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