WEBVTT 00:00:04.000 --> 00:00:11.517 Okay, we're talking about something called compound inequalities, which you should 00:00:11.517 --> 00:00:12.267 have 00:00:12.200 --> 00:00:16.120 learned last year, but we're going to kind of review a little bit. 00:00:16.120 --> 00:00:18.160 Let me explain a little more of this. 00:00:18.720 --> 00:00:20.200 Compound inequalities are two or more. 00:00:20.560 --> 00:00:25.560 We can have more than two, but typically it's just two inequalities. 00:00:27.520 --> 00:00:28.880 And then there's two different types. 00:00:29.440 --> 00:00:34.140 One's called an OR, one's called an AND compound inequality. 00:00:35.120 --> 00:00:43.704 For the AND compound inequality, a number is a solution if it's a solution to both 00:00:43.704 --> 00:00:44.454 inequalities. 00:00:45.700 --> 00:00:46.600 That's the AND. 00:00:47.500 --> 00:00:52.439 And the OR inequality is a number is a solution to the compound if it's a solution 00:00:52.439 --> 00:00:53.189 to one 00:00:53.020 --> 00:00:54.420 or the other inequality. 00:00:56.060 --> 00:00:57.720 So let me give you an example. 00:00:57.720 --> 00:01:08.960 So we have AND and OR inequalities. 00:01:08.960 --> 00:01:11.020 They will either say the words 00:01:11.020 --> 00:01:14.877 They will either use the word AND or the word OR. There is another way to write an 00:01:14.877 --> 00:01:17.020 AND inequality where you will not see the word AND. 00:01:24.340 --> 00:01:30.900 solution to the compound inequality the number must be a solution to this and 00:01:30.980 --> 00:01:33.100 and also a solution to this. 00:01:36.420 --> 00:01:41.020 So these are already solved for x. 00:01:49.680 --> 00:01:54.180 So all the solutions to x greater than or equal to 4 00:01:58.840 --> 00:02:01.780 Are all of these numbers correct? 00:02:06.940 --> 00:02:14.980 And solutions to this, x is less than 10, are all these solutions? 00:02:20.880 --> 00:02:23.780 So solutions to this compound inequality 00:02:24.620 --> 00:02:30.440 are just the numbers that are solutions to this and this. 00:02:31.580 --> 00:02:33.440 So what on the graph is going to be a solution? 00:02:34.940 --> 00:02:35.840 5. 00:02:36.860 --> 00:02:39.360 Whatever is in both green and blue. 00:02:40.980 --> 00:02:41.880 OK. 00:02:41.180 --> 00:02:52.320 So the numbers are in both green and blue would be is four in both colors? 00:02:53.380 --> 00:03:02.780 Yes. Yes and then everything here what about ten is ten in both colors? No. No 00:03:02.780 --> 00:03:10.120 because of the open circle there. Okay so this is the solution set to that 00:03:10.120 --> 00:03:13.080 compound and inequality. 00:03:13.080 --> 00:03:20.140 And so the solution set, of course that's the graph of the solution set. 00:03:20.520 --> 00:03:26.520 The actual solution set in interval notation is all the numbers between 4 and 10, 00:03:26.520 --> 00:03:29.060 including 4 but not including 10: [4, 10). 00:03:29.060 --> 00:03:35.760 If this compound inequality could be written in one line, 00:03:37.020 --> 00:03:51.940 Okay. If, uh, that's not the best way of writing it. Let's do it this way. 00:03:58.240 --> 00:03:59.540 That's this guy right here, right? 00:04:00.680 --> 00:04:05.100 And if I flip this guy around, same thing as 4 is less than 00:04:05.100 --> 00:04:06.000 or equal to x. 00:04:07.520 --> 00:04:10.560 So we can just put this guy right here. 00:04:15.740 --> 00:04:18.040 So if you ever see something like this, 00:04:19.080 --> 00:04:20.620 that's a compound inequality. 