the 2 numbers from this method."},{"Start":"16:00.560 ","End":"16:02.360","Text":"Now, what do we do with these 2 numbers?"},{"Start":"16:02.360 ","End":"16:04.954","Text":"They\u0027re used to break up the middle piece,"},{"Start":"16:04.954 ","End":"16:08.030","Text":"they add up to give b in general and 6 here,"},{"Start":"16:08.030 ","End":"16:11.710","Text":"so we rewrite this trinomial,"},{"Start":"16:11.710 ","End":"16:13.710","Text":"in case you\u0027ve forgotten trinomial,"},{"Start":"16:13.710 ","End":"16:16.350","Text":"because of 3 separate terms in it."},{"Start":"16:16.350 ","End":"16:19.395","Text":"We write it as x^2 and instead of the 6,"},{"Start":"16:19.395 ","End":"16:21.960","Text":"this gives us instructions how to break it up,"},{"Start":"16:21.960 ","End":"16:25.530","Text":"plus 7x minus 1x,"},{"Start":"16:25.530 ","End":"16:30.790","Text":"leaving write it as 1x minus 7."},{"Start":"16:31.100 ","End":"16:34.574","Text":"Once we have it broken up this way,"},{"Start":"16:34.574 ","End":"16:36.990","Text":"we then use group factorization,"},{"Start":"16:36.990 ","End":"16:38.865","Text":"which this is a pair, this is a pair,"},{"Start":"16:38.865 ","End":"16:43.920","Text":"so this one is x times x plus 7."},{"Start":"16:43.920 ","End":"16:50.640","Text":"This would be minus 1 times x, plus 7."},{"Start":"16:50.640 ","End":"16:56.760","Text":"Finally, if I have x of these minus 1 of these are altogether go to x minus 1 of these,"},{"Start":"16:56.760 ","End":"17:02.340","Text":"or you might just prefer to put x plus 7 out front and get x minus 1 either way."},{"Start":"17:02.340 ","End":"17:05.145","Text":"This is the decomposition of this,"},{"Start":"17:05.145 ","End":"17:08.400","Text":"and in most exercises I don\u0027t do this because its time-consuming,"},{"Start":"17:08.400 ","End":"17:10.694","Text":"but for a change I did it."},{"Start":"17:10.694 ","End":"17:13.320","Text":"Now, let\u0027s go back here,"},{"Start":"17:13.320 ","End":"17:16.604","Text":"and while I\u0027m talking about factorizations,"},{"Start":"17:16.604 ","End":"17:18.629","Text":"there\u0027s one other 1 that\u0027s obvious."},{"Start":"17:18.629 ","End":"17:20.955","Text":"The 4x plus 28,"},{"Start":"17:20.955 ","End":"17:24.930","Text":"I\u0027m just somewhere else, did I say,"},{"Start":"17:24.930 ","End":"17:28.900","Text":"is 4 times x plus 7."},{"Start":"17:30.560 ","End":"17:32.669","Text":"With this in hand,"},{"Start":"17:32.669 ","End":"17:41.595","Text":"we go back here and tackle it and write x minus 7 over x minus 1,"},{"Start":"17:41.595 ","End":"17:50.325","Text":"minus 3x over 4 times x plus 7."},{"Start":"17:50.325 ","End":"17:52.845","Text":"Then we bring this over to this side,"},{"Start":"17:52.845 ","End":"17:59.865","Text":"minus 3 over whatever it is here in a minute,"},{"Start":"17:59.865 ","End":"18:03.000","Text":"then bigger or equal to 0."},{"Start":"18:03.000 ","End":"18:06.540","Text":"This is going over with a minus and bigger than 0,"},{"Start":"18:06.540 ","End":"18:11.100","Text":"and what we put here is what is written here, but not that."},{"Start":"18:11.100 ","End":"18:13.230","Text":"I\u0027ll use the factorization of it,"},{"Start":"18:13.230 ","End":"18:19.785","Text":"so it\u0027s x minus 1 times x plus 7."},{"Start":"18:19.785 ","End":"18:24.940","Text":"At this point we want 1 single fraction,"},{"Start":"18:25.460 ","End":"18:28.200","Text":"so I want a common denominator,"},{"Start":"18:28.200 ","End":"18:30.690","Text":"I put everything over that common denominator."},{"Start":"18:30.690 ","End":"18:34.319","Text":"Now, if I look at the factors, the bits I have and pieces,"},{"Start":"18:34.319 ","End":"18:36.989","Text":"I have an x minus 1 here and here,"},{"Start":"18:36.989 ","End":"18:40.545","Text":"I have an x plus 7 here and here, and I have a 4."},{"Start":"18:40.545 ","End":"18:42.480","Text":"It\u0027s sufficient to take those,"},{"Start":"18:42.480 ","End":"18:44.055","Text":"and I\u0027ve covered everything."},{"Start":"18:44.055 ","End":"18:47.020","Text":"If I take a dividing line,"},{"Start":"18:47.360 ","End":"18:51.120","Text":"I can write the common denominator down here,"},{"Start":"18:51.120 ","End":"18:52.440","Text":"so what did we say?"},{"Start":"18:52.440 ","End":"18:56.700","Text":"We want 4, we want an x minus 1,"},{"Start":"18:56.700 ","End":"18:58.530","Text":"and we want an x plus 7."},{"Start":"18:58.530 ","End":"19:00.540","Text":"There\u0027s no need to take things twice,"},{"Start":"19:00.540 ","End":"19:05.169","Text":"and each of these denominators goes into this common denominator."},{"Start":"19:05.750 ","End":"19:09.254","Text":"When I add and subtract and so on,"},{"Start":"19:09.254 ","End":"19:13.200","Text":"I just need to keep that common denominator and add and subtract the numerators."},{"Start":"19:13.200 ","End":"19:15.210","Text":"Now, what are the numerators?"},{"Start":"19:15.210 ","End":"19:17.010","Text":"One way to keep track is that,"},{"Start":"19:17.010 ","End":"19:20.820","Text":"see I put a little separator here and I say,"},{"Start":"19:20.820 ","End":"19:24.240","Text":"what is the missing factor in the denominator in each case?"},{"Start":"19:24.240 ","End":"19:28.065","Text":"Here and to here I\u0027m missing the 4 and the x plus 7,"},{"Start":"19:28.065 ","End":"19:32.295","Text":"so I write 4 times x plus 7,"},{"Start":"19:32.295 ","End":"19:34.965","Text":"and afterwards I have to multiply this by this."},{"Start":"19:34.965 ","End":"19:40.965","Text":"That tells me what the numerator for this fraction is after the common denominator."},{"Start":"19:40.965 ","End":"19:45.030","Text":"Here, I\u0027m missing the x minus 1,"},{"Start":"19:45.030 ","End":"19:50.790","Text":"here, I\u0027m just missing the 4."},{"Start":"19:50.790 ","End":"19:53.235","Text":"So this times this,"},{"Start":"19:53.235 ","End":"19:55.950","Text":"this times this, this times this with the appropriate sign,"},{"Start":"19:55.950 ","End":"20:03.240","Text":"and I get 4x plus 7x minus 7."},{"Start":"20:03.240 ","End":"20:06.314","Text":"That\u0027s from this over this,"},{"Start":"20:06.314 ","End":"20:08.445","Text":"minus little over this,"},{"Start":"20:08.445 ","End":"20:11.954","Text":"and I\u0027m doing the numerators now."},{"Start":"20:11.954 ","End":"20:13.740","Text":"x minus 1 times 3x,"},{"Start":"20:13.740 ","End":"20:16.305","Text":"I better put the 3x first, it looks better,"},{"Start":"20:16.305 ","End":"20:18.345","Text":"x minus 1,"},{"Start":"20:18.345 ","End":"20:22.170","Text":"then this is just numbers,"},{"Start":"20:22.170 ","End":"20:25.780","Text":"4 times 3 is 12."},{"Start":"20:25.850 ","End":"20:30.179","Text":"We\u0027re still with bigger or equal to 0."},{"Start":"20:30.179 ","End":"20:35.430","Text":"The next step I\u0027m going to just open up all the brackets here,"},{"Start":"20:35.430 ","End":"20:38.145","Text":"and did a quick copy paste,"},{"Start":"20:38.145 ","End":"20:43.155","Text":"while you were looking for the rest of it to save time."},{"Start":"20:43.155 ","End":"20:46.570","Text":"Let\u0027s just do the numerator now,"},{"Start":"20:47.480 ","End":"20:49.500","Text":"x plus 7,"},{"Start":"20:49.500 ","End":"20:51.690","Text":"times x minus 7,"},{"Start":"20:51.690 ","End":"20:53.969","Text":"that\u0027s another thing I can do at the side,"},{"Start":"20:53.969 ","End":"20:56.520","Text":"x plus 7 times x minus 7."},{"Start":"20:56.520 ","End":"20:58.890","Text":"Remember the difference of squares formula,"},{"Start":"20:58.890 ","End":"21:02.400","Text":"a minus b, a plus b is a^2 minus b^2."},{"Start":"21:02.400 ","End":"21:05.805","Text":"This is x^2 minus 7^2,"},{"Start":"21:05.805 ","End":"21:09.494","Text":"and we all know that 7^2 is 49."},{"Start":"21:09.494 ","End":"21:13.395","Text":"Here we have 4 times x^2 minus 49,"},{"Start":"21:13.395 ","End":"21:19.230","Text":"so 4x^2 minus 4 times 49 would be, let\u0027s see,"},{"Start":"21:19.230 ","End":"21:23.564","Text":"it\u0027s 50 less 1,"},{"Start":"21:23.564 ","End":"21:26.055","Text":"200 less 4, 196, I make it,"},{"Start":"21:26.055 ","End":"21:31.815","Text":"minus 3x times x is minus 3x^2,"},{"Start":"21:31.815 ","End":"21:33.885","Text":"minus 3x times minus 1,"},{"Start":"21:33.885 ","End":"21:36.135","Text":"is plus 3x,"},{"Start":"21:36.135 ","End":"21:41.160","Text":"and then minus 12 still bigger or equal to 0."},{"Start":"21:41.160 ","End":"21:42.690","Text":"That\u0027s just the expansion,"},{"Start":"21:42.690 ","End":"21:45.525","Text":"now we\u0027re going to collect the like terms."},{"Start":"21:45.525 ","End":"21:51.540","Text":"Once again, I did a quick copy paste for the bit that I know is not changing,"},{"Start":"21:51.540 ","End":"21:53.549","Text":"and so let\u0027s see."},{"Start":"21:53.549 ","End":"21:55.830","Text":"We collect x^2,"},{"Start":"21:55.830 ","End":"22:01.170","Text":"4x^2 minus 3x^2 is just x^2."},{"Start":"22:01.170 ","End":"22:04.245","Text":"As for x\u0027s, I only have x\u0027s here,"},{"Start":"22:04.245 ","End":"22:07.815","Text":"so that\u0027s plus 3x,"},{"Start":"22:07.815 ","End":"22:15.225","Text":"and now numbers minus 196 minus 12."},{"Start":"22:15.225 ","End":"22:17.150","Text":"Let\u0027s see, because they\u0027re both minuses,"},{"Start":"22:17.150 ","End":"22:20.555","Text":"like an addition 196 plus 12 is 208,"},{"Start":"22:20.555 ","End":"22:25.565","Text":"so it\u0027s minus 208, bigger or equals to 0."},{"Start":"22:25.565 ","End":"22:28.949","Text":"Let\u0027s see if we can factorize this."},{"Start":"22:30.270 ","End":"22:36.909","Text":"This time, I\u0027m going to give you the answer,"},{"Start":"22:36.909 ","End":"22:41.590","Text":"and I need some more space,"},{"Start":"22:41.780 ","End":"22:44.520","Text":"just scroll a bit,"},{"Start":"22:44.520 ","End":"22:49.259","Text":"and I\u0027ll give you the answer."},{"Start":"22:49.259 ","End":"22:56.504","Text":"It is x plus 16 times x minus 13."},{"Start":"22:56.504 ","End":"22:59.130","Text":"I will write the rest of it in a moment,"},{"Start":"22:59.130 ","End":"23:02.280","Text":"I just want to verify,"},{"Start":"23:02.280 ","End":"23:04.230","Text":"if I didn\u0027t show you how I got there,"},{"Start":"23:04.230 ","End":"23:08.835","Text":"at least I can show you that it\u0027s true x times x is x^2, fine,"},{"Start":"23:08.835 ","End":"23:14.910","Text":"plus 16x minus 13x is 3x, 16 minus 13."},{"Start":"23:14.910 ","End":"23:17.715","Text":"On your calculator or in your head,"},{"Start":"23:17.715 ","End":"23:21.100","Text":"16 times 13 is 208 and it\u0027s a plus,"},{"Start":"23:21.100 ","End":"23:22.895","Text":"minus, so it\u0027s a minus."},{"Start":"23:22.895 ","End":"23:24.560","Text":"That\u0027s what we get here,"},{"Start":"23:24.560 ","End":"23:31.219","Text":"I\u0027m just copying the common bit,"},{"Start":"23:31.219 ","End":"23:35.160","Text":"let\u0027s see, that should be about right."},{"Start":"23:35.910 ","End":"23:38.950","Text":"Great. This is the point where we want to be."},{"Start":"23:38.950 ","End":"23:41.004","Text":"We have a rational expression,"},{"Start":"23:41.004 ","End":"23:46.525","Text":"and numerator and denominator completely factorized."},{"Start":"23:46.525 ","End":"23:48.909","Text":"Dividing lines a bit long,"},{"Start":"23:48.909 ","End":"23:51.800","Text":"let\u0027s just trim it there."},{"Start":"23:53.310 ","End":"24:00.339","Text":"Now, we look for the values of x,"},{"Start":"24:00.339 ","End":"24:02.169","Text":"which makes the numerator 0,"},{"Start":"24:02.169 ","End":"24:04.540","Text":"and then the denominator 0."},{"Start":"24:04.540 ","End":"24:08.380","Text":"Start with numerator. For this to be 0,"},{"Start":"24:08.380 ","End":"24:10.735","Text":"x has to be minus 16."},{"Start":"24:10.735 ","End":"24:13.570","Text":"For this factor to be 0,"},{"Start":"24:13.570 ","End":"24:15.910","Text":"we need the x as 13."},{"Start":"24:15.910 ","End":"24:18.460","Text":"Here, we need x to be 1,"},{"Start":"24:18.460 ","End":"24:19.869","Text":"if we want this to be 0,"},{"Start":"24:19.869 ","End":"24:21.726","Text":"and here, minus 7."},{"Start":"24:21.726 ","End":"24:24.640","Text":"There\u0027s full points,"},{"Start":"24:24.640 ","End":"24:28.510","Text":"but we will distinguish between the ones in the numerator and the denominator,"},{"Start":"24:28.510 ","End":"24:31.160","Text":"and you\u0027ll see what follows."},{"Start":"24:31.680 ","End":"24:35.245","Text":"Let me do a sketch."},{"Start":"24:35.245 ","End":"24:37.570","Text":"I\u0027ve got too much clutter here,"},{"Start":"24:37.570 ","End":"24:40.629","Text":"I don\u0027t need this."},{"Start":"24:40.629 ","End":"24:43.105","Text":"This will be enough space."},{"Start":"24:43.105 ","End":"24:46.330","Text":"Here we have a number line,"},{"Start":"24:46.330 ","End":"24:51.924","Text":"and now these, I\u0027m going to place on it in the correct order."},{"Start":"24:51.924 ","End":"24:56.065","Text":"Minus 16 is the smallest,"},{"Start":"24:56.065 ","End":"24:59.965","Text":"then minus 7 next,"},{"Start":"24:59.965 ","End":"25:04.614","Text":"then 1, and then 13."},{"Start":"25:04.614 ","End":"25:09.319","Text":"Now, I want to distinguish the enumerator, denominator."},{"Start":"25:11.640 ","End":"25:14.995","Text":"The 1 and the minus 7,"},{"Start":"25:14.995 ","End":"25:21.174","Text":"our values which we\u0027re not allowed to substitute for this expression."},{"Start":"25:21.174 ","End":"25:24.760","Text":"Because we can never have a 0 on the denominator,"},{"Start":"25:24.760 ","End":"25:27.849","Text":"so 1 and minus 7 are illegal, let\u0027s call them that."},{"Start":"25:27.849 ","End":"25:30.130","Text":"They\u0027re not in the domain, another way of saying it."},{"Start":"25:30.130 ","End":"25:34.376","Text":"I\u0027m going to indicate that by making a little hole"},{"Start":"25:34.376 ","End":"25:38.890","Text":"next to the 1 to show that this is illegal,"},{"Start":"25:38.890 ","End":"25:41.799","Text":"and next to the minus 7."},{"Start":"25:41.799 ","End":"25:47.320","Text":"There we are. I\u0027ll further emphasize that these are actually missing"},{"Start":"25:47.320 ","End":"25:53.020","Text":"by putting a little hollow circle, oval."},{"Start":"25:53.020 ","End":"25:58.030","Text":"In contrast, the minus 16 and the 13 came from the numerator."},{"Start":"25:58.030 ","End":"26:00.280","Text":"No problem with numerator being 0,"},{"Start":"26:00.280 ","End":"26:03.920","Text":"and that\u0027s what gives the expression a value of 0."},{"Start":"26:03.930 ","End":"26:08.125","Text":"That\u0027s here, and that\u0027s here."