1 00:00:00,000 --> 00:00:00,410 2 00:00:00,410 --> 00:00:03,470 Let's do some compound inequality problems, and these 3 00:00:03,470 --> 00:00:07,450 are just inequality problems that have more than one set of 4 00:00:07,450 --> 00:00:08,060 constraints. 5 00:00:08,060 --> 00:00:10,019 You're going to see what I'm talking about in a second. 6 00:00:10,019 --> 00:00:15,109 So the first problem I have is negative 5 is less than or 7 00:00:15,109 --> 00:00:22,300 equal to x minus 4, which is also less than or equal to 13. 8 00:00:22,300 --> 00:00:25,640 So we have two sets of constraints on the set of x's 9 00:00:25,640 --> 00:00:27,426 that satisfy these equations. 10 00:00:27,425 --> 00:00:31,269 x minus 4 has to be greater than or equal to negative 5 11 00:00:31,269 --> 00:00:36,240 and x minus 4 has to be less than or equal to 13. 12 00:00:36,240 --> 00:00:40,179 So we could rewrite this compound inequality as 13 00:00:40,179 --> 00:00:49,039 negative 5 has to be less than or equal to x minus 4, and x 14 00:00:49,039 --> 00:00:57,799 minus 4 needs to be less than or equal to 13. 15 00:00:57,799 --> 00:00:59,989 And then we could solve each of these separately, and then 16 00:00:59,990 --> 00:01:02,219 we have to remember this "and" there to think about the 17 00:01:02,219 --> 00:01:05,069 solution set because it has to be things that satisfy this 18 00:01:05,069 --> 00:01:07,200 equation and this equation. 19 00:01:07,200 --> 00:01:09,659 So let's solve each of them individually. 20 00:01:09,659 --> 00:01:12,429 So this one over here, we can add 4 to both 21 00:01:12,430 --> 00:01:13,680 sides of the equation. 22 00:01:13,680 --> 00:01:17,090 23 00:01:17,090 --> 00:01:21,840 The left-hand side, negative 5 plus 4, is negative 1. 24 00:01:21,840 --> 00:01:26,120 Negative 1 is less than or equal to x, right? 25 00:01:26,120 --> 00:01:28,850 These 4's just cancel out here and you're just left with an x 26 00:01:28,849 --> 00:01:30,619 on this right-hand side. 27 00:01:30,620 --> 00:01:37,120 So the left, this part right here, simplifies to x needs to 28 00:01:37,120 --> 00:01:40,780 be greater than or equal to negative 1 or negative 1 is 29 00:01:40,780 --> 00:01:42,280 less than or equal to x. 30 00:01:42,280 --> 00:01:43,579 So we can also write it like this. 31 00:01:43,579 --> 00:01:46,109 X needs to be greater than or equal to negative 1. 32 00:01:46,109 --> 00:01:46,950 These are equivalent. 33 00:01:46,950 --> 00:01:48,799 I just swapped the sides. 34 00:01:48,799 --> 00:01:51,506 Now let's do this other condition here in green. 35 00:01:51,506 --> 00:01:55,959 36 00:01:55,959 --> 00:01:57,909 Let's add 4 to both sides of this equation. 37 00:01:57,909 --> 00:02:01,759 38 00:02:01,760 --> 00:02:04,329 The left-hand side, we just get an x. 39 00:02:04,329 --> 00:02:07,379 And then the right-hand side, we get 13 plus 40 00:02:07,379 --> 00:02:09,840 14, which is 17. 41 00:02:09,840 --> 00:02:13,800 So we get x is less than or equal to 17. 42 00:02:13,800 --> 00:02:17,969 So our two conditions, x has to be greater than or equal to 43 00:02:17,969 --> 00:02:22,310 negative 1 and less than or equal to 17. 44 00:02:22,310 --> 00:02:24,460 So we could write this again as a compound 45 00:02:24,460 --> 00:02:25,700 inequality if we want. 46 00:02:25,699 --> 00:02:29,469 We can say that the solution set, that x has to be less 47 00:02:29,469 --> 00:02:34,949 than or equal to 17 and greater than or equal to 48 00:02:34,949 --> 00:02:35,549 negative 1. 