WEBVTT 00:00:00.040 --> 00:00:04.040 Welcome to the presentation on solving inequalities 00:00:04.040 --> 00:00:07.000 or I guess you could call them algebraic inequalities. 00:00:07.000 --> 00:00:09.020 So let's get started. 00:00:09.030 --> 00:00:12.050 If I were to tell you that, well, let's just say 00:00:12.050 --> 00:00:17.090 x > 5, right? 00:00:18.000 --> 00:00:22.080 So x could be 5.01, it could be 5.5, it could be a million. 00:00:22.090 --> 00:00:26.030 It just can't be 4 or 3 or 0 or -8. 00:00:26.040 --> 00:00:28.010 And actually, just for convenience, 00:00:28.020 --> 00:00:31.000 let's actually draw that on the number line. 00:00:31.010 --> 00:00:33.030 That's the number line. 00:00:33.030 --> 00:00:36.090 And if this is 5, x can't be equal to 5 00:00:37.000 --> 00:00:39.090 so we draw a big circle here and then we would color 00:00:40.000 --> 00:00:42.010 in all the values that x could be. 00:00:42.020 --> 00:00:45.080 So x could be 5.000001, 00:00:45.090 --> 00:00:48.060 it just has to be a little bit bigger than 5 00:00:48.070 --> 00:00:50.090 and any of those would satisfy, right? 00:00:51.000 --> 00:00:53.060 So let's just write some numbers that satisfy. 00:00:53.070 --> 00:00:56.010 6 would satisfy it, 10 would satisfy it, 00:00:56.020 --> 00:00:57.090 100 would satisfy it. 00:00:58.000 --> 00:01:01.000 Now, if I were to multiply or, I guess, divide, 00:01:01.010 --> 00:01:03.080 both sides of this, I guess we could say equation, 00:01:03.090 --> 00:01:09.080 or this inequality, by -1, I wanna understand what happens. 00:01:09.090 --> 00:01:15.090 So what's the relation between -x and -5? 00:01:17.040 --> 00:01:19.000 And when I say, what's the relation, 00:01:19.000 --> 00:01:24.030 is it greater than or is it less than -5? 00:01:24.030 --> 00:01:27.723 Well, 6 is a value that works for x, 00:01:27.723 --> 00:01:33.030 so -6, is that greater than or less than -5? 00:01:33.030 --> 00:01:36.090 -6 is less than -5, right? 00:01:37.000 --> 00:01:41.010 So let me draw the number line here. 00:01:41.010 --> 00:01:44.040 If we have -5 here-- let's just draw a circle around it 00:01:44.050 --> 00:01:46.070 because we know it's not gonna equal to -5 00:01:46.080 --> 00:01:48.050 because we're just deciding 00:01:48.060 --> 00:01:50.010 between greater than or less than. 00:01:50.020 --> 00:01:54.040 So we're saying 6 works for x, so -6 is here, right? 00:01:54.040 --> 00:01:56.040 -6. 00:01:56.040 --> 00:01:58.608 So -6 is less than -5 00:01:58.608 --> 00:02:03.019 So is -10, so is -100, so is -1,000,000, right? 00:02:03.019 --> 00:02:08.100 So, turns out -X is less than -5 00:02:08.100 --> 00:02:11.332 So this is really all you have to remember 00:02:11.332 --> 00:02:14.444 When you are working with inequalities in algebra 00:02:14.444 --> 00:02:18.205 Inequalities you can treat them just the way 00:02:18.205 --> 00:02:21.100 A > or a < sign you can treat them just the way you would treat an = sign 00:02:21.100 --> 00:02:24.750 The only difference is: if you multiply or divide 00:02:24.750 --> 00:02:29.819 both sides of the equation by a negative number 00:02:29.819 --> 00:02:30.651 You swap it 00:02:30.651 --> 00:02:31.766 That's all you have to remember 00:02:31.782 --> 00:02:34.325 Let's do some problems and hopefully that will bring the point home 00:02:34.325 --> 00:02:38.064 If you ever forget, you just have to try, you just have to remember this 00:02:38.064 --> 00:02:41.276 X is > 5, well then -X < -5 00:02:41.276 --> 00:02:42.243 And keep trying out numbers 00:02:42.243 --> 00:02:45.518 That's what is going to give you the best intuition 00:02:45.518 --> 00:02:46.941 Let's do some problems 00:02:46.941 --> 00:02:56.136 So if I say that 3X + 2 is less than or equal to 1 00:02:56.136 --> 00:02:58.076 Well, this is a pretty easy equation to solve 00:02:58.076 --> 00:03:01.395 We say 3X, let's subtract 2 from both sides 00:03:01.395 --> 00:03:03.099 When you do add or subtract 00:03:03.099 --> 00:03:04.877 You don't do anything to the inequality 00:03:04.877 --> 00:03:07.657 So if you subtract 2 from both sides 00:03:07.657 --> 00:03:12.447 You get 3X is less than or equal to -1, right? 