},{"Start":"26:08.125 ","End":"26:11.860","Text":"0 undefined, undefined 0,"},{"Start":"26:11.860 ","End":"26:14.454","Text":"and in-between on these 5 intervals,"},{"Start":"26:14.454 ","End":"26:17.319","Text":"we have 5 separate bits from here to here."},{"Start":"26:17.319 ","End":"26:24.295","Text":"The value of the expression will be always positive or always negative,"},{"Start":"26:24.295 ","End":"26:26.319","Text":"it won\u0027t change in the middle of an interval."},{"Start":"26:26.319 ","End":"26:30.145","Text":"The way we tell for each of these 5 intervals what it is,"},{"Start":"26:30.145 ","End":"26:32.739","Text":"is by using sample values,"},{"Start":"26:32.739 ","End":"26:35.139","Text":"that\u0027s the easiest way to go."},{"Start":"26:35.139 ","End":"26:39.280","Text":"You could choose any values usually for convenience of computation."},{"Start":"26:39.280 ","End":"26:43.509","Text":"For example, here I would choose the 0,"},{"Start":"26:43.509 ","End":"26:45.010","Text":"and between 1 and 13,"},{"Start":"26:45.010 ","End":"26:47.320","Text":"I don\u0027t know, I\u0027ll take just 2."},{"Start":"26:47.320 ","End":"26:50.030","Text":"From 13 onwards,"},{"Start":"26:50.580 ","End":"26:54.190","Text":"I\u0027ll take 15, could take 14."},{"Start":"26:54.190 ","End":"27:01.030","Text":"I think between minus 7 and minus 16 why don\u0027t I take minus 10."},{"Start":"27:01.030 ","End":"27:03.745","Text":"Pretty arbitrary, doesn\u0027t matter."},{"Start":"27:03.745 ","End":"27:05.770","Text":"Beyond minus 16,"},{"Start":"27:05.770 ","End":"27:11.659","Text":"I\u0027ll be conservative, go with minus 17."},{"Start":"27:12.660 ","End":"27:15.009","Text":"It doesn\u0027t matter which point you choose,"},{"Start":"27:15.009 ","End":"27:16.720","Text":"you\u0027re going to get the same plus or minus."},{"Start":"27:16.720 ","End":"27:18.909","Text":"We get different values, but you\u0027ll get the same plus or"},{"Start":"27:18.909 ","End":"27:22.370","Text":"minus for each of them. Let\u0027s see."},{"Start":"27:22.740 ","End":"27:25.254","Text":"I don\u0027t have to start from here,"},{"Start":"27:25.254 ","End":"27:27.290","Text":"I can start from 0, which is I\u0027m going to get these,"},{"Start":"27:27.290 ","End":"27:28.615","Text":"you went out the way."},{"Start":"27:28.615 ","End":"27:30.445","Text":"I put 0 in,"},{"Start":"27:30.445 ","End":"27:34.209","Text":"I\u0027ve got 16 minus 13,"},{"Start":"27:34.209 ","End":"27:36.175","Text":"minus 1 and 7."},{"Start":"27:36.175 ","End":"27:38.290","Text":"The false positives, so I can ignore it."},{"Start":"27:38.290 ","End":"27:42.925","Text":"I basically get plus, minus, minus, plus."},{"Start":"27:42.925 ","End":"27:45.219","Text":"I counted 2 minuses there,"},{"Start":"27:45.219 ","End":"27:46.666","Text":"it\u0027s an even number,"},{"Start":"27:46.666 ","End":"27:48.940","Text":"that makes it plus."},{"Start":"27:48.940 ","End":"27:53.965","Text":"For a change, I\u0027ll use a variation on this snake method where I don\u0027t sketch it,"},{"Start":"27:53.965 ","End":"28:01.479","Text":"I just put a plus to indicate that this interval up to here is plus."},{"Start":"28:01.479 ","End":"28:04.135","Text":"The case of minus 10,"},{"Start":"28:04.135 ","End":"28:06.400","Text":"this will be positive,"},{"Start":"28:06.400 ","End":"28:11.005","Text":"negative, negative,"},{"Start":"28:11.005 ","End":"28:15.350","Text":"and negative, it will make that 3 negatives, so a negative for minus 10."},{"Start":"28:21.720 ","End":"28:29.886","Text":"For 0"},{"Start":"28:29.886 ","End":"28:31.189","Text":", we get plus, minus, minus, plus, 2 minuses which means that it\u0027s a positive at 0."},{"Start":"28:31.189 ","End":"28:31.190","Text":"At 2, we have this 2 plus 16 is positive, 2 minus 13 is negative,"},{"Start":"28:31.190 ","End":"28:31.191","Text":"2 minus 1, negative."},{"Start":"28:31.191 ","End":"28:37.400","Text":"Just"},{"Start":"28:42.200 ","End":"28:45.344","Text":"one negative,"},{"Start":"28:45.344 ","End":"28:50.530","Text":"which is this one."},{"Start":"28:54.210 ","End":"28:58.359","Text":"That\u0027s a minus above the 2,"},{"Start":"28:58.359 ","End":"29:00.655","Text":"and the 15,"},{"Start":"29:00.655 ","End":"29:02.169","Text":"that will be positive,"},{"Start":"29:02.169 ","End":"29:04.494","Text":"positive, they\u0027ll all be positive."},{"Start":"29:04.494 ","End":"29:12.535","Text":"That\u0027s a plus. Now I\u0027m not just saying that at minus 17,"},{"Start":"29:12.535 ","End":"29:14.829","Text":"it\u0027s plus so that the minus 10 is minus."},{"Start":"29:14.829 ","End":"29:20.620","Text":"At the whole interval, sometimes I even put some faint lines just so we can"},{"Start":"29:20.620 ","End":"29:28.195","Text":"see these are the 5 intervals between the lines."},{"Start":"29:28.195 ","End":"29:29.829","Text":"All of this is positive,"},{"Start":"29:29.829 ","End":"29:32.289","Text":"all of this is negative and so on."},{"Start":"29:32.289 ","End":"29:34.854","Text":"Now, what do we want?"},{"Start":"29:34.854 ","End":"29:41.180","Text":"We want the places where it\u0027s positive or 0."},{"Start":"29:41.220 ","End":"29:45.895","Text":"That\u0027s what bigger or equal to 0 means."},{"Start":"29:45.895 ","End":"29:47.350","Text":"Positive is good,"},{"Start":"29:47.350 ","End":"29:48.985","Text":"and 0 is good."},{"Start":"29:48.985 ","End":"29:52.434","Text":"No negatives, and no undefines."},{"Start":"29:52.434 ","End":"29:55.430","Text":"Let\u0027s see. Positive,"},{"Start":"29:55.860 ","End":"29:59.335","Text":"here I have a positive."},{"Start":"29:59.335 ","End":"30:02.479","Text":"That\u0027s a terrible color."},{"Start":"30:03.090 ","End":"30:07.705","Text":"Positive, positive is good,"},{"Start":"30:07.705 ","End":"30:11.064","Text":"positive is good, and 0 is good also."},{"Start":"30:11.064 ","End":"30:15.970","Text":"This is where it\u0027s 0, and this is where it\u0027s 0."},{"Start":"30:15.970 ","End":"30:20.119","Text":"Don\u0027t take the undefines, whatever you do."},{"Start":"30:21.330 ","End":"30:26.665","Text":"If we look at it and we want to write it in terms of x,"},{"Start":"30:26.665 ","End":"30:33.699","Text":"this positive area interval with this point minus"},{"Start":"30:33.699 ","End":"30:41.605","Text":"16 all together gives us everything up to minus 16."},{"Start":"30:41.605 ","End":"30:49.209","Text":"This gives us x less than or equal to minus 16."},{"Start":"30:49.209 ","End":"30:52.869","Text":"I was going to highlight or shade but it\u0027s too messy as it is."},{"Start":"30:52.869 ","End":"30:56.859","Text":"We\u0027re just talking about this interval,"},{"Start":"30:56.859 ","End":"31:01.630","Text":"it goes on forever up to here including this point, that\u0027s that."},{"Start":"31:01.630 ","End":"31:07.430","Text":"Then this bit, this plus is from minus 7-1."},{"Start":"31:07.770 ","End":"31:11.725","Text":"But we don\u0027t include either of the endpoints,"},{"Start":"31:11.725 ","End":"31:18.790","Text":"so this becomes minus 7 less than x, less than 1."},{"Start":"31:18.790 ","End":"31:23.020","Text":"This bit here, the plus together with the 0, remember,"},{"Start":"31:23.020 ","End":"31:25.609","Text":"this is plus or 0,"},{"Start":"31:25.609 ","End":"31:30.505","Text":"would be from 13 and onwards, but including the 13."},{"Start":"31:30.505 ","End":"31:36.220","Text":"This is just 13 less than x."},{"Start":"31:36.220 ","End":"31:40.989","Text":"You know what? I prefer to write it as x bigger than 13."},{"Start":"31:40.989 ","End":"31:44.560","Text":"It just somehow looks better this way."},{"Start":"31:44.560 ","End":"31:49.344","Text":"Then we put or here and here,"},{"Start":"31:49.344 ","End":"31:51.534","Text":"and that gives us our answer."},{"Start":"31:51.534 ","End":"31:56.754","Text":"I think it deserves highlighting, so that\u0027s it."},{"Start":"31:56.754 ","End":"31:58.435","Text":"This was part b,"},{"Start":"31:58.435 ","End":"32:01.999","Text":"and so we\u0027re done with this exercise."}],"Thumbnail":null,"ID":5081}],"ID":3877},{"Name":"Absolute Value Inequalities","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Inequalities with Absolute Value","Duration":"12m 37s","ChapterTopicVideoID":5082,"CourseChapterTopicPlaylistID":3878,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.415","Text":"In this clip, I\u0027ll be explaining about Inequalities with Absolute Value."},{"Start":"00:05.415 ","End":"00:08.865","Text":"I\u0027m going to do this by way of examples."},{"Start":"00:08.865 ","End":"00:11.265","Text":"Let\u0027s start with the first example."},{"Start":"00:11.265 ","End":"00:17.520","Text":"Absolute value of x minus 3 is less than 4."},{"Start":"00:17.520 ","End":"00:22.050","Text":"There\u0027s a standard formula for this kind of exercise."},{"Start":"00:22.050 ","End":"00:24.840","Text":"Whenever you have the absolute value of something,"},{"Start":"00:24.840 ","End":"00:27.659","Text":"say a is less than b,"},{"Start":"00:27.659 ","End":"00:32.505","Text":"then that\u0027s equivalent to a double inequality where"},{"Start":"00:32.505 ","End":"00:38.000","Text":"a is in the middle and it\u0027s between b and minus b."},{"Start":"00:38.000 ","End":"00:41.435","Text":"You can accept this on faith or I can give you a brief explanation."},{"Start":"00:41.435 ","End":"00:43.055","Text":"Let\u0027s take, for example,"},{"Start":"00:43.055 ","End":"00:45.350","Text":"absolute value of a less than 4."},{"Start":"00:45.350 ","End":"00:51.170","Text":"It looks something like this and absolute value of a is the distance of a from 0."},{"Start":"00:51.170 ","End":"00:54.350","Text":"If I have a number line and I have 0,"},{"Start":"00:54.350 ","End":"00:59.210","Text":"and I\u0027m looking for a number whose distance from 0 is less than 4."},{"Start":"00:59.210 ","End":"01:00.410","Text":"It\u0027s like a radius of 4,"},{"Start":"01:00.410 ","End":"01:05.660","Text":"so it can be at most up to 4 and down to minus 4,"},{"Start":"01:05.660 ","End":"01:09.395","Text":"all these numbers have a distance less than 4 from 0,"},{"Start":"01:09.395 ","End":"01:11.755","Text":"not including the endpoints."},{"Start":"01:11.755 ","End":"01:14.070","Text":"That\u0027s a quick explanation."},{"Start":"01:14.070 ","End":"01:16.085","Text":"If I apply it here,"},{"Start":"01:16.085 ","End":"01:18.250","Text":"then my a is x minus 3,"},{"Start":"01:18.250 ","End":"01:20.015","Text":"and I put this in the middle,"},{"Start":"01:20.015 ","End":"01:24.530","Text":"and I sandwich it between 2 inequalities of less than,"},{"Start":"01:24.530 ","End":"01:27.650","Text":"so less than 4 and greater than minus 4."},{"Start":"01:27.650 ","End":"01:30.085","Text":"That\u0027s a double inequality."},{"Start":"01:30.085 ","End":"01:32.420","Text":"This one is a fairly simple one."},{"Start":"01:32.420 ","End":"01:35.240","Text":"It\u0027s linear and there\u0027s only x in the middle term."},{"Start":"01:35.240 ","End":"01:38.570","Text":"We can work on all 3 parts simultaneously."},{"Start":"01:38.570 ","End":"01:41.029","Text":"I would add 3 to everything."},{"Start":"01:41.029 ","End":"01:45.830","Text":"Then I say x is less than 4 plus 3 is 7,"},{"Start":"01:45.830 ","End":"01:50.015","Text":"and greater than minus 4 plus 3 is minus 1,"},{"Start":"01:50.015 ","End":"01:52.565","Text":"and that would be the answer."},{"Start":"01:52.565 ","End":"01:54.994","Text":"We started out with an easy example."},{"Start":"01:54.994 ","End":"01:58.625","Text":"Now another example, this time,"},{"Start":"01:58.625 ","End":"02:07.445","Text":"absolute value of 5 minus 2x and will take a less than or equal to this time."},{"Start":"02:07.445 ","End":"02:11.210","Text":"Now, the formula is very similar to this one."},{"Start":"02:11.210 ","End":"02:13.715","Text":"I\u0027ll just copy past it from here."},{"Start":"02:13.715 ","End":"02:17.195","Text":"The only difference is that if I have a less than or equal to here,"},{"Start":"02:17.195 ","End":"02:20.780","Text":"then this is also less than or equal to"},{"Start":"02:20.780 ","End":"02:24.740","Text":"here and here and it\u0027s the same explanation like before."},{"Start":"02:24.740 ","End":"02:29.400","Text":"If I wanted something of distance 4 from 0,"},{"Start":"02:29.400 ","End":"02:33.375","Text":"including 4 then I would also include the endpoints."},{"Start":"02:33.375 ","End":"02:35.840","Text":"This explanation works here too."},{"Start":"02:35.840 ","End":"02:38.150","Text":"We can just accept it on faith."},{"Start":"02:38.150 ","End":"02:41.135","Text":"These are the 2 formulas that we need essentially the same,"},{"Start":"02:41.135 ","End":"02:43.280","Text":"less than, less than,"},{"Start":"02:43.280 ","End":"02:45.725","Text":"less than or equal to, it\u0027s less than or equal to."},{"Start":"02:45.725 ","End":"02:49.940","Text":"In our case, we get minus 3,"},{"Start":"02:49.940 ","End":"02:56.070","Text":"less than or equal to 5 minus 2x less than or equal to 3."},{"Start":"02:56.070 ","End":"02:59.555","Text":"This is a double inequality but without absolute value."},{"Start":"02:59.555 ","End":"03:04.100","Text":"Once again, we have a linear expression with x and x only appears in the middle,"},{"Start":"03:04.100 ","End":"03:08.165","Text":"so we can work on all 3 parts simultaneously."},{"Start":"03:08.165 ","End":"03:11.780","Text":"Let\u0027s start out by subtracting 5 everywhere."},{"Start":"03:11.780 ","End":"03:14.375","Text":"We\u0027ve got minus 8,"},{"Start":"03:14.375 ","End":"03:20.460","Text":"less than or equal to minus 2x less than or equal to minus 2,"},{"Start":"03:20.460 ","End":"03:22.230","Text":"and I subtract 5."},{"Start":"03:22.230 ","End":"03:25.715","Text":"Now I have to divide everything by minus 2,"},{"Start":"03:25.715 ","End":"03:28.430","Text":"but minus 2 is negative,"},{"Start":"03:28.430 ","End":"03:30.775","Text":"so we have to watch out."},{"Start":"03:30.775 ","End":"03:36.095","Text":"We\u0027re going to reverse the direction of the inequality when we divide by a negative."},{"Start":"03:36.095 ","End":"03:42.155","Text":"We get that x is bigger or equal to minus 2 over minus 2 is 1,"},{"Start":"03:42.155 ","End":"03:45.990","Text":"minus 8 over minus 2 is 4,"},{"Start":"03:45.990 ","End":"03:47.780","Text":"and this is the answer,"},{"Start":"03:47.780 ","End":"03:51.620","Text":"but it\u0027s more common to have the numbers"},{"Start":"03:51.620 ","End":"03:56.270","Text":"increase from left to right so we can just write the equivalent to this,"},{"Start":"03:56.270 ","End":"03:59.315","Text":"that 1 is less than or equal to x,"},{"Start":"03:59.315 ","End":"04:01.475","Text":"less than or equal to 4,"},{"Start":"04:01.475 ","End":"04:04.460","Text":"x is between 1 and 4 inclusive."