49 00:02:35,550 --> 00:02:38,750 It has to satisfy both of these conditions. 50 00:02:38,750 --> 00:02:43,675 So what would that look like on a number line? 51 00:02:43,675 --> 00:02:46,250 So let's put our number line right there. 52 00:02:46,250 --> 00:02:48,590 Let's say that this is 17. 53 00:02:48,590 --> 00:02:50,090 Maybe that's 18. 54 00:02:50,090 --> 00:02:51,039 You keep going down. 55 00:02:51,039 --> 00:02:52,179 Maybe this is 0. 56 00:02:52,180 --> 00:02:55,620 I'm obviously skipping a bunch of stuff in between. 57 00:02:55,620 --> 00:02:58,650 Then we would have a negative 1 right there, maybe a 58 00:02:58,650 --> 00:02:59,920 negative 2. 59 00:02:59,919 --> 00:03:03,629 So x is greater than or equal to negative 1, so we would 60 00:03:03,629 --> 00:03:04,609 start at negative 1. 61 00:03:04,610 --> 00:03:07,000 We're going to circle it in because we have a greater than 62 00:03:07,000 --> 00:03:08,680 or equal to. 63 00:03:08,680 --> 00:03:13,439 And then x is greater than that, but it has to be less 64 00:03:13,439 --> 00:03:17,579 than or equal to 17. 65 00:03:17,580 --> 00:03:21,170 So it could be equal to 17 or less than 17. 66 00:03:21,169 --> 00:03:23,649 So this right here is a solution set, everything that 67 00:03:23,650 --> 00:03:25,710 I've shaded in orange. 68 00:03:25,710 --> 00:03:28,849 And if we wanted to write it in interval notation, it would 69 00:03:28,849 --> 00:03:35,030 be x is between negative 1 and 17, and it can also equal 70 00:03:35,030 --> 00:03:37,120 negative 1, so we put a bracket, and it 71 00:03:37,120 --> 00:03:39,509 can also equal 17. 72 00:03:39,509 --> 00:03:43,349 So this is the interval notation for this compound 73 00:03:43,349 --> 00:03:45,329 inequality right there. 74 00:03:45,330 --> 00:03:46,580 Let's do another one. 75 00:03:46,580 --> 00:03:49,920 76 00:03:49,919 --> 00:03:51,979 Let me get a good problem here. 77 00:03:51,979 --> 00:03:56,619 Let's say that we have negative 12. 78 00:03:56,620 --> 00:03:58,640 I'm going to change the problem a little bit from the 79 00:03:58,639 --> 00:04:00,269 one that I've found here. 80 00:04:00,270 --> 00:04:08,230 Negative 12 is less than 2 minus 5x, which is less than 81 00:04:08,229 --> 00:04:10,229 or equal to 7. 82 00:04:10,229 --> 00:04:12,949 I want to do a problem that has just the less than and a 83 00:04:12,949 --> 00:04:14,629 less than or equal to. 84 00:04:14,629 --> 00:04:16,709 The problem in the book that I'm looking at has an equal 85 00:04:16,709 --> 00:04:18,600 sign here, but I want to remove that intentionally 86 00:04:18,600 --> 00:04:20,500 because I want to show you when you have a hybrid 87 00:04:20,500 --> 00:04:22,389 situation, when you have a little bit of both. 88 00:04:22,389 --> 00:04:28,310 So first we can separate this into two normal inequalities. 89 00:04:28,310 --> 00:04:31,939 You have this inequality right there. 90 00:04:31,939 --> 00:04:37,529 We know that negative 12 needs to be less than 2 minus 5x. 91 00:04:37,529 --> 00:04:43,229 That has to be satisfied, and-- let me do it in another 92 00:04:43,230 --> 00:04:46,810 color-- this inequality also needs to be satisfied. 