00:04:22.000 --> 00:04:25.460 4 is less than or equal to x, which is less than 10. 00:04:25.760 --> 00:04:26.660 Was it less than 10? 00:04:27.860 --> 00:04:28.760 OK. 00:04:28.680 --> 00:04:31.860 This is two inequalities in one, a two for one. 00:04:33.000 --> 00:04:35.100 Okay, if you ever see two inequalities in one, 00:04:36.120 --> 00:04:38.380 it's a compound and inequality. 00:04:39.660 --> 00:04:42.900 Has to be a solution to this and a solution to this. 00:04:42.900 --> 00:04:45.560 Where or inequalities have to be written separately, 00:04:46.240 --> 00:04:47.880 you have to use the word or in between. 00:04:47.880 --> 00:05:00.590 So x is greater than or equal to 5, or x is less than 0. You always see the word OR 00:05:00.590 --> 00:05:01.800 in between. 00:05:01.800 --> 00:05:02.700 word or in between 00:05:34.480 --> 00:05:40.760 OK, so in orange is the solution to the x is less than zero, green is solution to x 00:05:40.900 --> 00:05:41.800 greater than or equal to 5. 00:05:43.460 --> 00:05:47.100 So the solutions to the compound inequality 00:05:47.100 --> 00:05:51.620 is the ones that are in green or orange. 00:05:52.120 --> 00:05:53.020 We draw these. 00:05:58.940 --> 00:06:03.700 And then to write this in interval notation, 00:06:03.700 --> 00:06:05.720 we have to write two separate intervals. 00:06:06.940 --> 00:06:20.212 So, from negative infinity to zero, and five to infinity, which are two different 00:06:20.212 --> 00:06:21.160 sets. 00:06:21.760 --> 00:06:23.520 And how do we join them into one set? 00:06:25.820 --> 00:06:27.420 Did I teach you guys this? 00:06:28.900 --> 00:06:29.800 It's a U. 00:06:30.560 --> 00:06:31.460 Yeah. 00:06:30.600 --> 00:06:34.520 So we put the this big U looking thing 00:06:35.280 --> 00:06:36.420 stands for union 00:06:37.160 --> 00:06:39.140 all that means is 00:06:43.300 --> 00:06:48.006 These are two separate sets. To combine them into one set, put a union sign between 00:06:48.006 --> 00:06:48.756 them. 00:06:49.360 --> 00:06:52.940 Now this is one big set even though it's two separate intervals 00:06:52.940 --> 00:06:54.900 So is this an or or and? 00:06:55.520 --> 00:06:56.420 How about an and? 00:06:56.800 --> 00:06:58.700 Definitely, we see the word or. 00:06:59.380 --> 00:07:03.617 So the solutions are all the solutions to this, or all the solutions to this. For 00:07:03.617 --> 00:07:06.000 each inequality, we have to do a little work. 00:07:06.000 --> 00:07:07.500 of these we got to do a tiny bit of work. 00:07:07.500 --> 00:07:10.520 So we do divide by three. 00:07:14.560 --> 00:07:16.700 Subtract six. 00:07:20.400 --> 00:07:21.620 Fourteen. 00:07:52.520 --> 00:07:59.380 So solutions to the compound inequalities is everything we see in red. 00:07:59.380 --> 00:08:09.940 So in interval notation: negative infinity to 4, union 14 to infinity. 00:08:09.940 --> 00:08:12.300 Is this an and or an or? 00:08:12.620 --> 00:08:13.520 And. 00:08:13.980 --> 00:08:15.800 And how do we know? 00:08:19.100 --> 00:08:22.180 You know we got this inequality 00:08:23.680 --> 00:08:25.600 We have this inequality a 00:08:27.760 --> 00:08:32.025 Solution it's a solution to the compound inequality if it's a solution to the left 00:08:32.025 --> 00:08:34.300 inequality, and it's a solution to the right 00:08:34.300 --> 00:08:38.418 Okay, the beauty about these is we don't have to write two separate ones. We could 00:08:38.418 --> 00:08:40.220 solve it all at the same time 00:08:40.220 --> 00:08:48.917 So first is we subtract 6 from all three sides, which is really from each side of 00:08:48.917 --> 00:08:49.940 the two 00:08:49.940 --> 00:08:50.940 separate inequalities. 