},{"Start":"04:04.460 ","End":"04:10.445","Text":"Now I\u0027m going to take a slightly harder example which is not linear."},{"Start":"04:10.445 ","End":"04:12.575","Text":"That will be Number 3."},{"Start":"04:12.575 ","End":"04:18.139","Text":"I\u0027ll take the absolute value of x^2 minus x"},{"Start":"04:18.139 ","End":"04:24.975","Text":"minus 4 is less than or equal to 2."},{"Start":"04:24.975 ","End":"04:29.210","Text":"I think we still can just see this formula here."},{"Start":"04:29.210 ","End":"04:35.330","Text":"There we are. What I have to do now is write it as 2 inequalities."},{"Start":"04:35.330 ","End":"04:41.030","Text":"I\u0027ve got x^2 minus x minus 4,"},{"Start":"04:41.030 ","End":"04:45.425","Text":"greater or equal to minus 2,"},{"Start":"04:45.425 ","End":"04:49.250","Text":"but less than or equal to 2."},{"Start":"04:49.250 ","End":"04:54.740","Text":"But this time, what\u0027s in here is not linear so I can\u0027t work on"},{"Start":"04:54.740 ","End":"05:00.515","Text":"all 3 sides simultaneously with just subtraction and division like we did before."},{"Start":"05:00.515 ","End":"05:05.435","Text":"We\u0027re going to actually have to break it apart into 2 separate inequalities."},{"Start":"05:05.435 ","End":"05:13.420","Text":"We get minus 2 less than or equal to x^2 minus x minus 4 for this bit,"},{"Start":"05:13.420 ","End":"05:18.380","Text":"and I\u0027m just writing it here to give myself more space,"},{"Start":"05:18.380 ","End":"05:23.580","Text":"x^2 minus x minus 4 is less than or equal to 2."},{"Start":"05:23.580 ","End":"05:24.915","Text":"That\u0027s from this inequality."},{"Start":"05:24.915 ","End":"05:29.570","Text":"I broke up the double inequality into an end of 2 separate ones."},{"Start":"05:29.570 ","End":"05:36.245","Text":"Now this one gives me that x^2 minus x."},{"Start":"05:36.245 ","End":"05:38.945","Text":"If I bring the minus 2 over here,"},{"Start":"05:38.945 ","End":"05:43.225","Text":"it\u0027s minus 4 plus 2 is minus 2."},{"Start":"05:43.225 ","End":"05:48.880","Text":"But then it\u0027s bigger or equal to 0 if I reverse sides."},{"Start":"05:48.880 ","End":"05:55.325","Text":"This one is x^2 minus x minus 6,"},{"Start":"05:55.325 ","End":"05:58.940","Text":"less than or equal to 0."},{"Start":"05:58.940 ","End":"06:02.510","Text":"To save time, I\u0027ll just factorize these for you."},{"Start":"06:02.510 ","End":"06:06.365","Text":"This one is x minus 2,"},{"Start":"06:06.365 ","End":"06:10.050","Text":"x plus 1 bigger or equal to 0."},{"Start":"06:10.050 ","End":"06:13.710","Text":"Let\u0027s just check, x times x is x squared minus 2x"},{"Start":"06:13.710 ","End":"06:18.030","Text":"plus x is minus x and minus 2 times 1 is minus 2."},{"Start":"06:18.030 ","End":"06:22.550","Text":"This one factorizes as x minus"},{"Start":"06:22.550 ","End":"06:28.980","Text":"3 x plus 2x^2 minus 3x plus 2x is this."},{"Start":"06:28.980 ","End":"06:31.150","Text":"Minus 3 times 2 is minus 6,"},{"Start":"06:31.150 ","End":"06:33.214","Text":"less than or equal to 0."},{"Start":"06:33.214 ","End":"06:35.575","Text":"Of course, there\u0027s the and in the middle."},{"Start":"06:35.575 ","End":"06:41.390","Text":"What we have to do is work on each one separately and then combine."},{"Start":"06:41.390 ","End":"06:43.895","Text":"Let\u0027s work on this one."},{"Start":"06:43.895 ","End":"06:48.200","Text":"The roots 2 and minus 1,"},{"Start":"06:48.200 ","End":"06:52.100","Text":"or I\u0027ll write them in order minus 1 and 2."},{"Start":"06:52.100 ","End":"06:59.580","Text":"I might as well for this one also say that the roots are minus 2 and 3."},{"Start":"06:59.580 ","End":"07:01.290","Text":"X plus 2, 0,"},{"Start":"07:01.290 ","End":"07:03.330","Text":"this is minus 2 and x minus 3,0."},{"Start":"07:03.330 ","End":"07:04.875","Text":"X is 3, and so on."},{"Start":"07:04.875 ","End":"07:08.310","Text":"I need a picture for each."},{"Start":"07:08.310 ","End":"07:10.020","Text":"Let\u0027s do this one."},{"Start":"07:10.020 ","End":"07:14.130","Text":"We need to mark the roots on the number line."},{"Start":"07:14.130 ","End":"07:17.325","Text":"I have minus 1 somewhere,"},{"Start":"07:17.325 ","End":"07:19.110","Text":"and a 2 somewhere,"},{"Start":"07:19.110 ","End":"07:22.805","Text":"and then we take some pool points."},{"Start":"07:22.805 ","End":"07:25.080","Text":"I\u0027ll take minus 2,"},{"Start":"07:25.080 ","End":"07:28.845","Text":"I\u0027ll take 0 and I\u0027ll take 3,"},{"Start":"07:28.845 ","End":"07:31.970","Text":"and we plug in these"},{"Start":"07:31.970 ","End":"07:37.384","Text":"sample points into the expression to see if we get positive, negative or 0."},{"Start":"07:37.384 ","End":"07:40.205","Text":"Just added some dotted lines there."},{"Start":"07:40.205 ","End":"07:43.670","Text":"We know that it\u0027s 0 here and here,"},{"Start":"07:43.670 ","End":"07:46.459","Text":"so that\u0027s 0 and here it\u0027s 0."},{"Start":"07:46.459 ","End":"07:48.170","Text":"In these 3 intervals,"},{"Start":"07:48.170 ","End":"07:50.310","Text":"it\u0027s always positive or always negative."},{"Start":"07:50.310 ","End":"07:52.730","Text":"That\u0027s why we use the sample point plug-in,"},{"Start":"07:52.730 ","End":"07:55.425","Text":"the minus 2,"},{"Start":"07:55.425 ","End":"07:57.120","Text":"and we get minus 2,"},{"Start":"07:57.120 ","End":"07:58.560","Text":"minus 2 is negative,"},{"Start":"07:58.560 ","End":"08:01.830","Text":"minus 2 plus 1 is also negative."},{"Start":"08:01.830 ","End":"08:04.595","Text":"Negative times negative is positive."},{"Start":"08:04.595 ","End":"08:07.639","Text":"I won\u0027t draw the whole snake diagram,"},{"Start":"08:07.639 ","End":"08:09.800","Text":"I\u0027ll just write the pluses and minuses."},{"Start":"08:09.800 ","End":"08:13.490","Text":"At 0, 0 minus 2 is negative,"},{"Start":"08:13.490 ","End":"08:15.155","Text":"0 plus 1 is positive."},{"Start":"08:15.155 ","End":"08:17.060","Text":"Here we are negative,"},{"Start":"08:17.060 ","End":"08:19.680","Text":"and when x is 3,"},{"Start":"08:19.680 ","End":"08:20.910","Text":"3 minus 2 is positive,"},{"Start":"08:20.910 ","End":"08:23.340","Text":"3 plus 1 is positive."},{"Start":"08:23.340 ","End":"08:25.235","Text":"Here it\u0027s positive."},{"Start":"08:25.235 ","End":"08:28.325","Text":"Now, I want to look what I need."},{"Start":"08:28.325 ","End":"08:33.820","Text":"The bigger or equal to 0 means that we\u0027re collecting positive and 0,"},{"Start":"08:33.820 ","End":"08:37.725","Text":"positive, positive, 0, and 0."},{"Start":"08:37.725 ","End":"08:39.660","Text":"What this gives us,"},{"Start":"08:39.660 ","End":"08:41.890","Text":"if I sketch it is,"},{"Start":"08:41.890 ","End":"08:44.710","Text":"this interval together with this point."},{"Start":"08:44.710 ","End":"08:48.510","Text":"There\u0027s a solid dot here at minus 1,"},{"Start":"08:48.510 ","End":"08:51.225","Text":"and this goes on indefinitely there."},{"Start":"08:51.225 ","End":"08:54.050","Text":"Here I take this point here,"},{"Start":"08:54.050 ","End":"08:57.230","Text":"the 2, and to the right."},{"Start":"08:57.230 ","End":"09:00.410","Text":"If I write this in terms of x,"},{"Start":"09:00.410 ","End":"09:05.690","Text":"this says that x less than or equal to 1,"},{"Start":"09:05.690 ","End":"09:09.795","Text":"or x bigger or equal to 2."},{"Start":"09:09.795 ","End":"09:13.850","Text":"This is the solution just for the first inequality."},{"Start":"09:13.850 ","End":"09:16.085","Text":"Now we have to complete the second."},{"Start":"09:16.085 ","End":"09:17.884","Text":"On this number line,"},{"Start":"09:17.884 ","End":"09:25.590","Text":"I mark minus 2 and 3 and some dotted lines."},{"Start":"09:25.590 ","End":"09:29.210","Text":"Now we will need to take sample points."},{"Start":"09:29.210 ","End":"09:32.875","Text":"I\u0027ll take a sample point minus 3,"},{"Start":"09:32.875 ","End":"09:37.050","Text":"0, 4 If I substitute minus 3."},{"Start":"09:37.050 ","End":"09:38.490","Text":"Minus 3, minus 3 is negative,"},{"Start":"09:38.490 ","End":"09:40.485","Text":"minus 3 plus 2 is negative."},{"Start":"09:40.485 ","End":"09:42.135","Text":"Here it\u0027s positive."},{"Start":"09:42.135 ","End":"09:44.700","Text":"Of course at minus 2, it\u0027s 0."},{"Start":"09:44.700 ","End":"09:46.130","Text":"Minus 2 plus 2 is 0."},{"Start":"09:46.130 ","End":"09:47.480","Text":"But that\u0027s how we got these."},{"Start":"09:47.480 ","End":"09:49.760","Text":"These are the roots and here is 0,"},{"Start":"09:49.760 ","End":"09:54.515","Text":"of course also and if I try the sample point 0,"},{"Start":"09:54.515 ","End":"09:58.595","Text":"I get a negative and a positive which is a negative."},{"Start":"09:58.595 ","End":"10:00.905","Text":"This applies to this whole interval."},{"Start":"10:00.905 ","End":"10:04.370","Text":"For both of these are positive and I get"},{"Start":"10:04.370 ","End":"10:10.545","Text":"a positive all the way from 3 to infinity anyway."},{"Start":"10:10.545 ","End":"10:13.310","Text":"Now we have to look at what we\u0027re collecting."},{"Start":"10:13.310 ","End":"10:16.040","Text":"We\u0027re interested in less than or equal to 0,"},{"Start":"10:16.040 ","End":"10:19.565","Text":"which means we are collecting negatives and 0s."},{"Start":"10:19.565 ","End":"10:23.315","Text":"There\u0027s a negative, there\u0027s a 0, there\u0027s a 0."},{"Start":"10:23.315 ","End":"10:31.085","Text":"If I write down what this means as a picture it\u0027s from here to here,"},{"Start":"10:31.085 ","End":"10:34.730","Text":"including, that\u0027s why it\u0027s solid dots."},{"Start":"10:34.730 ","End":"10:37.220","Text":"As an expression in x,"},{"Start":"10:37.220 ","End":"10:42.289","Text":"it is minus 2 less than or equal to x,"},{"Start":"10:42.289 ","End":"10:44.950","Text":"less than or equal to 3."},{"Start":"10:44.950 ","End":"10:48.650","Text":"This is the solution to the second inequality,"},{"Start":"10:48.650 ","End":"10:51.905","Text":"just like this was the solution to the first inequality."},{"Start":"10:51.905 ","End":"10:55.460","Text":"What I have to do now is to combine"},{"Start":"10:55.460 ","End":"10:59.900","Text":"them with an and then we\u0027ll do that on a separate diagram."},{"Start":"10:59.900 ","End":"11:03.170","Text":"I want to sketch these 2 on here,"},{"Start":"11:03.170 ","End":"11:08.495","Text":"so I need to mark the points, minus 2."},{"Start":"11:08.495 ","End":"11:10.205","Text":"This should be a minus,"},{"Start":"11:10.205 ","End":"11:18.770","Text":"so minus 1 and the next in order is 2 and then we have a 3."},{"Start":"11:18.770 ","End":"11:23.565","Text":"This one I sketch as let\u0027s see,"},{"Start":"11:23.565 ","End":"11:27.435","Text":"minus 1 and onwards."},{"Start":"11:27.435 ","End":"11:34.725","Text":"The other part from 2 also to the right and onwards indefinitely."},{"Start":"11:34.725 ","End":"11:39.660","Text":"Then this one is from minus 2"},{"Start":"11:39.660 ","End":"11:45.045","Text":"up to 3 inclusive say,"},{"Start":"11:45.045 ","End":"11:49.730","Text":"this, and now we just have to apply the and,"},{"Start":"11:49.730 ","End":"11:51.545","Text":"which means the overlap."},{"Start":"11:51.545 ","End":"11:53.840","Text":"I\u0027ll do that in a different color."},{"Start":"11:53.840 ","End":"11:58.025","Text":"This part here including the endpoints,"},{"Start":"11:58.025 ","End":"12:00.590","Text":"in this part here including the endpoints."},{"Start":"12:00.590 ","End":"12:02.780","Text":"That\u0027s the overlap, that\u0027s the and."},{"Start":"12:02.780 ","End":"12:07.790","Text":"The answer can be written as minus 2 less than or"},{"Start":"12:07.790 ","End":"12:12.730","Text":"equal to x less than or equal to minus 1."},{"Start":"12:12.730 ","End":"12:17.000","Text":"This bit here should have drawn the and on the graph."},{"Start":"12:17.000 ","End":"12:19.190","Text":"Let\u0027s see here."},{"Start":"12:19.190 ","End":"12:22.200","Text":"Solid dot line,"},{"Start":"12:22.200 ","End":"12:29.045","Text":"and the other bit which we write with an or is 2 less than or equal to x,"},{"Start":"12:29.045 ","End":"12:31.370","Text":"less than or equal to 3."},{"Start":"12:31.370 ","End":"12:34.415","Text":"This is the final answer to the question,"},{"Start":"12:34.415 ","End":"12:38.370","Text":"and I think it\u0027s time to take a break."}],"Thumbnail":null,"ID":5082},{"Watched":false,"Name":"Inequalities with Absolute Value (continued)","Duration":"12m 30s","ChapterTopicVideoID":5083,"CourseChapterTopicPlaylistID":3878,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.270","Text":"Here we are again with Inequalities with Absolute Value."},{"Start":"00:03.270 ","End":"00:06.570","Text":"I\u0027ll just write the word continued here."},{"Start":"00:06.570 ","End":"00:11.850","Text":"Last time we mostly did this inequality."},{"Start":"00:11.850 ","End":"00:16.170","Text":"And we had a variation of this where if this was less than or equal to"},{"Start":"00:16.170 ","End":"00:21.240","Text":"and here also less than or equal to almost the same thing."},{"Start":"00:21.240 ","End":"00:23.520","Text":"Now we\u0027re going to see what happens when we have"},{"Start":"00:23.520 ","End":"00:28.080","Text":"a greater than inequality and the answer is this."},{"Start":"00:28.080 ","End":"00:30.690","Text":"If we have an expression of the form,"},{"Start":"00:30.690 ","End":"00:34.575","Text":"absolute value of a is bigger than b."},{"Start":"00:34.575 ","End":"00:39.090","Text":"What that gives us is not this double inequality,"},{"Start":"00:39.090 ","End":"00:46.805","Text":"but an all situation we have that either a is less than"},{"Start":"00:46.805 ","End":"00:55.610","Text":"minus b or a is greater than b. I\u0027ll give you a brief explanation."},{"Start":"00:55.610 ","End":"00:58.145","Text":"I mean, suppose b was 3,"},{"Start":"00:58.145 ","End":"01:01.610","Text":"and if I say the absolute value of a is bigger than 3,"},{"Start":"01:01.610 ","End":"01:05.135","Text":"I mean the distance of a from 0 is bigger than 3."},{"Start":"01:05.135 ","End":"01:10.095","Text":"There\u0027s two places that I can be further away from 0 than 3,"},{"Start":"01:10.095 ","End":"01:11.900","Text":"it\u0027s either on the right side,"},{"Start":"01:11.900 ","End":"01:14.000","Text":"in which case it\u0027s bigger than 3,"},{"Start":"01:14.000 ","End":"01:15.560","Text":"or it\u0027s on the left side,"},{"Start":"01:15.560 ","End":"01:18.445","Text":"in which case it\u0027s less than minus 3."