93 00:04:46,810 --> 00:04:50,740 2 minus 5x has to be less than 7 and greater than 12, less 94 00:04:50,740 --> 00:04:56,530 than or equal to 7 and greater than negative 12, so and 2 95 00:04:56,529 --> 00:05:02,139 minus 5x has to be less than or equal to 7. 96 00:05:02,139 --> 00:05:05,289 So let's just solve this the way we solve everything. 97 00:05:05,290 --> 00:05:08,050 Let's get this 2 onto the left-hand side here. 98 00:05:08,050 --> 00:05:11,730 So let's subtract 2 from both sides of this equation. 99 00:05:11,730 --> 00:05:15,500 So if you subtract 2 from both sides of this equation, the 100 00:05:15,500 --> 00:05:19,560 left-hand side becomes negative 14, is less than-- 101 00:05:19,560 --> 00:05:23,829 these cancel out-- less than negative 5x. 102 00:05:23,829 --> 00:05:27,139 Now let's divide both sides by negative 5. 103 00:05:27,139 --> 00:05:29,360 And remember, when you multiply or divide by a 104 00:05:29,360 --> 00:05:32,139 negative number, the inequality swaps around. 105 00:05:32,139 --> 00:05:35,879 So if you divide both sides by negative 5, you get a negative 106 00:05:35,879 --> 00:05:39,980 14 over negative 5, and you have an x on the right-hand 107 00:05:39,980 --> 00:05:43,140 side, if you divide that by negative 5, and this swaps 108 00:05:43,139 --> 00:05:47,919 from a less than sign to a greater than sign. 109 00:05:47,920 --> 00:05:53,560 The negatives cancel out, so you get 14/5 is greater than 110 00:05:53,560 --> 00:05:58,579 x, or x is less than 14/5, which is-- what is this? 111 00:05:58,579 --> 00:06:01,379 This is 2 and 4/5. 112 00:06:01,379 --> 00:06:04,319 x is less than 2 and 4/5. 113 00:06:04,319 --> 00:06:08,089 I just wrote this improper fraction as a mixed number. 114 00:06:08,089 --> 00:06:10,619 Now let's do the other constraint 115 00:06:10,620 --> 00:06:12,629 over here in magenta. 116 00:06:12,629 --> 00:06:15,209 So let's subtract 2 from both sides of this equation, just 117 00:06:15,209 --> 00:06:16,799 like we did before. 118 00:06:16,800 --> 00:06:19,910 And actually, you can do these simultaneously, but it becomes 119 00:06:19,910 --> 00:06:20,770 kind of confusing. 120 00:06:20,769 --> 00:06:23,359 So to avoid careless mistakes, I encourage you to separate it 121 00:06:23,360 --> 00:06:24,650 out like this. 122 00:06:24,649 --> 00:06:27,299 So if you subtract 2 from both sides of the equation, the 123 00:06:27,300 --> 00:06:30,680 left-hand side becomes negative 5x. 124 00:06:30,680 --> 00:06:33,100 The right-hand side, you have less than or equal to. 125 00:06:33,100 --> 00:06:37,620 The right-hand side becomes 7 minus 2, becomes 5. 126 00:06:37,620 --> 00:06:40,780 Now, you divide both sides by negative 5. 127 00:06:40,779 --> 00:06:42,369 On the left-hand side, you get an x. 128 00:06:42,370 --> 00:06:46,449 On the right-hand side, 5 divided by negative 5 is 129 00:06:46,449 --> 00:06:47,599 negative 1. 130 00:06:47,600 --> 00:06:50,439 And since we divided by a negative number, we swap the 131 00:06:50,439 --> 00:06:51,379 inequality. 132 00:06:51,379 --> 00:06:53,310 It goes from less than or equal to, to greater 133 00:06:53,310 --> 00:06:54,610 than or equal to. 134 00:06:54,610 --> 00:06:56,819 So we have our two constraints. 135 00:06:56,819 --> 00:07:01,509 x has to be less than 2 and 4/5, and it has to be greater 136 00:07:01,509 --> 00:07:03,719 than or equal to negative 1. 