1 00:00:00,040 --> 00:00:04,040 Welcome to the presentation on solving inequalities 2 00:00:04,040 --> 00:00:07,000 or I guess you could call them algebraic inequalities. 3 00:00:07,000 --> 00:00:09,020 So let's get started. 4 00:00:09,030 --> 00:00:12,050 If I were to tell you that, well, let's just say 5 00:00:12,050 --> 00:00:17,090 x > 5, right? 6 00:00:18,000 --> 00:00:22,080 So x could be 5.01, it could be 5.5, it could be a million. 7 00:00:22,090 --> 00:00:26,030 It just can't be 4 or 3 or 0 or -8. 8 00:00:26,040 --> 00:00:28,010 And actually, just for convenience, 9 00:00:28,020 --> 00:00:31,000 let's actually draw that on the number line. 10 00:00:31,010 --> 00:00:33,030 That's the number line. 11 00:00:33,030 --> 00:00:36,090 And if this is 5, x can't be equal to 5 12 00:00:37,000 --> 00:00:39,090 so we draw a big circle here and then we would color 13 00:00:40,000 --> 00:00:42,010 in all the values that x could be. 14 00:00:42,020 --> 00:00:45,080 So x could be 5.000001, 15 00:00:45,090 --> 00:00:48,060 it just has to be a little bit bigger than 5 16 00:00:48,070 --> 00:00:50,090 and any of those would satisfy, right? 17 00:00:51,000 --> 00:00:53,060 So let's just write some numbers that satisfy. 18 00:00:53,070 --> 00:00:56,010 6 would satisfy it, 10 would satisfy it, 19 00:00:56,020 --> 00:00:57,090 100 would satisfy it. 20 00:00:58,000 --> 00:01:01,000 Now, if I were to multiply or, I guess, divide, 21 00:01:01,010 --> 00:01:03,080 both sides of this, I guess we could say equation, 22 00:01:03,090 --> 00:01:09,080 or this inequality, by -1, I wanna understand what happens. 23 00:01:09,090 --> 00:01:15,090 So what's the relation between -x and -5? 24 00:01:17,040 --> 00:01:19,000 And when I say, what's the relation, 25 00:01:19,000 --> 00:01:24,030 is it greater than or is it less than -5? 26 00:01:24,030 --> 00:01:27,723 Well, 6 is a value that works for x, 27 00:01:27,723 --> 00:01:33,030 so -6, is that greater than or less than -5? 28 00:01:33,030 --> 00:01:36,090 -6 is less than -5, right? 29 00:01:37,000 --> 00:01:41,010 So let me draw the number line here. 30 00:01:41,010 --> 00:01:44,040 If we have -5 here-- let's just draw a circle around it 31 00:01:44,050 --> 00:01:46,070 because we know it's not gonna equal to -5 32 00:01:46,080 --> 00:01:48,050 because we're just deciding 33 00:01:48,060 --> 00:01:50,010 between greater than or less than. 34 00:01:50,020 --> 00:01:54,040 So we're saying 6 works for x, so -6 is here, right? 35 00:01:54,040 --> 00:01:56,040 -6. 36 00:01:56,040 --> 00:01:58,608 So -6 is less than -5 37 00:01:58,608 --> 00:02:03,019 So is -10, so is -100, so is -1,000,000, right? 38 00:02:03,019 --> 00:02:08,100 So, turns out -X is less than -5 39 00:02:08,100 --> 00:02:11,332 So this is really all you have to remember 40 00:02:11,332 --> 00:02:14,444 When you are working with inequalities in algebra 41 00:02:14,444 --> 00:02:18,205 Inequalities you can treat them just the way 42 00:02:18,205 --> 00:02:21,100 A > or a < sign you can treat them just the way you would treat an = sign 43 00:02:21,100 --> 00:02:24,750 The only difference is: if you multiply or divide 44 00:02:24,750 --> 00:02:29,819 both sides of the equation by a negative number 45 00:02:29,819 --> 00:02:30,651 You swap it 46 00:02:30,651 --> 00:02:31,766 That's all you have to remember 47 00:02:31,782 --> 00:02:34,325 Let's do some problems and hopefully that will bring the point home 48 00:02:34,325 --> 00:02:38,064 If you ever forget, you just have to try, you just have to remember this 49 00:02:38,064 --> 00:02:41,276 X is > 5, well then -X < -5 50 00:02:41,276 --> 00:02:42,243 And keep trying out numbers 51 00:02:42,243 --> 00:02:45,518 That's what is going to give you the best intuition 52 00:02:45,518 --> 00:02:46,941 Let's do some problems 53 00:02:46,941 --> 00:02:56,136 So if I say that 3X + 2 is less than or equal to 1 54 00:02:56,136 --> 00:02:58,076 Well, this is a pretty easy equation to solve 55 00:02:58,076 --> 00:03:01,395 We say 3X, let's subtract 2 from both sides 56 00:03:01,395 --> 00:03:03,099 When you do add or subtract 57 00:03:03,099 --> 00:03:04,877 You don't do anything to the inequality 58 00:03:04,877 --> 00:03:07,657 So if you subtract 2 from both sides 59 00:03:07,657 --> 00:03:12,447 You get 3X is less than or equal to -1, right? 