},{"Start":"01:18.445 ","End":"01:21.435","Text":"This is basically what we get."},{"Start":"01:21.435 ","End":"01:27.830","Text":"As before, there\u0027s a variation that if here it was bigger or equal to,"},{"Start":"01:27.830 ","End":"01:32.480","Text":"then here also would be the non-strict inequalities,"},{"Start":"01:32.480 ","End":"01:35.615","Text":"a would be less than or equal to minus b and here,"},{"Start":"01:35.615 ","End":"01:38.185","Text":"a bigger or equal to b."},{"Start":"01:38.185 ","End":"01:41.930","Text":"Let me take a couple of examples to illustrate"},{"Start":"01:41.930 ","End":"01:46.415","Text":"this new formula or pair of formulas as you want to look at it."},{"Start":"01:46.415 ","End":"01:49.520","Text":"I just copy two exercises from above and we\u0027ll change"},{"Start":"01:49.520 ","End":"01:53.645","Text":"the direction instead of less than and less than or equal to."},{"Start":"01:53.645 ","End":"01:55.730","Text":"I\u0027ll put greater than,"},{"Start":"01:55.730 ","End":"01:59.520","Text":"greater than or equal to and let\u0027s see what we get."},{"Start":"01:59.520 ","End":"02:04.250","Text":"Using this formula, with x minus 3 being a,"},{"Start":"02:04.250 ","End":"02:11.790","Text":"this one implies that either x minus 3 is less than"},{"Start":"02:11.790 ","End":"02:19.905","Text":"minus 4 or x minus 3 is bigger than 4."},{"Start":"02:19.905 ","End":"02:25.655","Text":"If I just modify each of these by adding 3 to both sides,"},{"Start":"02:25.655 ","End":"02:30.680","Text":"then this one gives me that x is less than minus"},{"Start":"02:30.680 ","End":"02:38.220","Text":"1 and the other one gives me x bigger than 7."},{"Start":"02:38.220 ","End":"02:42.105","Text":"This is the answer and its in oral form."},{"Start":"02:42.105 ","End":"02:45.950","Text":"The next example, again,"},{"Start":"02:45.950 ","End":"02:52.835","Text":"I just follow the formula for the second case in the bigger or equal to k. I get that"},{"Start":"02:52.835 ","End":"03:00.885","Text":"either 5 minus 2x is less than or equal to minus 3,"},{"Start":"03:00.885 ","End":"03:07.740","Text":"or 5 minus 2x is bigger or equal to 3."},{"Start":"03:07.740 ","End":"03:12.960","Text":"Let\u0027s see. We can subtract 5 and get minus"},{"Start":"03:12.960 ","End":"03:18.945","Text":"2x less than or equal to minus 8."},{"Start":"03:18.945 ","End":"03:21.800","Text":"You what I\u0027ll work them both simultaneously in here,"},{"Start":"03:21.800 ","End":"03:25.535","Text":"if I subtract 5 minus 2x,"},{"Start":"03:25.535 ","End":"03:30.700","Text":"bigger or equal to 3 minus 5 is minus 2."},{"Start":"03:30.700 ","End":"03:37.865","Text":"Now, I\u0027m going to divide both sides by minus 2 but minus 2 is negative."},{"Start":"03:37.865 ","End":"03:41.510","Text":"I change the sign to bigger or equal to,"},{"Start":"03:41.510 ","End":"03:48.225","Text":"and I divide by minus 2 gives me x bigger or equal to 4 minus 8,"},{"Start":"03:48.225 ","End":"03:50.460","Text":"minus 2 is 4."},{"Start":"03:50.460 ","End":"03:56.945","Text":"The minus two is negative so I changed the sign this time from bigger or equal to,"},{"Start":"03:56.945 ","End":"03:58.765","Text":"less than or equal to."},{"Start":"03:58.765 ","End":"04:03.975","Text":"I get x less than or equal to 1."},{"Start":"04:03.975 ","End":"04:06.355","Text":"So not to forget the or."},{"Start":"04:06.355 ","End":"04:08.495","Text":"So this would be the answer,"},{"Start":"04:08.495 ","End":"04:12.620","Text":"x bigger or equal to 4 or x less than or equal to 1."},{"Start":"04:12.620 ","End":"04:16.775","Text":"I don\u0027t know why, but I would prefer to write it the other way round."},{"Start":"04:16.775 ","End":"04:25.985","Text":"That x is less than or equal to 1 or x bigger or equal to 4 somehow looks nicer to me."},{"Start":"04:25.985 ","End":"04:31.280","Text":"Why don\u0027t I highlight the solution and I should have done that here also,"},{"Start":"04:31.280 ","End":"04:32.880","Text":"and that\u0027s the answer."},{"Start":"04:32.880 ","End":"04:35.989","Text":"We\u0027ve illustrated the second formula."},{"Start":"04:35.989 ","End":"04:38.915","Text":"Of course, if it wasn\u0027t something linear,"},{"Start":"04:38.915 ","End":"04:43.035","Text":"like had x^2 then it wouldn\u0027t be a z, z."},{"Start":"04:43.035 ","End":"04:46.775","Text":"We\u0027ll see maybe some examples of that later on."},{"Start":"04:46.775 ","End":"04:49.730","Text":"I\u0027d like to get back to this pair of formulas."},{"Start":"04:49.730 ","End":"04:52.970","Text":"Well, there\u0027s 2 or 4 formulas depending how you look at it."},{"Start":"04:52.970 ","End":"04:59.180","Text":"I wanted to discuss some special cases which don\u0027t occur that often."},{"Start":"04:59.180 ","End":"05:01.010","Text":"But up till now,"},{"Start":"05:01.010 ","End":"05:02.660","Text":"in all our examples,"},{"Start":"05:02.660 ","End":"05:06.215","Text":"we have taken b to be a positive number."},{"Start":"05:06.215 ","End":"05:09.905","Text":"These formulas are most useful when b is a positive number,"},{"Start":"05:09.905 ","End":"05:13.115","Text":"and they work for all values."},{"Start":"05:13.115 ","End":"05:15.829","Text":"But if b is 0 or negative,"},{"Start":"05:15.829 ","End":"05:17.810","Text":"we don\u0027t usually use these."},{"Start":"05:17.810 ","End":"05:22.655","Text":"It usually turns out to be much simpler and I\u0027ll give you some examples."},{"Start":"05:22.655 ","End":"05:29.480","Text":"Let\u0027s first of all take the example of b=0."},{"Start":"05:29.480 ","End":"05:32.120","Text":"Let\u0027s suppose I have an example."},{"Start":"05:32.120 ","End":"05:36.710","Text":"X minus 3 in absolute value is less than,"},{"Start":"05:36.710 ","End":"05:39.619","Text":"and we said b is 0 less than 0."},{"Start":"05:39.619 ","End":"05:43.505","Text":"I think we could solve this without using any formulas because look,"},{"Start":"05:43.505 ","End":"05:48.500","Text":"absolute value of anything is always bigger or equal to 0."},{"Start":"05:48.500 ","End":"05:52.310","Text":"I mean, in general, whatever a is,"},{"Start":"05:52.310 ","End":"05:55.520","Text":"it\u0027s always going to be true that the absolute value of"},{"Start":"05:55.520 ","End":"06:00.335","Text":"a is bigger or equal to 0, whatever a is."},{"Start":"06:00.335 ","End":"06:02.750","Text":"Absolute value of something,"},{"Start":"06:02.750 ","End":"06:04.700","Text":"if it\u0027s bigger or equal to 0,"},{"Start":"06:04.700 ","End":"06:07.505","Text":"it can\u0027t be less than 0."},{"Start":"06:07.505 ","End":"06:12.245","Text":"In this case, we could immediately write no solution."},{"Start":"06:12.245 ","End":"06:15.815","Text":"If you decided that you wanted to do it with the formula."},{"Start":"06:15.815 ","End":"06:17.540","Text":"I mean, you could, I mean,"},{"Start":"06:17.540 ","End":"06:20.450","Text":"what you would get would be c,"},{"Start":"06:20.450 ","End":"06:27.710","Text":"you\u0027d get x minus 3 is on the one hand less than 0."},{"Start":"06:27.710 ","End":"06:31.250","Text":"The other hand bigger than minus 0,"},{"Start":"06:31.250 ","End":"06:34.160","Text":"minus 0 is just 0."},{"Start":"06:34.160 ","End":"06:36.850","Text":"Let\u0027s just drop the minus."},{"Start":"06:36.850 ","End":"06:38.510","Text":"If you looked at this pair,"},{"Start":"06:38.510 ","End":"06:40.895","Text":"you\u0027d get x bigger than 3,"},{"Start":"06:40.895 ","End":"06:43.355","Text":"or 3 less than x."},{"Start":"06:43.355 ","End":"06:44.450","Text":"If you looked at this pair,"},{"Start":"06:44.450 ","End":"06:46.520","Text":"you\u0027d get x less than 3."},{"Start":"06:46.520 ","End":"06:49.850","Text":"You\u0027d see immediately, since there\u0027s end between them,"},{"Start":"06:49.850 ","End":"06:52.610","Text":"we can\u0027t have x bigger than 3 and less than 3."},{"Start":"06:52.610 ","End":"06:55.054","Text":"We\u0027d see straight away that there\u0027s no solution."},{"Start":"06:55.054 ","End":"07:01.025","Text":"What I\u0027m saying is that there\u0027s no need to use the formula in simple cases like this."},{"Start":"07:01.025 ","End":"07:02.570","Text":"Did it copy paste here?"},{"Start":"07:02.570 ","End":"07:07.235","Text":"Let\u0027s discuss all the 4 cases and see we\u0027ve got all bases covered."},{"Start":"07:07.235 ","End":"07:11.240","Text":"What if I change this one to say less than or equal to 0?"},{"Start":"07:11.240 ","End":"07:15.380","Text":"Well, like I said, the absolute value is always bigger or equal to 0."},{"Start":"07:15.380 ","End":"07:17.900","Text":"If it\u0027s also less than or equal to 0,"},{"Start":"07:17.900 ","End":"07:19.354","Text":"then without the formulas,"},{"Start":"07:19.354 ","End":"07:24.155","Text":"we could say that absolute value of x minus 3 has to equal 0."},{"Start":"07:24.155 ","End":"07:27.530","Text":"Because if it\u0027s bigger or equal to 0 and also less than or equal to 0."},{"Start":"07:27.530 ","End":"07:33.965","Text":"Then from this, we would get that the only possibility is that x minus 3 is 0,"},{"Start":"07:33.965 ","End":"07:39.380","Text":"because the only thing was absolute value of 0 is 0 and then we would say x equals 3."},{"Start":"07:39.380 ","End":"07:41.150","Text":"There\u0027s one single solution."},{"Start":"07:41.150 ","End":"07:43.040","Text":"It\u0027s not an inequality, it\u0027s not a range,"},{"Start":"07:43.040 ","End":"07:46.655","Text":"it\u0027s just one solution for this inequality."},{"Start":"07:46.655 ","End":"07:53.105","Text":"What if I said absolute value of x minus 3 is bigger than 0?"},{"Start":"07:53.105 ","End":"08:00.005","Text":"Well, absolute value of x minus 3 is always bigger or equal to 0."},{"Start":"08:00.005 ","End":"08:03.410","Text":"If I want it to be strictly bigger than 0,"},{"Start":"08:03.410 ","End":"08:10.080","Text":"then what I get is everything except x equals 3."},{"Start":"08:10.080 ","End":"08:17.420","Text":"What you get here is every value of x is true except for 3."},{"Start":"08:17.420 ","End":"08:19.575","Text":"I could have put a middle step here and said look,"},{"Start":"08:19.575 ","End":"08:21.740","Text":"this is always bigger or equal to 0."},{"Start":"08:21.740 ","End":"08:25.220","Text":"To say that it\u0027s bigger than 0 means to rule out 0\u0027s."},{"Start":"08:25.220 ","End":"08:26.300","Text":"I could have said, well,"},{"Start":"08:26.300 ","End":"08:29.300","Text":"x minus 3 is not equal to 0."},{"Start":"08:29.300 ","End":"08:34.595","Text":"The only time it\u0027s 0 is when x is 3 so we\u0027re ruling that out and asked for the last one."},{"Start":"08:34.595 ","End":"08:37.700","Text":"The absolute value already is bigger or equal to 0."},{"Start":"08:37.700 ","End":"08:41.675","Text":"This doesn\u0027t add anything for any x I put here."},{"Start":"08:41.675 ","End":"08:45.500","Text":"This would be true always,"},{"Start":"08:45.500 ","End":"08:48.320","Text":"Let\u0027s say for every x is a solution,"},{"Start":"08:48.320 ","End":"08:52.475","Text":"every x, so all x, maybe that\u0027s better."},{"Start":"08:52.475 ","End":"08:54.350","Text":"This is true for all x."},{"Start":"08:54.350 ","End":"08:56.720","Text":"Now as I say, in each one of these cases,"},{"Start":"08:56.720 ","End":"08:59.375","Text":"you could blindly follow the formula."},{"Start":"08:59.375 ","End":"09:01.730","Text":"But when you get a 0 here,"},{"Start":"09:01.730 ","End":"09:04.700","Text":"it\u0027s just by common sense because we know the absolute value of"},{"Start":"09:04.700 ","End":"09:08.705","Text":"x is bigger or equal to 0 and then we get these."},{"Start":"09:08.705 ","End":"09:10.685","Text":"Again just to be thorough,"},{"Start":"09:10.685 ","End":"09:15.335","Text":"what I\u0027m going to do is take the case where b is negative."},{"Start":"09:15.335 ","End":"09:18.065","Text":"B is negative."},{"Start":"09:18.065 ","End":"09:20.480","Text":"This time for a, I\u0027ll take something different."},{"Start":"09:20.480 ","End":"09:26.915","Text":"Let\u0027s take 2x minus 4 less than something negative,"},{"Start":"09:26.915 ","End":"09:29.730","Text":"let\u0027s say minus 6."},{"Start":"09:29.730 ","End":"09:34.250","Text":"The common sense way without using formulas would just be to say,"},{"Start":"09:34.250 ","End":"09:36.395","Text":"okay, have an absolute value."},{"Start":"09:36.395 ","End":"09:40.310","Text":"Absolute value of something is always bigger or equal to 0,"},{"Start":"09:40.310 ","End":"09:43.740","Text":"non-negative, it can\u0027t be less than minus 6."},{"Start":"09:43.740 ","End":"09:47.660","Text":"I would write right away, no solution."},{"Start":"09:47.660 ","End":"09:53.195","Text":"But if you insisted on using the formula, it would work."},{"Start":"09:53.195 ","End":"09:57.170","Text":"I\u0027ll show you that I don\u0027t recommend doing it this way, just blindly."},{"Start":"09:57.170 ","End":"10:01.040","Text":"But yeah, you could look at the formula here and say, okay,"},{"Start":"10:01.040 ","End":"10:07.595","Text":"I have absolute value of a less than b. I could say minus b,"},{"Start":"10:07.595 ","End":"10:09.785","Text":"which is 6,"},{"Start":"10:09.785 ","End":"10:12.290","Text":"is less than a,"},{"Start":"10:12.290 ","End":"10:14.900","Text":"which is 2x minus 4,"},{"Start":"10:14.900 ","End":"10:17.690","Text":"and less than b,"},{"Start":"10:17.690 ","End":"10:19.205","Text":"which is minus 6."},{"Start":"10:19.205 ","End":"10:23.270","Text":"Already, something\u0027s looking wrong here because 6 is not less than minus 6."},{"Start":"10:23.270 ","End":"10:26.915","Text":"But suppose you persevered and solve this pair,"},{"Start":"10:26.915 ","End":"10:29.345","Text":"you\u0027d get from the first two,"},{"Start":"10:29.345 ","End":"10:32.225","Text":"it gets 10 is less than 2x,"},{"Start":"10:32.225 ","End":"10:35.740","Text":"5 is less than x."},{"Start":"10:35.740 ","End":"10:37.310","Text":"If you solve this pair,"},{"Start":"10:37.310 ","End":"10:44.345","Text":"we\u0027d get 2x minus 4 less than minus 6 2x less than minus 2,"},{"Start":"10:44.345 ","End":"10:48.080","Text":"and you\u0027d get x less than minus 1."},{"Start":"10:48.080 ","End":"10:50.665","Text":"The b in the end and in the middle."},{"Start":"10:50.665 ","End":"10:57.380","Text":"I want to find x which at the same time is bigger than 5 and less than minus 1."},{"Start":"10:57.380 ","End":"11:00.080","Text":"Just as before we got something impossible,"},{"Start":"11:00.080 ","End":"11:02.405","Text":"we also find that this is impossible,"},{"Start":"11:02.405 ","End":"11:04.850","Text":"I\u0027m back to the no solution."