137 00:07:03,720 --> 00:07:05,600 So we could write it like this. 138 00:07:05,600 --> 00:07:10,300 x has to be greater than or equal to negative 1, so that 139 00:07:10,300 --> 00:07:13,389 would be the lower bound on our interval, and it has to be 140 00:07:13,389 --> 00:07:14,939 less than 2 and 4/5. 141 00:07:14,939 --> 00:07:20,719 142 00:07:20,720 --> 00:07:22,590 And notice, not less than or equal to. 143 00:07:22,589 --> 00:07:24,509 That's why I wanted to show you, you have the parentheses 144 00:07:24,509 --> 00:07:26,810 there because it can't be equal to 2 and 4/5. 145 00:07:26,810 --> 00:07:29,579 x has to be less than 2 and 4/5. 146 00:07:29,579 --> 00:07:31,502 Or we could write this way. 147 00:07:31,502 --> 00:07:37,009 x has to be less than 2 and 4/5, that's just this 148 00:07:37,009 --> 00:07:40,539 inequality, swapping the sides, and it has to be 149 00:07:40,540 --> 00:07:44,670 greater than or equal to negative 1. 150 00:07:44,670 --> 00:07:47,210 So these two statements are equivalent. 151 00:07:47,209 --> 00:07:52,039 And if I were to draw it on a number line, it 152 00:07:52,040 --> 00:07:53,490 would look like this. 153 00:07:53,490 --> 00:08:00,410 So you have a negative 1, you have 2 and 4/5 over here. 154 00:08:00,410 --> 00:08:01,850 Obviously, you'll have stuff in between. 155 00:08:01,850 --> 00:08:03,580 Maybe, you know, 0 sitting there. 156 00:08:03,579 --> 00:08:06,639 We have to be greater than or equal to negative 1, so we can 157 00:08:06,639 --> 00:08:08,099 be equal to negative 1. 158 00:08:08,100 --> 00:08:10,220 And we're going to be greater than negative 1, but we also 159 00:08:10,220 --> 00:08:12,700 have to be less than 2 and 4/5. 160 00:08:12,699 --> 00:08:14,779 So we can't include 2 and 4/5 there. 161 00:08:14,779 --> 00:08:18,099 We can't be equal to 2 and 4/5, so we can only be less 162 00:08:18,100 --> 00:08:22,590 than, so we put a empty circle around 2 and 4/5 and then we 163 00:08:22,589 --> 00:08:24,959 fill in everything below that, all the way down to negative 164 00:08:24,959 --> 00:08:27,579 1, and we include negative 1 because we have this less than 165 00:08:27,579 --> 00:08:29,120 or equal sign. 166 00:08:29,120 --> 00:08:31,819 So the last two problems I did are kind of "and" problems. 167 00:08:31,819 --> 00:08:34,418 You have to meet both of these constraints. 168 00:08:34,418 --> 00:08:36,024 Now, let's do an "or" problem. 169 00:08:36,024 --> 00:08:38,798 170 00:08:38,798 --> 00:08:42,620 So let's say I have these inequalities. 171 00:08:42,620 --> 00:08:49,909 Let's say I'm given-- let's say that 4x minus 1 needs to 172 00:08:49,909 --> 00:08:58,769 be greater than or equal to 7, or 9x over 2 needs to 173 00:08:58,769 --> 00:09:00,289 be less than 3. 174 00:09:00,289 --> 00:09:03,459 So now when we're saying "or," an x that would satisfy these 175 00:09:03,460 --> 00:09:06,400 are x's that satisfy either of these equations. 176 00:09:06,399 --> 00:09:09,250 In the last few videos or in the last few problems, we had 177 00:09:09,250 --> 00:09:11,769 to find x's that satisfied both of these equations. 178 00:09:11,769 --> 00:09:14,259 Here, this is much more lenient. 179 00:09:14,259 --> 00:09:16,779 We just have to satisfy one of these two. 180 00:09:16,779 --> 00:09:19,139 So let's figure out the solution sets for both of 181 00:09:19,139 --> 00:09:21,679 these and then we figure out essentially their union, their 182 00:09:21,679 --> 00:09:22,979 combination, all of the things that'll 183 00:09:22,980 --> 00:09:25,100 satisfy either of these. 