60 00:03:12,447 --> 00:03:16,534 And then now we are going to divide both sides by 3 61 00:03:16,534 --> 00:03:22,292 We get X is less than or equal to -1/3 62 00:03:22,292 --> 00:03:24,011 Look we didn't change anything 63 00:03:24,011 --> 00:03:26,704 Because we divided both sides by a positive 3 64 00:03:26,704 --> 00:03:31,952 Alright? We could have done this equation in a slightly different way 65 00:03:31,952 --> 00:03:35,439 What if we subtracted 1 from both sides 66 00:03:35,439 --> 00:03:37,584 So this is another way of solving it 67 00:03:37,584 --> 00:03:42,392 What if we said 3X + 1 is equal to or less than 0 68 00:03:42,392 --> 00:03:44,464 I just subtracted 1 from both sides 69 00:03:44,464 --> 00:03:46,645 And now I'll subtract 3X from both sides 70 00:03:46,645 --> 00:03:51,457 I'll get 1 is less than or equal to -3X 71 00:03:51,457 --> 00:03:53,479 I subtracted 3X from here 72 00:03:53,479 --> 00:03:54,779 So I'll subtract 3X from here 73 00:03:54,779 --> 00:03:58,258 Now I'll have to divide both sides by a negative number 74 00:03:58,258 --> 00:04:02,347 Right? Because I'm going to divide both sides by -3 75 00:04:02,347 --> 00:04:05,472 So I get -1/3 on this side 76 00:04:05,472 --> 00:04:07,252 And based on what we had just learned 77 00:04:07,267 --> 00:04:08,485 Since we are dividing by a negative number 78 00:04:08,485 --> 00:04:10,255 We want to swap the inequality right? 79 00:04:10,255 --> 00:04:11,778 It was less than or equal to 80 00:04:11,778 --> 00:04:15,160 And it's going to be greater than or equal to X 81 00:04:15,160 --> 00:04:19,181 Now did we get the same answer when did both in two different ways? 82 00:04:22,571 --> 00:04:25,917 And here we got -1/3 is greater than or equal to X 83 00:04:19,181 --> 00:04:22,571 Here we got X is less than or equal to negative 1/3 84 00:04:25,917 --> 00:04:27,401 That's the same answer right? 85 00:04:27,401 --> 00:04:29,584 X is less than or equal to negative 1/3 86 00:04:29,584 --> 00:04:31,957 That's what I find to be the cool thing about algebra. 87 00:04:31,957 --> 00:04:33,964 You can tackle a problem in a bunch of two different ways. 88 00:04:33,964 --> 00:04:37,560 You should get the right answer as long as I guess you do it right. 89 00:04:37,560 --> 00:04:41,890 Let's do a couple more problems. 90 00:04:41,890 --> 00:04:46,859 Erase the thing. Here you go. Let's do a slightly harder one. 91 00:04:46,859 --> 00:04:56,751 Let's say - 8x + 7 > 5x +2 92 00:04:56,751 --> 00:05:02,138 Let's subtract 5x from both sides. 93 00:05:02,138 --> 00:05:06,085 - 13x + 7 > 2 94 00:05:06,085 --> 00:05:09,754 Now we can subtract 7 from both sides, 95 00:05:09,754 --> 00:05:13,469 -13x > -5. 96 00:05:13,469 --> 00:05:17,311 Now we're gonna divide both sides of this equation by -13. 97 00:05:17,311 --> 00:05:19,460 Well, very easy. 98 00:05:19,460 --> 00:05:24,940 It's just x, and on this side -5/-13 = 5/13, right? 99 00:05:24,940 --> 00:05:26,705 The negatives cancel out. 100 00:05:26,705 --> 00:05:30,068 And since we divided by a negative, 101 00:05:30,068 --> 00:05:31,849 we switch the sign. 102 00:05:31,849 --> 00:05:34,292 x is less than 5/13 103 00:05:34,292 --> 00:05:35,719 And once again, just like the beginning, 104 00:05:35,719 --> 00:05:37,966 if you don't believe me, try out some numbers. 105 00:05:37,966 --> 00:05:39,080 I remember when I first learned this 106 00:05:39,080 --> 00:05:40,734 I didn't believe the teacher so I did try out numbers 107 00:05:40,749 --> 00:05:44,538 and that's how I got convinced that it works 108 00:05:44,538 --> 00:05:47,050 when you multiply or divide both sides of this equation 109 00:05:47,050 --> 00:05:50,220 by a negative sign, you swap the inequality. 110 00:05:50,220 --> 00:05:53,040 And remember: that's only when you multiply or divide, 111 00:05:53,050 --> 00:05:55,781 not when you add or subtract. 112 00:05:55,781 --> 00:05:57,665 I think that should give you 113 00:05:57,665 --> 00:05:59,810 a good idea of how to do these problems. 114 00:05:59,810 --> 00:06:01,080 There's really not much new here. 115 00:06:01,080 --> 00:06:05,290 You do an inequality or-- I guess you could call this 116 00:06:05,290 --> 00:06:08,000 an inequality equation-- you do it exactly the same way 117 00:06:08,010 --> 00:06:10,085 you would do a normal linear equation. 118 00:06:10,085 --> 00:06:14,475 The only difference being is if you multiply or you divide 119 00:06:14,475 --> 00:06:16,030 both sides of the equation by a negative number, 120 00:06:16,040 --> 00:06:19,259 then you swap the inequality. 121 00:06:19,259 --> 00:06:22,050 I think you're ready now to try some practice problems. 122 00:06:22,060 --> 00:06:24,040 Have fun.