},{"Start":"11:04.850 ","End":"11:07.400","Text":"What I\u0027m saying is when we have special cases that"},{"Start":"11:07.400 ","End":"11:09.995","Text":"b is not positive if it\u0027s 0 or negative,"},{"Start":"11:09.995 ","End":"11:13.430","Text":"you look at each case and use common sense where"},{"Start":"11:13.430 ","End":"11:18.890","Text":"the most important thing is that the absolute value of anything is bigger or equal to 0."},{"Start":"11:18.890 ","End":"11:21.935","Text":"If you just remember this, it\u0027ll go easy."},{"Start":"11:21.935 ","End":"11:24.424","Text":"Just for completeness, I\u0027ll do all the cases."},{"Start":"11:24.424 ","End":"11:27.770","Text":"Take less than or equal to, greater than,"},{"Start":"11:27.770 ","End":"11:29.270","Text":"greater than or equal to,"},{"Start":"11:29.270 ","End":"11:32.815","Text":"and just using common sense will solve all these."},{"Start":"11:32.815 ","End":"11:37.225","Text":"Absolute value is always bigger or equal to 0."},{"Start":"11:37.225 ","End":"11:40.130","Text":"There\u0027s no way it\u0027s going to be less than or equal"},{"Start":"11:40.130 ","End":"11:42.935","Text":"to minus 6 can\u0027t even be equal minus 6."},{"Start":"11:42.935 ","End":"11:45.995","Text":"Here we had a couple of borderline cases,"},{"Start":"11:45.995 ","End":"11:48.005","Text":"but here there\u0027s nothing border line,"},{"Start":"11:48.005 ","End":"11:51.290","Text":"so there\u0027s still no solution."},{"Start":"11:51.290 ","End":"11:56.450","Text":"In these two, absolute value is bigger or equal to 0."},{"Start":"11:56.450 ","End":"12:00.260","Text":"So it\u0027s certainly going to be bigger than minus 6."},{"Start":"12:00.260 ","End":"12:02.645","Text":"Bigger or equal to minus 6 in this case,"},{"Start":"12:02.645 ","End":"12:05.635","Text":"the answer would be all x."},{"Start":"12:05.635 ","End":"12:08.525","Text":"We\u0027re just using common sense,"},{"Start":"12:08.525 ","End":"12:10.520","Text":"I\u0027m not forgetting that,"},{"Start":"12:10.520 ","End":"12:16.310","Text":"remembering that absolute value of something is always bigger or equal to 0 so much for"},{"Start":"12:16.310 ","End":"12:24.785","Text":"special cases and in practice will usually get it with this B and being positive."},{"Start":"12:24.785 ","End":"12:28.490","Text":"Let\u0027s move on. On second thought is looking at the time,"},{"Start":"12:28.490 ","End":"12:30.870","Text":"I think we\u0027ll take a break here."}],"Thumbnail":null,"ID":5083},{"Watched":false,"Name":"Exercise 1","Duration":"3m 42s","ChapterTopicVideoID":5084,"CourseChapterTopicPlaylistID":3878,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.350","Text":"This exercise is made up of 4 separate inequalities,"},{"Start":"00:04.350 ","End":"00:06.540","Text":"each involving the absolute value."},{"Start":"00:06.540 ","End":"00:09.765","Text":"In fact, notice that they\u0027re all less than."},{"Start":"00:09.765 ","End":"00:13.155","Text":"There is a formula that I\u0027m going to use that you"},{"Start":"00:13.155 ","End":"00:17.305","Text":"learned in the theory part and it is as follows."},{"Start":"00:17.305 ","End":"00:20.025","Text":"Well, first I\u0027ll scroll down to the first exercise."},{"Start":"00:20.025 ","End":"00:27.089","Text":"Yeah, and it goes like this absolute value of a less than b, in general,"},{"Start":"00:27.089 ","End":"00:28.725","Text":"it could be numbers or variables,"},{"Start":"00:28.725 ","End":"00:31.200","Text":"is equivalent to saying,"},{"Start":"00:31.200 ","End":"00:36.910","Text":"well, we need it in 1 direction that a is between -b and b."},{"Start":"00:37.670 ","End":"00:44.740","Text":"This actually is the formula that we\u0027re going to use in all 4 parts in this exercise."},{"Start":"00:44.740 ","End":"00:46.175","Text":"Let\u0027s begin here."},{"Start":"00:46.175 ","End":"00:49.820","Text":"A will take as x-4 and b is 5."},{"Start":"00:49.820 ","End":"00:58.340","Text":"We just write minus 5 less than x minus 4 less than 5."},{"Start":"00:58.340 ","End":"01:00.590","Text":"This is one of those cases where we can actually work on"},{"Start":"01:00.590 ","End":"01:04.025","Text":"all 3 parts of the inequalities simultaneously."},{"Start":"01:04.025 ","End":"01:07.415","Text":"Let\u0027s add 4 to all 3 sides."},{"Start":"01:07.415 ","End":"01:08.960","Text":"If we add 4,"},{"Start":"01:08.960 ","End":"01:18.670","Text":"then we get x less than 5 plus 4 is 9 and greater than minus 5 plus 4 is minus 1."},{"Start":"01:18.670 ","End":"01:21.725","Text":"That\u0027s the answer. Very simple."},{"Start":"01:21.725 ","End":"01:23.150","Text":"Onto the next."},{"Start":"01:23.150 ","End":"01:25.324","Text":"Here we\u0027ll use the same formula."},{"Start":"01:25.324 ","End":"01:29.810","Text":"Only a this time will be x plus 3."},{"Start":"01:29.810 ","End":"01:32.780","Text":"We get that x plus 3,"},{"Start":"01:32.780 ","End":"01:35.945","Text":"the middle bit, and then bigger than, less than."},{"Start":"01:35.945 ","End":"01:37.600","Text":"Here we have the 7,"},{"Start":"01:37.600 ","End":"01:39.895","Text":"and here we have minus 7."},{"Start":"01:39.895 ","End":"01:41.975","Text":"Once again, it\u0027s straightforward,"},{"Start":"01:41.975 ","End":"01:44.255","Text":"which only have x in the middle and it\u0027s linear."},{"Start":"01:44.255 ","End":"01:46.935","Text":"We can just work on all 3 together."},{"Start":"01:46.935 ","End":"01:48.660","Text":"Subtract 3 this time."},{"Start":"01:48.660 ","End":"01:50.480","Text":"If we subtract 3 from everything here,"},{"Start":"01:50.480 ","End":"01:51.995","Text":"we\u0027re left with just x."},{"Start":"01:51.995 ","End":"01:53.345","Text":"Here we get 4,"},{"Start":"01:53.345 ","End":"01:55.265","Text":"and here we get minus 10."},{"Start":"01:55.265 ","End":"01:57.160","Text":"That\u0027s the answer."},{"Start":"01:57.160 ","End":"02:00.830","Text":"I\u0027m just going to copy this before I start the last part."},{"Start":"02:00.830 ","End":"02:03.245","Text":"We might need this again."},{"Start":"02:03.245 ","End":"02:08.040","Text":"Here we are in part c. We have the same formula."},{"Start":"02:08.040 ","End":"02:09.920","Text":"This time there\u0027s a 3x."},{"Start":"02:09.920 ","End":"02:12.725","Text":"We\u0027ll see if that makes any big difference."},{"Start":"02:12.725 ","End":"02:22.620","Text":"Using this we get 3x minus 6 which is our a is between 18 and minus 18."},{"Start":"02:22.620 ","End":"02:25.115","Text":"Once again, we can work on everything simultaneously."},{"Start":"02:25.115 ","End":"02:33.303","Text":"Add 6 you get 3x less than 24, minus 12."},{"Start":"02:33.303 ","End":"02:35.945","Text":"This time we have to also do a division."},{"Start":"02:35.945 ","End":"02:37.545","Text":"Division by 3,"},{"Start":"02:37.545 ","End":"02:39.290","Text":"3 is a positive number,"},{"Start":"02:39.290 ","End":"02:42.350","Text":"so we keep the direction of the inequality."},{"Start":"02:42.350 ","End":"02:49.745","Text":"x less than 24 over 3 is 8, minus 12 over 3 is 4 and this is our answer."},{"Start":"02:49.745 ","End":"02:52.055","Text":"Similarly in the last one,"},{"Start":"02:52.055 ","End":"02:56.580","Text":"we have the 5x plus 16 is the a from the formula,"},{"Start":"02:56.580 ","End":"02:58.620","Text":"the b is for 41,"},{"Start":"02:58.620 ","End":"03:05.175","Text":"so it\u0027s between b and -b. Subtract 16 from everything."},{"Start":"03:05.175 ","End":"03:10.025","Text":"What do we get? Collecting 16 gives us 25,"},{"Start":"03:10.025 ","End":"03:14.606","Text":"subtracting 16 gives us minus 57,"},{"Start":"03:14.606 ","End":"03:16.780","Text":"then dividing by 5,"},{"Start":"03:16.780 ","End":"03:18.530","Text":"which is a positive number,"},{"Start":"03:18.530 ","End":"03:19.970","Text":"I say it each time,"},{"Start":"03:19.970 ","End":"03:21.980","Text":"because if it\u0027s negative, we need to reverse."},{"Start":"03:21.980 ","End":"03:26.730","Text":"We have x less than 25 over 5 is 5,"},{"Start":"03:27.160 ","End":"03:29.970","Text":"57 over 5 doesn\u0027t come out a whole number."},{"Start":"03:29.970 ","End":"03:33.375","Text":"5 into 55 goes 11 times,"},{"Start":"03:33.375 ","End":"03:37.731","Text":"just 2 left over, so it\u0027s -11 and 2 fifths."},{"Start":"03:37.731 ","End":"03:42.610","Text":"That\u0027s the solution to d. We are done."}],"Thumbnail":null,"ID":5084},{"Watched":false,"Name":"Exercise 2 parts a,b","Duration":"4m 4s","ChapterTopicVideoID":5085,"CourseChapterTopicPlaylistID":3878,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.340","Text":"In this exercise, we have 4 different inequalities to solve."},{"Start":"00:05.340 ","End":"00:09.690","Text":"They all are the form absolute value of something is less than something."},{"Start":"00:09.690 ","End":"00:14.550","Text":"What we\u0027ll need is the formula and I\u0027ll copy it from a previous exercise."},{"Start":"00:14.550 ","End":"00:16.225","Text":"Here it is, I won\u0027t go over,"},{"Start":"00:16.225 ","End":"00:17.760","Text":"we\u0027ve done it enough times already,"},{"Start":"00:17.760 ","End":"00:19.260","Text":"but others leave it there."},{"Start":"00:19.260 ","End":"00:21.240","Text":"Let\u0027s start with part a. I actually, I did,"},{"Start":"00:21.240 ","End":"00:26.565","Text":"I move this down where it\u0027s going to be more useful to me is further down here."},{"Start":"00:26.565 ","End":"00:30.540","Text":"Very well, and now it starts with part a."},{"Start":"00:30.540 ","End":"00:33.930","Text":"Using this formula, we start with the middle bit,"},{"Start":"00:33.930 ","End":"00:38.670","Text":"14-x and then inequalities,"},{"Start":"00:38.670 ","End":"00:42.255","Text":"and then here 7, here minus 7."},{"Start":"00:42.255 ","End":"00:47.045","Text":"We can work on the 3 parts of the inequalities simultaneously."},{"Start":"00:47.045 ","End":"00:49.909","Text":"Let\u0027s subtract 14 from everything."},{"Start":"00:49.909 ","End":"00:52.115","Text":"I just have minus x,"},{"Start":"00:52.115 ","End":"00:57.485","Text":"subtracting 14-7, subtracting 14-21."},{"Start":"00:57.485 ","End":"01:01.850","Text":"Now we divide both sides by a negative number."},{"Start":"01:01.850 ","End":"01:07.020","Text":"We\u0027re going to divide by -1 negative,"},{"Start":"01:07.020 ","End":"01:08.375","Text":"so we switched signs."},{"Start":"01:08.375 ","End":"01:10.010","Text":"I write the signs first,"},{"Start":"01:10.010 ","End":"01:15.165","Text":"so I don\u0027t forget afterward switching from less than to bigger than."},{"Start":"01:15.165 ","End":"01:16.790","Text":"Then dividing by -1,"},{"Start":"01:16.790 ","End":"01:20.210","Text":"just change the sign of everything here it\u0027s x here it\u0027s plus 7,"},{"Start":"01:20.210 ","End":"01:22.550","Text":"and here it\u0027s plus 21."},{"Start":"01:22.550 ","End":"01:24.690","Text":"This is the answer,"},{"Start":"01:24.690 ","End":"01:29.270","Text":"but some people like to look at it the other way."},{"Start":"01:29.270 ","End":"01:30.980","Text":"It\u0027s an alternative form,"},{"Start":"01:30.980 ","End":"01:32.915","Text":"instead of looking at it from here,"},{"Start":"01:32.915 ","End":"01:34.550","Text":"to look at it from here,"},{"Start":"01:34.550 ","End":"01:41.300","Text":"and then you get 7\u003cx\u003c21 is completely equivalent."},{"Start":"01:41.300 ","End":"01:44.405","Text":"It\u0027s just a matter of looking at it from the other direction."},{"Start":"01:44.405 ","End":"01:47.480","Text":"Either 1 of these 2 is your answer."},{"Start":"01:47.480 ","End":"01:49.835","Text":"Now let\u0027s go on to the next one."},{"Start":"01:49.835 ","End":"01:52.234","Text":"Keep the formula in view."},{"Start":"01:52.234 ","End":"01:54.380","Text":"Just apply it again."},{"Start":"01:54.380 ","End":"01:57.755","Text":"The bit between bars goes first,"},{"Start":"01:57.755 ","End":"01:59.780","Text":"then we put these inequalities,"},{"Start":"01:59.780 ","End":"02:03.860","Text":"and then here we put as is and then here with a minus."},{"Start":"02:03.860 ","End":"02:06.155","Text":"That\u0027s the mechanical way of doing it."},{"Start":"02:06.155 ","End":"02:09.995","Text":"This time we cannot and do it all in one go."},{"Start":"02:09.995 ","End":"02:12.815","Text":"I mean, work with all 3 parts simultaneously."},{"Start":"02:12.815 ","End":"02:19.040","Text":"There are variables, so we split it up into this part and this part."},{"Start":"02:19.040 ","End":"02:20.610","Text":"I have 2 things."},{"Start":"02:20.610 ","End":"02:28.385","Text":"Minus x is less than 12 minus x from this inequality and from this inequality,"},{"Start":"02:28.385 ","End":"02:37.160","Text":"I get, I\u0027ll put it over here,12 minus x is less than x. I\u0027ll put an and in-between them."},{"Start":"02:37.160 ","End":"02:40.085","Text":"This has to hold true and this has to hold true."},{"Start":"02:40.085 ","End":"02:41.975","Text":"In the first inequality,"},{"Start":"02:41.975 ","End":"02:43.700","Text":"if I bring this,"},{"Start":"02:43.700 ","End":"02:46.020","Text":"the xs to the left, they cancel out."},{"Start":"02:46.020 ","End":"02:52.459","Text":"Basically, this leaves me with 0 less than 12 without the x and as a statement,"},{"Start":"02:52.459 ","End":"02:54.140","Text":"it happens to be true."},{"Start":"02:54.140 ","End":"02:59.150","Text":"You could say, all x and some people even would say, I\u0027m not going to write it."},{"Start":"02:59.150 ","End":"03:01.490","Text":"x is between minus infinity and infinity,"},{"Start":"03:01.490 ","End":"03:03.650","Text":"meaning there\u0027s no restriction on x."},{"Start":"03:03.650 ","End":"03:07.985","Text":"Now the other one, 12 minus x less than x."},{"Start":"03:07.985 ","End":"03:10.684","Text":"We bring the xs to the right,"},{"Start":"03:10.684 ","End":"03:14.985","Text":"12 is less than 2x, and I usually bring the xs to the left, it doesn\u0027t matter,"},{"Start":"03:14.985 ","End":"03:17.850","Text":"divide by 2 which is positive,"},{"Start":"03:17.850 ","End":"03:21.090","Text":"6 is less than x, the x bigger than 6,"},{"Start":"03:21.090 ","End":"03:24.500","Text":"which is just the other way of looking at this same thing."},{"Start":"03:24.500 ","End":"03:26.870","Text":"We have 2 bits that we have to connect."},{"Start":"03:26.870 ","End":"03:32.750","Text":"We have to connect this bit with this bit by means of this and the thing that you"},{"Start":"03:32.750 ","End":"03:35.480","Text":"do if you have and with something that\u0027s always"},{"Start":"03:35.480 ","End":"03:39.215","Text":"true with it\u0027s true for all xs you just ignore this part,"},{"Start":"03:39.215 ","End":"03:42.990","Text":"x is bigger than 6, that\u0027s it, x bigger than 6."