184 00:09:25,100 --> 00:09:27,009 So on this one, on the one on the left, we can 185 00:09:27,009 --> 00:09:29,490 add 1 to both sides. 186 00:09:29,490 --> 00:09:31,440 You add 1 to both sides. 187 00:09:31,440 --> 00:09:35,480 The left-hand side just becomes 4x is greater than or 188 00:09:35,480 --> 00:09:39,840 equal to 7 plus 1 is 8. 189 00:09:39,840 --> 00:09:42,120 Divide both sides by 4. 190 00:09:42,120 --> 00:09:46,120 You get x is greater than or equal to 2. 191 00:09:46,120 --> 00:09:48,789 Or let's do this one. 192 00:09:48,789 --> 00:09:51,769 Let's see, if we multiply both sides of this equation by 2/9, 193 00:09:51,769 --> 00:09:53,069 what do we get? 194 00:09:53,070 --> 00:09:56,070 If you multiply both sides by 2/9, it's a positive number, 195 00:09:56,070 --> 00:09:58,570 so we don't have to do anything to the inequality. 196 00:09:58,570 --> 00:10:06,760 These cancel out, and you get x is less than 3 times 2/9. 197 00:10:06,759 --> 00:10:10,610 3/9 is the same thing as 1/3, so x needs to 198 00:10:10,610 --> 00:10:12,460 be less than 2/3. 199 00:10:12,460 --> 00:10:17,280 So or x is less than 2/3. 200 00:10:17,279 --> 00:10:18,839 So that's our solution set. 201 00:10:18,840 --> 00:10:23,240 x needs to be greater than or equal to 2, or less than 2/3. 202 00:10:23,240 --> 00:10:24,340 So this is interesting. 203 00:10:24,340 --> 00:10:27,920 Let me plot the solution set on the number line. 204 00:10:27,919 --> 00:10:31,240 205 00:10:31,240 --> 00:10:33,379 So that is our number line. 206 00:10:33,379 --> 00:10:41,629 Maybe this is 0, this is 1, this is 2, 3, maybe that is 207 00:10:41,629 --> 00:10:42,939 negative 1. 208 00:10:42,940 --> 00:10:46,910 So x can be greater than or equal to 2. 209 00:10:46,909 --> 00:10:49,959 So we could start-- let me do it in another color. 210 00:10:49,960 --> 00:10:53,320 We can start at 2 here and it would be greater than or equal 211 00:10:53,320 --> 00:10:58,690 to 2, so include everything greater than or equal to 2. 212 00:10:58,690 --> 00:11:01,530 That's that condition right there. 213 00:11:01,529 --> 00:11:03,439 Or x could be less than 2/3. 214 00:11:03,440 --> 00:11:06,550 215 00:11:06,549 --> 00:11:11,120 So 2/3 is going to be right around here, right? 216 00:11:11,120 --> 00:11:13,679 That is 2/3. 217 00:11:13,679 --> 00:11:16,849 x could be less than 2/3. 218 00:11:16,850 --> 00:11:19,149 And this is interesting. 219 00:11:19,149 --> 00:11:21,490 Because if we pick one of these numbers, it's going to 220 00:11:21,490 --> 00:11:23,019 satisfy this inequality. 221 00:11:23,019 --> 00:11:24,840 If we pick one of these numbers, it's going to satisfy 222 00:11:24,840 --> 00:11:25,850 that inequality. 223 00:11:25,850 --> 00:11:28,500 If we had an "and" here, there would have been no numbers 224 00:11:28,500 --> 00:11:32,419 that satisfy it because you can't be both greater than 2 225 00:11:32,419 --> 00:11:34,629 and less than 2/3. 226 00:11:34,629 --> 00:11:37,019 So the only way that there's any solution set here is 227 00:11:37,019 --> 00:11:40,740 because it's "or." You can satisfy one of the two 228 00:11:40,740 --> 00:11:41,919 inequalities. 229 00:11:41,919 --> 00:11:44,479 Anyway, hopefully you, found that fun. 230 00:11:44,480 --> 00:11:44,932