00:08:53.440 --> 00:09:02.100 So we get negative 10 is less than or equal to 2x, which is less than 4. 00:09:04.880 --> 00:09:09.969 Then we use the multiplication property of inequalities—or division—and divide all 00:09:09.969 --> 00:09:11.820 three sides by 2. 00:09:11.820 --> 00:09:14.000 by two, divide by two, divide by two. 00:09:39.680 --> 00:09:42.180 So this one, x is less than 2. 00:09:50.760 --> 00:09:51.660 That. 00:09:54.580 --> 00:10:00.620 This one, which same thing as x is greater than or equal to negative five. 00:10:10.760 --> 00:10:17.080 So the solution set has to be both in green and red here. 00:10:28.480 --> 00:10:35.811 So, that set of numbers, which is only one interval, so that's all the numbers 00:10:35.811 --> 00:10:36.561 between 00:10:36.300 --> 00:10:40.700 negative five and two excluding two, including negative five. 00:10:55.860 --> 00:10:56.760 Questions on this so far? 00:10:56.360 --> 00:10:58.920 So is the solution set where they overlap? 00:11:00.720 --> 00:11:02.780 Yes, but also when they overlap. 00:11:02.780 --> 00:11:05.320 Okay, it's not just where they overlap 00:11:06.520 --> 00:11:11.020 it's all the solutions to this which is everything in blue or 00:11:13.280 --> 00:11:17.780 Everything in purple whatever color that is 00:11:21.900 --> 00:11:23.060 Which is 00:11:23.060 --> 00:11:24.740 All real numbers 00:11:25.760 --> 00:11:30.307 We're gonna be careful just because it overlaps the solution set is not where it 00:11:30.307 --> 00:11:31.520 overlaps. It's very important 00:11:31.520 --> 00:11:34.240 And it's a solution to this, or it's a solution to this. 00:11:34.740 --> 00:11:40.380 So the solution set is all real numbers. 00:11:41.820 --> 00:11:43.920 This would be the graph of the solution set. 00:11:45.260 --> 00:11:48.240 In interval notation: negative infinity to positive infinity. 00:11:49.080 --> 00:11:52.180 So be real careful whether it's an or or an and. 00:11:53.860 --> 00:11:55.440 Just because it overlaps. 00:11:55.440 --> 00:11:58.920 That does not mean the solution set is only where they overlap. 00:12:01.400 --> 00:12:02.300 Okay. 00:12:01.720 --> 00:12:03.800 So is the solution set those two intervals? 00:12:05.100 --> 00:12:06.000 No. 00:12:05.780 --> 00:12:06.680 No. 00:12:07.820 --> 00:12:14.460 It has to be both in purple and in blue and there are no numbers in purple and blue. 00:12:15.180 --> 00:12:17.020 So what is the solution set? 00:12:19.200 --> 00:12:22.160 There are no solutions but what is the solution set? 00:12:25.460 --> 00:12:26.360 The empty set. 00:12:26.420 --> 00:12:29.620 Either that or we write it with curly brackets. 00:12:34.480 --> 00:12:36.140 Okay, be real careful. 00:12:37.460 --> 00:12:39.420 The AND and the OR make a big difference. 00:12:42.580 --> 00:12:47.120 There are no numbers, there are both solutions to this and solutions to this. 00:12:47.380 --> 00:12:50.200 So there are no solutions; the solution set is empty. 00:12:50.840 --> 00:12:54.140 Let me pass out the homework and stop here.