},{"Start":"03:43.000 ","End":"03:46.050","Text":"Now highlight that."},{"Start":"03:46.790 ","End":"03:53.900","Text":"Let me just copy the formula down here it\u0027s always useful to have at hand."},{"Start":"03:53.900 ","End":"03:59.770","Text":"There we go. Now we can serve as for c and d. In this one,"},{"Start":"03:59.770 ","End":"04:02.360","Text":"it\u0027s a bit different because it\u0027s going to be a quadratic here,"},{"Start":"04:02.360 ","End":"04:05.550","Text":"but this is still the same logic."}],"Thumbnail":null,"ID":5085},{"Watched":false,"Name":"Exercise 2 parts c,d","Duration":"12m 8s","ChapterTopicVideoID":5086,"CourseChapterTopicPlaylistID":3878,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.880","Text":"In this exercise, we have 4 different inequalities to solve."},{"Start":"00:05.880 ","End":"00:09.990","Text":"They all are the form absolute value of something is less than something."},{"Start":"00:09.990 ","End":"00:14.745","Text":"What we\u0027ll need is the formula and I\u0027ll copy it from a previous exercise."},{"Start":"00:14.745 ","End":"00:16.350","Text":"Here it is. I won\u0027t go over it."},{"Start":"00:16.350 ","End":"00:17.760","Text":"We\u0027ve done it enough times already,"},{"Start":"00:17.760 ","End":"00:19.470","Text":"but I\u0027ll just leave it there."},{"Start":"00:19.470 ","End":"00:23.670","Text":"Let\u0027s start with part a. I better move this down here where it\u0027s going"},{"Start":"00:23.670 ","End":"00:27.765","Text":"to be more useful to me as further down here. Very well."},{"Start":"00:27.765 ","End":"00:30.540","Text":"Now let\u0027s start with part a."},{"Start":"00:30.540 ","End":"00:33.930","Text":"Using this formula, we start with the middle bit,"},{"Start":"00:33.930 ","End":"00:36.360","Text":"14 minus x,"},{"Start":"00:36.360 ","End":"00:39.374","Text":"and then inequalities and then here,"},{"Start":"00:39.374 ","End":"00:42.255","Text":"7, here, minus 7."},{"Start":"00:42.255 ","End":"00:47.045","Text":"We can work on the 3 parts of the inequalities simultaneously."},{"Start":"00:47.045 ","End":"00:49.909","Text":"Let\u0027s subtract 14 from everything."},{"Start":"00:49.909 ","End":"00:52.115","Text":"I just have minus x,"},{"Start":"00:52.115 ","End":"00:55.160","Text":"subtracting 14 minus 7,"},{"Start":"00:55.160 ","End":"00:58.130","Text":"subtracting 14 minus 21."},{"Start":"00:58.130 ","End":"01:01.850","Text":"Now we divide both sides by a negative number."},{"Start":"01:01.850 ","End":"01:07.025","Text":"We\u0027re going to divide by minus 1 negative."},{"Start":"01:07.025 ","End":"01:08.375","Text":"So we switched signs."},{"Start":"01:08.375 ","End":"01:11.951","Text":"I write the signs first so I don\u0027t forget afterwards."},{"Start":"01:11.951 ","End":"01:15.165","Text":"Switching from less than to bigger than."},{"Start":"01:15.165 ","End":"01:16.790","Text":"Then dividing by minus 1,"},{"Start":"01:16.790 ","End":"01:18.080","Text":"just change the sign of everything."},{"Start":"01:18.080 ","End":"01:20.420","Text":"Here it\u0027s x, here it\u0027s plus 7,"},{"Start":"01:20.420 ","End":"01:22.550","Text":"and here it\u0027s plus 21."},{"Start":"01:22.550 ","End":"01:24.245","Text":"This is the answer."},{"Start":"01:24.245 ","End":"01:29.270","Text":"But some people like to look at it the other way."},{"Start":"01:29.270 ","End":"01:33.130","Text":"An alternative form is instead of looking at it from here,"},{"Start":"01:33.130 ","End":"01:34.550","Text":"to look at it from here,"},{"Start":"01:34.550 ","End":"01:39.940","Text":"and then you get 7 less than x, less than 21."},{"Start":"01:39.940 ","End":"01:41.300","Text":"It\u0027s completely equivalent."},{"Start":"01:41.300 ","End":"01:44.425","Text":"It\u0027s just a matter of looking at it from the other direction."},{"Start":"01:44.425 ","End":"01:47.480","Text":"Either one of these 2 is your answer."},{"Start":"01:47.480 ","End":"01:49.835","Text":"Now let\u0027s go on to the next one."},{"Start":"01:49.835 ","End":"01:52.234","Text":"Keep the formula in view."},{"Start":"01:52.234 ","End":"01:53.943","Text":"Just apply it."},{"Start":"01:53.943 ","End":"01:57.755","Text":"Again, the bit between bars goes first,"},{"Start":"01:57.755 ","End":"01:59.780","Text":"then we put these inequalities,"},{"Start":"01:59.780 ","End":"02:02.210","Text":"and then here we put as is,"},{"Start":"02:02.210 ","End":"02:03.890","Text":"and then here with a minus."},{"Start":"02:03.890 ","End":"02:06.155","Text":"That\u0027s the mechanical way of doing it."},{"Start":"02:06.155 ","End":"02:09.995","Text":"This time, we cannot and do it all in one go."},{"Start":"02:09.995 ","End":"02:14.045","Text":"I mean, work with all 3 parts simultaneously. There are variables."},{"Start":"02:14.045 ","End":"02:19.040","Text":"We split it up into this part and this part."},{"Start":"02:19.040 ","End":"02:20.630","Text":"I have 2 things."},{"Start":"02:20.630 ","End":"02:26.255","Text":"Minus x is less than 12 minus x from this inequality."},{"Start":"02:26.255 ","End":"02:30.710","Text":"From this inequality, I get, I\u0027ll put it over here,"},{"Start":"02:30.710 ","End":"02:37.180","Text":"12 minus x is less than x. I\u0027ll put an and in-between them."},{"Start":"02:37.180 ","End":"02:39.710","Text":"This has to hold true and this has to hold true."},{"Start":"02:39.710 ","End":"02:41.975","Text":"In the first inequality,"},{"Start":"02:41.975 ","End":"02:46.010","Text":"if I bring the x\u0027s to the left, they cancel out."},{"Start":"02:46.010 ","End":"02:52.459","Text":"Basically this leaves me with 0 less than 12 without the x and as a statement,"},{"Start":"02:52.459 ","End":"02:54.140","Text":"it happens to be true."},{"Start":"02:54.140 ","End":"02:56.675","Text":"You could say all x,"},{"Start":"02:56.675 ","End":"02:58.250","Text":"and some people even would say,"},{"Start":"02:58.250 ","End":"03:01.490","Text":"I\u0027m not going to write it, x is between minus infinity and infinity,"},{"Start":"03:01.490 ","End":"03:03.650","Text":"meaning there\u0027s no restriction on x."},{"Start":"03:03.650 ","End":"03:08.255","Text":"Now, the other one. 12 minus x less than x."},{"Start":"03:08.255 ","End":"03:10.349","Text":"We bring the x\u0027s to the right,"},{"Start":"03:10.349 ","End":"03:12.300","Text":"12 less than 2x,"},{"Start":"03:12.300 ","End":"03:14.025","Text":"and I usually bring the x\u0027s to the left,"},{"Start":"03:14.025 ","End":"03:17.580","Text":"it doesn\u0027t matter, divided by 2, which is positive."},{"Start":"03:17.580 ","End":"03:20.190","Text":"6 is less than x."},{"Start":"03:20.190 ","End":"03:21.650","Text":"The x bigger than 6,"},{"Start":"03:21.650 ","End":"03:24.600","Text":"which is just the other way of looking at this same thing."},{"Start":"03:24.600 ","End":"03:26.870","Text":"We have 2 bits that we have to connect."},{"Start":"03:26.870 ","End":"03:31.795","Text":"We have to connect this bit with this bit by means of this and."},{"Start":"03:31.795 ","End":"03:35.030","Text":"The thing that you do if you have and with something that\u0027s"},{"Start":"03:35.030 ","End":"03:39.020","Text":"always true where it\u0027s true for all x is that you just ignore this part."},{"Start":"03:39.020 ","End":"03:41.300","Text":"X is bigger than 6, that\u0027s it."},{"Start":"03:41.300 ","End":"03:43.765","Text":"X bigger than 6,"},{"Start":"03:43.765 ","End":"03:46.060","Text":"and I\u0027ll highlight that."},{"Start":"03:46.790 ","End":"03:53.900","Text":"Let me just copy the formula down here so it\u0027s useful to have at hand."},{"Start":"03:53.900 ","End":"03:55.580","Text":"There we go."},{"Start":"03:55.580 ","End":"03:59.780","Text":"Now we can solve as for c and d. In this one,"},{"Start":"03:59.780 ","End":"04:02.360","Text":"it\u0027s a bit different because it\u0027s going to be a quadratic here,"},{"Start":"04:02.360 ","End":"04:04.835","Text":"but this is still the same logic."},{"Start":"04:04.835 ","End":"04:06.920","Text":"If we apply this over here,"},{"Start":"04:06.920 ","End":"04:15.730","Text":"we get that x^2 minus x is between 6 and minus 6."},{"Start":"04:15.730 ","End":"04:19.520","Text":"As before, we split this into 2 inequalities."},{"Start":"04:19.520 ","End":"04:21.050","Text":"Instead of the double inequality,"},{"Start":"04:21.050 ","End":"04:30.785","Text":"we have minus 6 less than x squared minus x on the one hand and the other part,"},{"Start":"04:30.785 ","End":"04:34.915","Text":"x^2 minus x is less than 6,"},{"Start":"04:34.915 ","End":"04:36.860","Text":"and we do each bit separately."},{"Start":"04:36.860 ","End":"04:44.430","Text":"This one gives us x^6 minus x plus 6 is bigger than"},{"Start":"04:44.430 ","End":"04:48.245","Text":"0 and if you try to solve"},{"Start":"04:48.245 ","End":"04:52.280","Text":"this one using the quadratic equation in order to get its roots,"},{"Start":"04:52.280 ","End":"04:54.290","Text":"you will find that it doesn\u0027t have any roots."},{"Start":"04:54.290 ","End":"04:57.560","Text":"The b^2 minus 4ac is negative,"},{"Start":"04:57.560 ","End":"05:00.275","Text":"comes out minus 23 doesn\u0027t have a square root."},{"Start":"05:00.275 ","End":"05:02.825","Text":"There are no roots and as usual,"},{"Start":"05:02.825 ","End":"05:06.707","Text":"when there are no roots to the polynomial,"},{"Start":"05:06.707 ","End":"05:11.104","Text":"quadratic, it\u0027s either always positive or always negative."},{"Start":"05:11.104 ","End":"05:14.780","Text":"All you have to do is pick any value and I usually pick"},{"Start":"05:14.780 ","End":"05:18.791","Text":"0 to substitute just to see if it\u0027s always positive or always negative."},{"Start":"05:18.791 ","End":"05:23.970","Text":"0 gives us that this is 6, which is positive."},{"Start":"05:23.970 ","End":"05:26.670","Text":"This thing is always positive."},{"Start":"05:26.670 ","End":"05:30.230","Text":"The answer to the inequality is all x."},{"Start":"05:30.230 ","End":"05:31.520","Text":"That\u0027s this one."},{"Start":"05:31.520 ","End":"05:33.515","Text":"Now let\u0027s attend to this one."},{"Start":"05:33.515 ","End":"05:38.234","Text":"This one gives us that x^2 minus"},{"Start":"05:38.234 ","End":"05:44.180","Text":"x minus 6 is less than 0 and to save time,"},{"Start":"05:44.180 ","End":"05:46.430","Text":"I\u0027m going to give you the roots."},{"Start":"05:46.430 ","End":"05:51.525","Text":"The roots are 3 and minus 2 for x."},{"Start":"05:51.525 ","End":"05:58.245","Text":"This thing factorizes to be x minus 3,"},{"Start":"05:58.245 ","End":"06:01.005","Text":"x minus minus 2,"},{"Start":"06:01.005 ","End":"06:04.175","Text":"so x plus 2 less than 0."},{"Start":"06:04.175 ","End":"06:08.195","Text":"This helps us when we\u0027re substituting values to have it in this form."},{"Start":"06:08.195 ","End":"06:10.550","Text":"This time let\u0027s draw a number line."},{"Start":"06:10.550 ","End":"06:14.510","Text":"Here we are and let\u0027s pick a couple of points for these."},{"Start":"06:14.510 ","End":"06:17.420","Text":"The only thing is that they have to be the right way round."},{"Start":"06:17.420 ","End":"06:21.170","Text":"3 has to be on the right and minus 2 on the left."},{"Start":"06:21.170 ","End":"06:23.900","Text":"Now we\u0027ve got 3 separate intervals."},{"Start":"06:23.900 ","End":"06:26.210","Text":"This, this, and this."},{"Start":"06:26.210 ","End":"06:29.435","Text":"Let\u0027s pick a representative for each interval."},{"Start":"06:29.435 ","End":"06:35.015","Text":"In this interval, I\u0027ll choose 0, easiest to substitute."},{"Start":"06:35.015 ","End":"06:39.247","Text":"In this interval, just choose the next whole number along,"},{"Start":"06:39.247 ","End":"06:43.065","Text":"4, and here I\u0027ll choose minus 3."},{"Start":"06:43.065 ","End":"06:47.690","Text":"What I do is I substitute these numbers in here,"},{"Start":"06:47.690 ","End":"06:48.770","Text":"but better in here,"},{"Start":"06:48.770 ","End":"06:52.220","Text":"it\u0027s easier just to see positive or negative."},{"Start":"06:52.220 ","End":"06:54.560","Text":"If I substitute minus 3,"},{"Start":"06:54.560 ","End":"06:57.575","Text":"this is negative and this is also negative."},{"Start":"06:57.575 ","End":"07:01.760","Text":"Negative times negative is positive."},{"Start":"07:01.760 ","End":"07:10.270","Text":"If I substitute 0, I\u0027ve got minus times plus is minus and if I substitute 4,"},{"Start":"07:10.270 ","End":"07:12.590","Text":"this is positive and this is positive,"},{"Start":"07:12.590 ","End":"07:15.320","Text":"so plus, meaning bigger than 0,"},{"Start":"07:15.320 ","End":"07:17.620","Text":"less than 0 or bigger than 0."},{"Start":"07:17.620 ","End":"07:21.380","Text":"You should be able to see right away that the negative,"},{"Start":"07:21.380 ","End":"07:24.560","Text":"meaning less than 0, is between minus 2 and 3."},{"Start":"07:24.560 ","End":"07:28.880","Text":"But sometimes it helps to draw a little sketch that minus 3,"},{"Start":"07:28.880 ","End":"07:31.990","Text":"we draw something positive, and 0,"},{"Start":"07:31.990 ","End":"07:34.130","Text":"we draw something negative, and at 4,"},{"Start":"07:34.130 ","End":"07:40.800","Text":"we draw something positive and the roots are where the thing is 0."},{"Start":"07:40.800 ","End":"07:43.760","Text":"We just do a rough sketch just to get"},{"Start":"07:43.760 ","End":"07:47.600","Text":"an idea of where the thing is positive and negative."},{"Start":"07:47.600 ","End":"07:49.400","Text":"If we want less than 0,"},{"Start":"07:49.400 ","End":"07:50.955","Text":"we want this part."},{"Start":"07:50.955 ","End":"07:52.750","Text":"But as I say it,"},{"Start":"07:52.750 ","End":"07:57.460","Text":"drawing the sketch is optional and we can see it between minus 2 and 3,"},{"Start":"07:57.460 ","End":"08:01.900","Text":"and I\u0027ll just write that at the interval is minus 2, less than x,"},{"Start":"08:01.900 ","End":"08:03.190","Text":"less than 3,"},{"Start":"08:03.190 ","End":"08:08.755","Text":"not including the 2 and the 3 because these points it\u0027s 0 and we want strictly negative."},{"Start":"08:08.755 ","End":"08:11.425","Text":"That\u0027s the solution to the other part."},{"Start":"08:11.425 ","End":"08:15.065","Text":"Now we have to and these 2 together."},{"Start":"08:15.065 ","End":"08:18.040","Text":"Like I said, if you have something which is all x and you and it,"},{"Start":"08:18.040 ","End":"08:19.180","Text":"you just ignore that."