00:03:12.447 --> 00:03:16.534 And then now we are going to divide both sides by 3 00:03:16.534 --> 00:03:22.292 We get X is less than or equal to -1/3 00:03:22.292 --> 00:03:24.011 Look we didn't change anything 00:03:24.011 --> 00:03:26.704 Because we divided both sides by a positive 3 00:03:26.704 --> 00:03:31.952 Alright? We could have done this equation in a slightly different way 00:03:31.952 --> 00:03:35.439 What if we subtracted 1 from both sides 00:03:35.439 --> 00:03:37.584 So this is another way of solving it 00:03:37.584 --> 00:03:42.392 What if we said 3X + 1 is equal to or less than 0 00:03:42.392 --> 00:03:44.464 I just subtracted 1 from both sides 00:03:44.464 --> 00:03:46.645 And now I'll subtract 3X from both sides 00:03:46.645 --> 00:03:51.457 I'll get 1 is less than or equal to -3X 00:03:51.457 --> 00:03:53.479 I subtracted 3X from here 00:03:53.479 --> 00:03:54.779 So I'll subtract 3X from here 00:03:54.779 --> 00:03:58.258 Now I'll have to divide both sides by a negative number 00:03:58.258 --> 00:04:02.347 Right? Because I'm going to divide both sides by -3 00:04:02.347 --> 00:04:05.472 So I get -1/3 on this side 00:04:05.472 --> 00:04:07.252 And based on what we had just learned 00:04:07.267 --> 00:04:08.485 Since we are dividing by a negative number 00:04:08.485 --> 00:04:10.255 We want to swap the inequality right? 00:04:10.255 --> 00:04:11.778 It was less than or equal to 00:04:11.778 --> 00:04:15.160 And it's going to be greater than or equal to X 00:04:15.160 --> 00:04:19.181 Now did we get the same answer when did both in two different ways? 00:04:22.571 --> 00:04:25.917 And here we got -1/3 is greater than or equal to X 00:04:19.181 --> 00:04:22.571 Here we got X is less than or equal to negative 1/3 00:04:25.917 --> 00:04:27.401 That's the same answer right? 00:04:27.401 --> 00:04:29.584 X is less than or equal to negative 1/3 00:04:29.584 --> 00:04:31.957 That's what I find to be the cool thing about algebra. 00:04:31.957 --> 00:04:33.964 You can tackle a problem in a bunch of two different ways. 00:04:33.964 --> 00:04:37.560 You should get the right answer as long as I guess you do it right. 00:04:37.560 --> 00:04:41.890 Let's do a couple more problems. 00:04:41.890 --> 00:04:46.859 Erase the thing. Here you go. Let's do a slightly harder one. 00:04:46.859 --> 00:04:56.751 Let's say - 8x + 7 > 5x +2 00:04:56.751 --> 00:05:02.138 Let's subtract 5x from both sides. 00:05:02.138 --> 00:05:06.085 - 13x + 7 > 2 00:05:06.085 --> 00:05:09.754 Now we can subtract 7 from both sides, 00:05:09.754 --> 00:05:13.469 -13x > -5. 00:05:13.469 --> 00:05:17.311 Now we're gonna divide both sides of this equation by -13. 00:05:17.311 --> 00:05:19.460 Well, very easy. 00:05:19.460 --> 00:05:24.940 It's just x, and on this side -5/-13 = 5/13, right? 00:05:24.940 --> 00:05:26.705 The negatives cancel out. 00:05:26.705 --> 00:05:30.068 And since we divided by a negative, 00:05:30.068 --> 00:05:31.849 we switch the sign. 00:05:31.849 --> 00:05:34.292 x is less than 5/13 00:05:34.292 --> 00:05:35.719 And once again, just like the beginning, 00:05:35.719 --> 00:05:37.966 if you don't believe me, try out some numbers. 00:05:37.966 --> 00:05:39.080 I remember when I first learned this 00:05:39.080 --> 00:05:40.734 I didn't believe the teacher so I did try out numbers 00:05:40.749 --> 00:05:44.538 and that's how I got convinced that it works 00:05:44.538 --> 00:05:47.050 when you multiply or divide both sides of this equation 00:05:47.050 --> 00:05:50.220 by a negative sign, you swap the inequality. 00:05:50.220 --> 00:05:53.040 And remember: that's only when you multiply or divide, 00:05:53.050 --> 00:05:55.781 not when you add or subtract. 00:05:55.781 --> 00:05:57.665 I think that should give you 00:05:57.665 --> 00:05:59.810 a good idea of how to do these problems. 00:05:59.810 --> 00:06:01.080 There's really not much new here. 00:06:01.080 --> 00:06:05.290 You do an inequality or-- I guess you could call this 00:06:05.290 --> 00:06:08.000 an inequality equation-- you do it exactly the same way 00:06:08.010 --> 00:06:10.085 you would do a normal linear equation. 00:06:10.085 --> 00:06:14.475 The only difference being is if you multiply or you divide 00:06:14.475 --> 00:06:16.030 both sides of the equation by a negative number, 00:06:16.040 --> 00:06:19.259 then you swap the inequality. 00:06:19.259 --> 00:06:22.050 I think you're ready now to try some practice problems. 00:06:22.060 --> 00:06:24.040 Have fun.