},{"Start":"08:19.180 ","End":"08:21.775","Text":"It doesn\u0027t add any restriction to this."},{"Start":"08:21.775 ","End":"08:25.360","Text":"This on its own becomes the solution to the whole problem,"},{"Start":"08:25.360 ","End":"08:27.185","Text":"so I\u0027ll highlight it."},{"Start":"08:27.185 ","End":"08:29.565","Text":"We are done with this part,"},{"Start":"08:29.565 ","End":"08:33.420","Text":"so now we\u0027ll move on to the last part."},{"Start":"08:33.420 ","End":"08:35.930","Text":"We\u0027re still working with this scheme,"},{"Start":"08:35.930 ","End":"08:38.870","Text":"and we still have absolute value of x less than something,"},{"Start":"08:38.870 ","End":"08:47.426","Text":"so we use this to say that x^2 plus 2x is less than something and bigger than something."},{"Start":"08:47.426 ","End":"08:48.770","Text":"Here, the 8, here,"},{"Start":"08:48.770 ","End":"08:50.368","Text":"the minus 8,"},{"Start":"08:50.368 ","End":"08:55.160","Text":"and then we split it up as usual into 2 inequalities."},{"Start":"08:55.160 ","End":"09:01.710","Text":"The first one, minus 8 less than x^2 plus 2x and"},{"Start":"09:01.710 ","End":"09:09.300","Text":"the other one will be x^2 plus 2x less than 8."},{"Start":"09:09.300 ","End":"09:14.280","Text":"A slight rewrite, we get x^2 plus"},{"Start":"09:14.280 ","End":"09:19.950","Text":"2x plus 8 bigger than 0 and here,"},{"Start":"09:19.950 ","End":"09:27.410","Text":"we get x^2 plus 2x minus 8 less than 0."},{"Start":"09:27.410 ","End":"09:32.860","Text":"This one actually is almost the same as the one above and if you check,"},{"Start":"09:32.860 ","End":"09:34.675","Text":"it has no roots."},{"Start":"09:34.675 ","End":"09:38.420","Text":"If we substitute x equals 0,"},{"Start":"09:38.420 ","End":"09:40.400","Text":"for example, we get positive,"},{"Start":"09:40.400 ","End":"09:42.170","Text":"so it\u0027s always positive."},{"Start":"09:42.170 ","End":"09:43.790","Text":"Here, it\u0027s all x."},{"Start":"09:43.790 ","End":"09:46.235","Text":"Just look at part c, it\u0027s very similar,"},{"Start":"09:46.235 ","End":"09:48.385","Text":"just slightly different numbers."},{"Start":"09:48.385 ","End":"09:50.810","Text":"Here, as before we would"},{"Start":"09:50.810 ","End":"09:55.010","Text":"normally find the roots by solving the equation where this is 0,"},{"Start":"09:55.010 ","End":"09:56.600","Text":"but I\u0027ll do it for you,"},{"Start":"09:56.600 ","End":"10:01.895","Text":"the roots turn out to be minus 4 and 2."},{"Start":"10:01.895 ","End":"10:08.700","Text":"This means that this factorizes to be x minus minus 4,"},{"Start":"10:08.700 ","End":"10:14.050","Text":"plus 4, x minus 2 less than 0."},{"Start":"10:14.050 ","End":"10:16.040","Text":"Let\u0027s do it with a sketch."},{"Start":"10:16.040 ","End":"10:18.485","Text":"As I said, the sketch is optional."},{"Start":"10:18.485 ","End":"10:20.445","Text":"I\u0027ll draw a number line."},{"Start":"10:20.445 ","End":"10:24.500","Text":"Here it is. Let\u0027s put on these points in the right order."},{"Start":"10:24.500 ","End":"10:27.080","Text":"Somewhere, there\u0027s a minus 4."},{"Start":"10:27.080 ","End":"10:29.840","Text":"Somewhere, there\u0027s a 2 to the right of it."},{"Start":"10:29.840 ","End":"10:33.275","Text":"Then we pick some sample points."},{"Start":"10:33.275 ","End":"10:36.860","Text":"Here, I\u0027ll choose x equals 0."},{"Start":"10:36.860 ","End":"10:38.360","Text":"This by the way, is x,"},{"Start":"10:38.360 ","End":"10:40.265","Text":"and so it was here."},{"Start":"10:40.265 ","End":"10:43.705","Text":"Here, I\u0027ll just pick the next number along,"},{"Start":"10:43.705 ","End":"10:47.170","Text":"3, here, I\u0027ll pick minus 5."},{"Start":"10:47.170 ","End":"10:50.120","Text":"What I want to know is in each of the intervals,"},{"Start":"10:50.120 ","End":"10:51.410","Text":"what its sign is,"},{"Start":"10:51.410 ","End":"10:53.960","Text":"whether it\u0027s positive or negative."},{"Start":"10:53.960 ","End":"10:56.135","Text":"We\u0027ll go with the easy one, 0,"},{"Start":"10:56.135 ","End":"10:58.540","Text":"so I get 4 times minus 2,"},{"Start":"10:58.540 ","End":"11:04.475","Text":"positive times negative is negative or less than 0."},{"Start":"11:04.475 ","End":"11:06.075","Text":"When x is 3,"},{"Start":"11:06.075 ","End":"11:11.520","Text":"this is positive and 3 minus 2 is also positive, product of positives."},{"Start":"11:11.520 ","End":"11:13.650","Text":"When x is minus 5,"},{"Start":"11:13.650 ","End":"11:16.700","Text":"minus 5 plus 4 is negative and this is also negative."},{"Start":"11:16.700 ","End":"11:19.820","Text":"Negative times negative is positive."},{"Start":"11:19.820 ","End":"11:22.070","Text":"Let\u0027s skip the sketch this time."},{"Start":"11:22.070 ","End":"11:24.275","Text":"It\u0027s very much like the previous one."},{"Start":"11:24.275 ","End":"11:27.365","Text":"That it\u0027s positive here and negative here, positive here."},{"Start":"11:27.365 ","End":"11:28.865","Text":"Where it\u0027s negative,"},{"Start":"11:28.865 ","End":"11:30.230","Text":"negative is what we want."},{"Start":"11:30.230 ","End":"11:33.200","Text":"That\u0027s what less than 0 means, a negative."},{"Start":"11:33.200 ","End":"11:36.590","Text":"It\u0027s going to be between minus 4 and 2 and strictly,"},{"Start":"11:36.590 ","End":"11:41.075","Text":"negative means that we don\u0027t include minus 4 and 2."},{"Start":"11:41.075 ","End":"11:43.010","Text":"I\u0027ll write it over here."},{"Start":"11:43.010 ","End":"11:47.750","Text":"Minus 4 less than x less than 2."},{"Start":"11:47.750 ","End":"11:52.950","Text":"That\u0027s the second part and they\u0027re connected by and."},{"Start":"11:52.950 ","End":"11:55.970","Text":"As before when we have n with something that\u0027s all x,"},{"Start":"11:55.970 ","End":"11:57.800","Text":"we just take this one."},{"Start":"11:57.800 ","End":"12:01.430","Text":"What I\u0027m highlighting is the answer to the result of"},{"Start":"12:01.430 ","End":"12:05.668","Text":"the end and to the whole of this part d. That\u0027s the answer,"},{"Start":"12:05.668 ","End":"12:08.640","Text":"and d is the last one, so we are done."}],"Thumbnail":null,"ID":5086},{"Watched":false,"Name":"Exercise 3","Duration":"14m 59s","ChapterTopicVideoID":5087,"CourseChapterTopicPlaylistID":3878,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.860","Text":"Here we have 3 more inequalities to solve but this time each of"},{"Start":"00:04.860 ","End":"00:09.615","Text":"them is composed of 2 separate bits connected with AND,"},{"Start":"00:09.615 ","End":"00:11.400","Text":"and not only that,"},{"Start":"00:11.400 ","End":"00:17.010","Text":"we\u0027re used to the inequality with less than as in here, here, and here."},{"Start":"00:17.010 ","End":"00:20.655","Text":"But we also have inequalities with greater than,"},{"Start":"00:20.655 ","End":"00:26.070","Text":"so this formula which I brought from previous exercise is not going to be enough,"},{"Start":"00:26.070 ","End":"00:29.300","Text":"we\u0027re going to have another formula of what to do"},{"Start":"00:29.300 ","End":"00:33.200","Text":"when absolute value of a is bigger than b."},{"Start":"00:33.200 ","End":"00:34.400","Text":"Well, in this case,"},{"Start":"00:34.400 ","End":"00:37.405","Text":"it translates a bit differently or quite differently,"},{"Start":"00:37.405 ","End":"00:42.860","Text":"this becomes that either a is bigger than"},{"Start":"00:42.860 ","End":"00:50.105","Text":"b or a is less than minus b."},{"Start":"00:50.105 ","End":"00:52.550","Text":"Let\u0027s just leave it as a formula."},{"Start":"00:52.550 ","End":"00:56.285","Text":"You should have studied this and this is just a reminder,"},{"Start":"00:56.285 ","End":"00:59.180","Text":"and let\u0027s start on the first 1."},{"Start":"00:59.180 ","End":"01:00.565","Text":"We have 1 of each."},{"Start":"01:00.565 ","End":"01:02.310","Text":"Here\u0027s a less than,"},{"Start":"01:02.310 ","End":"01:11.115","Text":"so we use the first formula and we get 2x plus 1 less than 7,"},{"Start":"01:11.115 ","End":"01:14.635","Text":"and bigger than minus 7."},{"Start":"01:14.635 ","End":"01:17.450","Text":"That\u0027s for the first 1."},{"Start":"01:17.450 ","End":"01:20.645","Text":"For the second 1,"},{"Start":"01:20.645 ","End":"01:27.255","Text":"we get that x plus 1 is bigger than 3,"},{"Start":"01:27.255 ","End":"01:32.860","Text":"or x plus 1 is less than minus 3."},{"Start":"01:32.860 ","End":"01:35.570","Text":"Basically what we have,"},{"Start":"01:35.570 ","End":"01:41.270","Text":"and I\u0027ll just put brackets here to see that we have an AND here and then OR here,"},{"Start":"01:41.270 ","End":"01:44.990","Text":"we actually have 3 separate bits that we\u0027re going to have to combine."},{"Start":"01:44.990 ","End":"01:47.930","Text":"This inequality is fairly simple,"},{"Start":"01:47.930 ","End":"01:49.100","Text":"and let\u0027s solve it."},{"Start":"01:49.100 ","End":"01:51.950","Text":"We can work on all 3 sides simultaneously."},{"Start":"01:51.950 ","End":"01:57.120","Text":"Let\u0027s subtract 1 so we\u0027ve got 2x less than 6,"},{"Start":"01:57.120 ","End":"01:59.400","Text":"greater than minus 8,"},{"Start":"01:59.400 ","End":"02:00.840","Text":"and then divide by 2,"},{"Start":"02:00.840 ","End":"02:04.350","Text":"which is positive x less than 3,"},{"Start":"02:04.350 ","End":"02:06.790","Text":"bigger than minus 4."},{"Start":"02:06.790 ","End":"02:08.720","Text":"That\u0027s 1 part done."},{"Start":"02:08.720 ","End":"02:11.180","Text":"Now let\u0027s see. What does this 1 give us?"},{"Start":"02:11.180 ","End":"02:13.390","Text":"x plus 1 bigger than 3,"},{"Start":"02:13.390 ","End":"02:20.235","Text":"just subtract 1 gives us x bigger than 2 and this 1,"},{"Start":"02:20.235 ","End":"02:23.235","Text":"x plus 1 less than minus 3,"},{"Start":"02:23.235 ","End":"02:28.935","Text":"subtracting 1, we get x less than minus 4."},{"Start":"02:28.935 ","End":"02:30.990","Text":"There\u0027s an OR here,"},{"Start":"02:30.990 ","End":"02:36.735","Text":"and I\u0027ll put it in brackets and these 2 are connected with AND."},{"Start":"02:36.735 ","End":"02:39.995","Text":"This is 1 of those cases where I believe a sketch will really help,"},{"Start":"02:39.995 ","End":"02:43.790","Text":"make a better room first and let\u0027s see,"},{"Start":"02:43.790 ","End":"02:46.610","Text":"we need a number line. Here we are."},{"Start":"02:46.610 ","End":"02:52.730","Text":"The important points that we have to place are minus 4,"},{"Start":"02:52.730 ","End":"02:56.035","Text":"3, and 2."},{"Start":"02:56.035 ","End":"02:57.770","Text":"Let\u0027s put them in the right order."},{"Start":"02:57.770 ","End":"03:00.680","Text":"The leftmost is minus 4,"},{"Start":"03:00.680 ","End":"03:02.910","Text":"then we have 2 and 3,"},{"Start":"03:02.910 ","End":"03:05.280","Text":"so the next 1 would be 2,"},{"Start":"03:05.280 ","End":"03:07.650","Text":"and the next 1 is 3."},{"Start":"03:07.650 ","End":"03:10.230","Text":"Let\u0027s sketch this inequality,"},{"Start":"03:10.230 ","End":"03:13.020","Text":"x is between minus 4 and 3,"},{"Start":"03:13.020 ","End":"03:17.385","Text":"not inclusive, a hollow circle here,"},{"Start":"03:17.385 ","End":"03:21.110","Text":"a hollow circle here and a line to join them."},{"Start":"03:21.110 ","End":"03:24.005","Text":"That\u0027s this one. I\u0027ll use the same color here."},{"Start":"03:24.005 ","End":"03:29.360","Text":"As for this, I\u0027ll use the color green here."},{"Start":"03:29.360 ","End":"03:32.030","Text":"We need bigger than 2,"},{"Start":"03:32.030 ","End":"03:36.275","Text":"so it\u0027s from 2 and I\u0027ll draw the line in a moment to the right of it,"},{"Start":"03:36.275 ","End":"03:41.270","Text":"and less than 1 minua 4 means from here onto the left, and here we are."},{"Start":"03:41.270 ","End":"03:44.360","Text":"I\u0027ll just put some arrows at the end to show that it doesn\u0027t stop here,"},{"Start":"03:44.360 ","End":"03:46.115","Text":"it goes on endlessly."},{"Start":"03:46.115 ","End":"03:49.340","Text":"Now what we want is the AND,"},{"Start":"03:49.340 ","End":"03:51.200","Text":"which is the overlap,"},{"Start":"03:51.200 ","End":"03:52.685","Text":"what they have in common."},{"Start":"03:52.685 ","End":"03:53.720","Text":"If we think about it,"},{"Start":"03:53.720 ","End":"03:56.180","Text":"it\u0027s just this part here."},{"Start":"03:56.180 ","End":"04:01.260","Text":"It\u0027s just from 2 up to 3 not"},{"Start":"04:01.260 ","End":"04:09.215","Text":"inclusive and so this is our overlap and so the answer to this combination of AND,"},{"Start":"04:09.215 ","End":"04:17.990","Text":"OR is just that x is between 2 and 3 and this is our answer to part a."},{"Start":"04:17.990 ","End":"04:19.580","Text":"Let\u0027s go on to part b,"},{"Start":"04:19.580 ","End":"04:21.515","Text":"but I\u0027ll copy the formulas."},{"Start":"04:21.515 ","End":"04:24.065","Text":"Here we are now we can scroll."},{"Start":"04:24.065 ","End":"04:27.290","Text":"Once again, we have a less than and we have a greater than,"},{"Start":"04:27.290 ","End":"04:28.940","Text":"so for the first 1,"},{"Start":"04:28.940 ","End":"04:33.925","Text":"we write it as 2 separate inequalities."},{"Start":"04:33.925 ","End":"04:35.840","Text":"This 1 is 2 separate."},{"Start":"04:35.840 ","End":"04:39.920","Text":"This 1 is the double inequality that x plus"},{"Start":"04:39.920 ","End":"04:46.375","Text":"2 is less than x and greater than minus x,"},{"Start":"04:46.375 ","End":"04:50.480","Text":"and this 1 splits up into 2 separate ones."},{"Start":"04:50.480 ","End":"04:54.980","Text":"Minus x less than x plus 2,"},{"Start":"04:54.980 ","End":"04:59.899","Text":"AND x plus 2 is less than x."},{"Start":"04:59.899 ","End":"05:02.240","Text":"Then we have to take this 1 into account."},{"Start":"05:02.240 ","End":"05:05.105","Text":"I\u0027m just working on this 1 at the moment."},{"Start":"05:05.105 ","End":"05:08.735","Text":"But let\u0027s take a look at this 1 here."},{"Start":"05:08.735 ","End":"05:10.445","Text":"This is a strange one."},{"Start":"05:10.445 ","End":"05:13.040","Text":"This 1, if we try to solve it,"},{"Start":"05:13.040 ","End":"05:16.790","Text":"says that 2 is less than 0,"},{"Start":"05:16.790 ","End":"05:21.830","Text":"if we subtract x, it says 2 is less than 0, and that\u0027s impossible."},{"Start":"05:21.830 ","End":"05:24.860","Text":"That\u0027s definitely false for whatever,"},{"Start":"05:24.860 ","End":"05:26.560","Text":"x is not even in here,"},{"Start":"05:26.560 ","End":"05:29.810","Text":"so there cannot be any x if you add 2,"},{"Start":"05:29.810 ","End":"05:31.400","Text":"it\u0027s less than the thing itself."},{"Start":"05:31.400 ","End":"05:34.905","Text":"Once you have a false with an AND system,"},{"Start":"05:34.905 ","End":"05:38.195","Text":"the whole AND is false because you can\u0027t have something"},{"Start":"05:38.195 ","End":"05:42.245","Text":"that makes this true and this true if this already is not true,"},{"Start":"05:42.245 ","End":"05:44.470","Text":"and then we have another AND here,"},{"Start":"05:44.470 ","End":"05:46.395","Text":"so that\u0027s also not going to help."},{"Start":"05:46.395 ","End":"05:50.030","Text":"When you add an AND you\u0027re only making it stricter so"},{"Start":"05:50.030 ","End":"05:54.275","Text":"that this actually makes the whole thing false."},{"Start":"05:54.275 ","End":"05:57.320","Text":"If we have something AND something,"},{"Start":"05:57.320 ","End":"05:58.685","Text":"it doesn\u0027t matter what,"},{"Start":"05:58.685 ","End":"06:03.050","Text":"AND something, AND something,"},{"Start":"06:03.050 ","End":"06:04.985","Text":"and this 1 is false,"},{"Start":"06:04.985 ","End":"06:07.175","Text":"then the whole thing is false."},{"Start":"06:07.175 ","End":"06:11.690","Text":"We can already say that there is no solution or"},{"Start":"06:11.690 ","End":"06:18.135","Text":"no such x so some similar phrasing and now, that was quick."},{"Start":"06:18.135 ","End":"06:20.330","Text":"Let\u0027s go on to c. Once again,"},{"Start":"06:20.330 ","End":"06:22.880","Text":"I\u0027ll bring the formulas along."},{"Start":"06:22.880 ","End":"06:29.225","Text":"We have the formulas and so again we have a bigger than and we have a less than."},{"Start":"06:29.225 ","End":"06:31.400","Text":"Let\u0027s start with this 1,"},{"Start":"06:31.400 ","End":"06:34.040","Text":"which is a bigger than formula,"},{"Start":"06:34.040 ","End":"06:35.705","Text":"so we\u0027re going to use this."},{"Start":"06:35.705 ","End":"06:42.270","Text":"What we get is either x^2 plus 1 bigger than"},{"Start":"06:42.270 ","End":"06:49.885","Text":"2 or x^2 plus 1 less than minus 2."},{"Start":"06:49.885 ","End":"06:52.580","Text":"Let\u0027s see what the first one gives us."},{"Start":"06:52.580 ","End":"07:00.960","Text":"The first one gives us that x^2 is bigger than 1 just by"},{"Start":"07:00.960 ","End":"07:10.670","Text":"subtracting 1 and this one gives us that x^2 is less than minus 3."},{"Start":"07:10.670 ","End":"07:13.910","Text":"I\u0027m still just working on this inequality here."},{"Start":"07:13.910 ","End":"07:18.515","Text":"Perhaps just going to be to remind myself or just write this one also."},{"Start":"07:18.515 ","End":"07:20.210","Text":"This one has an or,"},{"Start":"07:20.210 ","End":"07:23.675","Text":"this one\u0027s going to have an and because it\u0027s a double inequality."},{"Start":"07:23.675 ","End":"07:31.430","Text":"It\u0027s x^2 minus 2x is either less than 3 or greater than minus 3"},{"Start":"07:31.430 ","End":"07:39.950","Text":"and that gives us 2 inequalities that minus 3 less than x^2 minus 2x,"},{"Start":"07:39.950 ","End":"07:45.725","Text":"and x^2 minus 2x less than 3."},{"Start":"07:45.725 ","End":"07:48.245","Text":"Of course, after we do all this there\u0027s still an"},{"Start":"07:48.245 ","End":"07:51.380","Text":"and between the answer to this combined,"},{"Start":"07:51.380 ","End":"07:54.815","Text":"maybe I\u0027ll put brackets here and this and this."},{"Start":"07:54.815 ","End":"07:57.950","Text":"Let\u0027s see what we have here."},{"Start":"07:57.950 ","End":"08:03.349","Text":"This one is impossible because x^2 can never be negative."},{"Start":"08:03.349 ","End":"08:05.450","Text":"It\u0027s always bigger or equal to 0."},{"Start":"08:05.450 ","End":"08:08.750","Text":"So this one is impossible."},{"Start":"08:08.750 ","End":"08:11.660","Text":"But there\u0027s an or here so we can\u0027t get any"},{"Start":"08:11.660 ","End":"08:14.780","Text":"x\u0027s from here but we can still get x\u0027s from here."},{"Start":"08:14.780 ","End":"08:17.465","Text":"What happens if x^2 is bigger than 1?"},{"Start":"08:17.465 ","End":"08:20.030","Text":"We can solve it with quadratic."},{"Start":"08:20.030 ","End":"08:21.875","Text":"Why not?"},{"Start":"08:21.875 ","End":"08:25.280","Text":"X^2 minus 1 bigger than 0."},{"Start":"08:25.280 ","End":"08:29.855","Text":"This is one of those difference of squares formulas x^2 minus 1 squared."},{"Start":"08:29.855 ","End":"08:32.810","Text":"I can immediately get x minus 1,"},{"Start":"08:32.810 ","End":"08:36.290","Text":"x plus 1 bigger than 0."},{"Start":"08:36.290 ","End":"08:38.090","Text":"The roots of this are"},{"Start":"08:38.090 ","End":"08:41.520","Text":"obviously 1 and minus 1. Let\u0027s write that 1 and minus 1 are the roots."},{"Start":"08:43.030 ","End":"08:46.280","Text":"If I draw a sketch for this one,"},{"Start":"08:46.280 ","End":"08:49.595","Text":"here\u0027s an x-axis number line,"},{"Start":"08:49.595 ","End":"08:55.955","Text":"let\u0027s put in the one on the right will be 1 and the one on the left will be minus 1."},{"Start":"08:55.955 ","End":"08:58.535","Text":"I\u0027m going to just cut corners here."},{"Start":"08:58.535 ","End":"09:01.820","Text":"Obviously, if I take something less than minus 1,"},{"Start":"09:01.820 ","End":"09:06.785","Text":"they\u0027re both going to be negative so here I\u0027m going to get a plus."},{"Start":"09:06.785 ","End":"09:08.870","Text":"If I take something in the middle like 0,"},{"Start":"09:08.870 ","End":"09:12.875","Text":"we\u0027ll get negative times positive will be negative."},{"Start":"09:12.875 ","End":"09:14.450","Text":"If I take something larger than 1,"},{"Start":"09:14.450 ","End":"09:17.368","Text":"they\u0027ll both be positive so it\u0027ll be positive."},{"Start":"09:17.368 ","End":"09:19.340","Text":"For those who want,"},{"Start":"09:19.340 ","End":"09:22.250","Text":"you can draw what is colloquially known as the snake,"},{"Start":"09:22.250 ","End":"09:29.015","Text":"which goes plus here and then minus here and plus here."},{"Start":"09:29.015 ","End":"09:32.694","Text":"We want the part that\u0027s bigger than 0,"},{"Start":"09:32.694 ","End":"09:36.110","Text":"which means that this part"},{"Start":"09:36.110 ","End":"09:40.144","Text":"here that\u0027s above the axis and this part here that\u0027s above the axis,"},{"Start":"09:40.144 ","End":"09:46.190","Text":"but of course not including these points and so we get that x is less than"},{"Start":"09:46.190 ","End":"09:52.670","Text":"minus 1 or x is bigger than 1."},{"Start":"09:52.670 ","End":"09:57.785","Text":"Like I said, or something impossible just doesn\u0027t add anything we forget about that."},{"Start":"09:57.785 ","End":"10:00.275","Text":"So far, that\u0027s just this bit."},{"Start":"10:00.275 ","End":"10:02.915","Text":"Now let\u0027s go for this bit."},{"Start":"10:02.915 ","End":"10:05.405","Text":"We have 2 quadratics."},{"Start":"10:05.405 ","End":"10:09.170","Text":"The first quadratic gives us, let\u0027s see."},{"Start":"10:09.170 ","End":"10:12.320","Text":"I\u0027ll write it a bit further over to give ourselves some space."},{"Start":"10:12.320 ","End":"10:19.550","Text":"This one gives us x^2 minus 2x plus 3, that\u0027s on the right,"},{"Start":"10:19.550 ","End":"10:24.950","Text":"so it\u0027s bigger than 0 and the other one gives"},{"Start":"10:24.950 ","End":"10:33.260","Text":"us x^2 minus 2x minus 3 less than 0."},{"Start":"10:33.260 ","End":"10:35.510","Text":"Since we\u0027ve done this kind of thing before,"},{"Start":"10:35.510 ","End":"10:38.885","Text":"I\u0027m going to again take shortcuts,"},{"Start":"10:38.885 ","End":"10:40.580","Text":"just not do all the steps."},{"Start":"10:40.580 ","End":"10:43.130","Text":"Turns out that this one has no roots."},{"Start":"10:43.130 ","End":"10:47.630","Text":"If you try to use the quadratic b^2 minus 4ac is negative,"},{"Start":"10:47.630 ","End":"10:48.950","Text":"it has no roots."},{"Start":"10:48.950 ","End":"10:54.065","Text":"We substitute any sample values such as 0 and it\u0027s positive."},{"Start":"10:54.065 ","End":"10:55.970","Text":"I\u0027m skipping the steps."},{"Start":"10:55.970 ","End":"10:58.580","Text":"This is true for all x."},{"Start":"10:58.580 ","End":"11:02.268","Text":"Some people say x from minus infinity to infinity,"},{"Start":"11:02.268 ","End":"11:04.355","Text":"every x on the number line."},{"Start":"11:04.355 ","End":"11:05.870","Text":"When we and it,"},{"Start":"11:05.870 ","End":"11:07.160","Text":"we basically ignore this,"},{"Start":"11:07.160 ","End":"11:09.125","Text":"it doesn\u0027t add anything."},{"Start":"11:09.125 ","End":"11:12.840","Text":"This one, we find that the roots are,"},{"Start":"11:12.840 ","End":"11:14.360","Text":"and I\u0027ll just give them to you,"},{"Start":"11:14.360 ","End":"11:16.865","Text":"won\u0027t go ahead and solve it and waste time,"},{"Start":"11:16.865 ","End":"11:21.845","Text":"the roots are minus 1 and 3."},{"Start":"11:21.845 ","End":"11:27.035","Text":"This one become x minus minus 1 which is plus 1,"},{"Start":"11:27.035 ","End":"11:30.815","Text":"x minus 3 less than 0."},{"Start":"11:30.815 ","End":"11:33.785","Text":"Quick sketch, number line."},{"Start":"11:33.785 ","End":"11:35.720","Text":"Mark the 2 points,"},{"Start":"11:35.720 ","End":"11:38.060","Text":"make sure you get them in the right order."},{"Start":"11:38.060 ","End":"11:39.620","Text":"Minus 1 to the left,"},{"Start":"11:39.620 ","End":"11:41.375","Text":"3 to the right."},{"Start":"11:41.375 ","End":"11:47.620","Text":"I\u0027ll just again cut some steps that if we take a value in the middle here,"},{"Start":"11:47.620 ","End":"11:54.380","Text":"say 0, we\u0027ll get something positive times negative which will be negative."},{"Start":"11:54.380 ","End":"11:58.910","Text":"If we take something here like 4 or 10 or whatever,"},{"Start":"11:58.910 ","End":"12:00.590","Text":"they\u0027re both going to come out positive,"},{"Start":"12:00.590 ","End":"12:01.925","Text":"so the product is positive."},{"Start":"12:01.925 ","End":"12:03.530","Text":"Here they\u0027re both going to come out negative."},{"Start":"12:03.530 ","End":"12:07.820","Text":"I think minus 2 then plus 1 is still negative and this is negative,"},{"Start":"12:07.820 ","End":"12:09.350","Text":"so it\u0027s going to be positive."},{"Start":"12:09.350 ","End":"12:16.565","Text":"What we want is the less than 0 so I won\u0027t bother drawing the so-called snake,"},{"Start":"12:16.565 ","End":"12:22.250","Text":"we\u0027ll just say that less than 0 is the negative,"},{"Start":"12:22.250 ","End":"12:25.610","Text":"but not including the minus 1,"},{"Start":"12:25.610 ","End":"12:31.055","Text":"not including the 3 between minus 1 and 3."},{"Start":"12:31.055 ","End":"12:38.135","Text":"If you write it in terms of x minus 1 less than x less than 3."},{"Start":"12:38.135 ","End":"12:40.925","Text":"Now we have to put all the bits together."},{"Start":"12:40.925 ","End":"12:44.345","Text":"These 2 have an and between them,"},{"Start":"12:44.345 ","End":"12:50.390","Text":"so the answer to this whole and the second part of this and this and this,"},{"Start":"12:50.390 ","End":"12:51.980","Text":"let me just write that."},{"Start":"12:51.980 ","End":"12:54.170","Text":"I\u0027m going to combine them with an and."},{"Start":"12:54.170 ","End":"12:56.435","Text":"What we get is just this one."},{"Start":"12:56.435 ","End":"12:59.765","Text":"It\u0027s minus 1, I\u0027ll highlight it."},{"Start":"12:59.765 ","End":"13:01.565","Text":"Not the final answer,"},{"Start":"13:01.565 ","End":"13:04.055","Text":"the answer to this one."},{"Start":"13:04.055 ","End":"13:05.960","Text":"Then we use another color."},{"Start":"13:05.960 ","End":"13:08.210","Text":"Let\u0027s say we use this color."},{"Start":"13:08.210 ","End":"13:11.840","Text":"The answer to this one is what we got here."},{"Start":"13:11.840 ","End":"13:15.395","Text":"Now we have to combine these 2 with an and,"},{"Start":"13:15.395 ","End":"13:17.330","Text":"so you choose a nice color,"},{"Start":"13:17.330 ","End":"13:20.975","Text":"these 2 are going to be combined with and,"},{"Start":"13:20.975 ","End":"13:23.240","Text":"and that will be our final answer."},{"Start":"13:23.240 ","End":"13:26.165","Text":"I\u0027ll just give us a bit more room here."},{"Start":"13:26.165 ","End":"13:31.790","Text":"Number line, what we want are the points minus 1,"},{"Start":"13:31.790 ","End":"13:35.060","Text":"1, and 3. Let\u0027s see."},{"Start":"13:35.060 ","End":"13:37.175","Text":"Put them here,"},{"Start":"13:37.175 ","End":"13:40.715","Text":"and we\u0027ll call them minus 1,"},{"Start":"13:40.715 ","End":"13:44.104","Text":"1, and 3 that\u0027s in the right order."},{"Start":"13:44.104 ","End":"13:47.990","Text":"This inequality, the yellow one,"},{"Start":"13:47.990 ","End":"13:54.410","Text":"says x between minus 1 and 3 so we\u0027ll have minus 1 and"},{"Start":"13:54.410 ","End":"14:01.415","Text":"3 and the line through them maybe not the greatest color and let\u0027s do the green one."},{"Start":"14:01.415 ","End":"14:08.030","Text":"What we want is less than minus 1 so it\u0027s from minus 1 and to the left,"},{"Start":"14:08.030 ","End":"14:09.658","Text":"I\u0027ll draw that in a minute,"},{"Start":"14:09.658 ","End":"14:10.797","Text":"or x bigger than 1,"},{"Start":"14:10.797 ","End":"14:13.340","Text":"so it\u0027s from here and to the right."},{"Start":"14:13.340 ","End":"14:16.580","Text":"There we go with the lines, go on indefinitely."},{"Start":"14:16.580 ","End":"14:23.360","Text":"The and means the overlap looks like this part here,"},{"Start":"14:23.360 ","End":"14:29.255","Text":"which basically just gives us from 1-3,"},{"Start":"14:29.255 ","End":"14:31.490","Text":"hollow circles not including."},{"Start":"14:31.490 ","End":"14:38.450","Text":"This will be our answer in picture form but in terms of x,"},{"Start":"14:38.450 ","End":"14:40.535","Text":"we have 1,"},{"Start":"14:40.535 ","End":"14:42.170","Text":"we have 3,"},{"Start":"14:42.170 ","End":"14:46.490","Text":"we have x in the middle and here these inequalities."},{"Start":"14:46.490 ","End":"14:52.055","Text":"This is the combined solution for these 2 ends."},{"Start":"14:52.055 ","End":"14:59.880","Text":"Maybe I should also underline it so we know that this is the answer to part c."}],"Thumbnail":null,"ID":5087}],"ID":3878}]

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1.1

1