WEBVTT 00:00:00.590 --> 00:00:03.880 Let's do some equations that deal with absolute values. 00:00:03.880 --> 00:00:05.119 And just as a bit of a review, 00:00:05.119 --> 00:00:07.650 when you take the absolute value of a number. 00:00:07.650 --> 00:00:10.680 Let's say I take the absolute value of -1. 00:00:10.680 --> 00:00:12.263 What you're really doing is 00:00:12.263 --> 00:00:16.090 you're saying, how far is that number from 0? 00:00:16.090 --> 00:00:20.620 And in the case of -1, if we draw a number line right there 00:00:20.620 --> 00:00:23.310 -- that's a very badly drawn number line. 00:00:23.310 --> 00:00:26.230 If we draw a number line right there, that's 0. 00:00:26.230 --> 00:00:28.470 You have a -1 right there. 00:00:28.470 --> 00:00:30.230 Well, it's 1 away from 0. 00:00:30.230 --> 00:00:33.250 So the absolute value of -1 is 1. 00:00:33.250 --> 00:00:38.850 And the absolute value of 1 is also 1 away from 0. 00:00:38.850 --> 00:00:40.610 It's also equal to 1. 00:00:40.610 --> 00:00:43.500 So on some level, absolute value is the distance from 0. 00:00:43.500 --> 00:00:45.587 But another, I guess simpler way to think of it, 00:00:45.587 --> 00:00:48.600 it always results in the positive version of the number. 00:00:48.600 --> 00:00:59.360 The absolute value of -7,346 is equal to 7,346. 00:00:59.360 --> 00:01:00.779 So with that in mind, let's try to 00:01:00.779 --> 00:01:05.050 solve some equations with absolute values in them. 00:01:05.050 --> 00:01:06.675 So let's say I have the equation 00:01:06.675 --> 00:01:14.500 the absolute value of x -5 is equal to 10. 00:01:14.500 --> 00:01:15.895 And one way you can interpret this, 00:01:15.895 --> 00:01:18.161 and I want you to think about this, this is actually saying 00:01:18.161 --> 00:01:23.120 that the distance between x and 5 is equal to 10. 00:01:23.120 --> 00:01:26.750 So how many numbers that are exactly 10 away from 5? 00:01:26.750 --> 00:01:29.430 And you can already think of the solution to this equation, 00:01:29.430 --> 00:01:31.960 but I'll show you how to solve it systematically. 00:01:31.960 --> 00:01:36.510 Now this is going to be true in two situations. 00:01:36.510 --> 00:01:41.800 Either x -5 is equal to +10. 00:01:41.800 --> 00:01:44.630 If this evaluates out to +10, 00:01:44.630 --> 00:01:46.610 then when you take the absolute value of it, 00:01:46.610 --> 00:01:48.380 you're going to get +10. 00:01:48.380 --> 00:01:53.130 Or x - 5 might evaluate to -10. 00:01:53.130 --> 00:01:58.700 If x - 5 evaluated to -10, when you take the absolute value of it, 00:01:58.700 --> 00:01:59.950 you would get 10 again. 00:01:59.950 --> 00:02:04.280 So x - 5 could also be equal to -10. 00:02:04.280 --> 00:02:07.730 Both of these would satisfy this equation. 00:02:07.730 --> 00:02:08.958 Now, to solve this one, 00:02:08.958 --> 00:02:11.500 add 5 to both sides of this equation. 00:02:11.500 --> 00:02:14.160 You get x is equal to 15. 00:02:14.160 --> 00:02:17.830 To solve this one, add 5 to both sides of this equation. 00:02:17.830 --> 00:02:20.900 x is equal to -5. 00:02:20.900 --> 00:02:21.963 So our solution, 00:02:21.963 --> 00:02:24.910 there's two x's that satisfy this equation. 00:02:24.910 --> 00:02:26.890 x could be 15. 00:02:26.890 --> 00:02:29.502 15 - 5 is 10, take the absolute value, 00:02:29.502 --> 00:02:32.690 you're going to get 10, or x could be -5. 00:02:32.690 --> 00:02:36.060 - 5 minus 5 is -10. 00:02:36.060 --> 00:02:39.020 Take the absolute value, you get 10. 00:02:39.020 --> 00:02:41.632 And notice, both of these numbers 00:02:41.632 --> 00:02:45.750 are exactly 10 away from the number 5. 00:02:45.750 --> 00:02:48.050 Let's do another one of these. 00:02:48.050 --> 00:02:51.130 Let's do another one. 00:02:51.130 --> 00:02:52.182 Let's say we have 00:02:52.182 --> 00:02:58.580 the absolute value of x + 2 is equal to 6. 00:02:58.580 --> 00:02:59.610 So what does that tell us? 00:02:59.610 --> 00:03:03.132 That tells us that either x + 2, 00:03:03.132 --> 00:03:07.030 that the thing inside the absolute value sign, is equal to 6. 00:03:07.030 --> 00:03:10.380 Or the thing inside of the absolute value sign, 00:03:10.380 --> 00:03:12.050 the x + 2, could also be -6. 00:03:12.050 --> 00:03:13.910 If this whole thing evaluated to -6, 00:03:13.910 --> 00:03:16.210 you take the absolute value, you'd get 6. 00:03:16.210 --> 00:03:20.340 So, or x + 2 could equal -6. 00:03:20.340 --> 00:03:22.880 And then if you subtract 2 from both sides of this equation, 00:03:22.880 --> 00:03:25.850 you get x could be equal to 4. 00:03:25.850 --> 00:03:29.780 If you subtract 2 from both sides of this equation, 00:03:29.780 --> 00:03:33.690 you get x could be equal to -8. 00:03:33.690 --> 00:03:37.240 So these are the two solutions to the equation. 00:03:37.240 --> 00:03:39.740 And just to kind of have it gel in your mind, 00:03:39.740 --> 00:03:42.500 that absolute value, you can kind of view it as a distance, 00:03:42.500 --> 00:03:43.940 you could rewrite this problem 00:03:43.940 --> 00:03:50.410 as the absolute value of x minus -2 is equal to 6. 00:03:50.410 --> 00:03:52.759 And so this is asking me, 00:03:52.759 --> 00:03:57.590 what are the x's that are exactly 6 away from -2? 00:03:57.590 --> 00:03:59.168 Remember, up here we said, 00:03:59.168 --> 00:04:03.560 what are the x's that are exactly 10 away from +5? 00:04:03.560 --> 00:04:05.990 Whatever number you're subtracting from +5, 00:04:05.990 --> 00:04:08.560 these are both 10 away from +5. 00:04:08.560 --> 00:04:09.515 This is asking, 00:04:09.515 --> 00:04:13.080 what is exactly 6 away from -2? 00:04:13.080 --> 00:04:15.510 And it's going to be 4, or -8. 00:04:15.510 --> 00:04:17.959 You could try those numbers out for yourself. 00:04:17.959 --> 00:04:20.459 Let's do another one of these. 00:04:20.459 --> 00:04:25.330 Let's do another one, and we'll do it in purple. 00:04:25.330 --> 00:04:30.190 Let's say we have the absolute value of 4x. 00:04:30.190 --> 00:04:31.430 I'm going to change this problem up a little bit. 00:04:31.430 --> 00:04:33.390 4x -1. 00:04:33.390 --> 00:04:36.583 The absolute value of 4x -1, is equal to-- 00:04:36.583 --> 00:04:40.200 actually, I'll just keep it-- is equal to 19. 00:04:40.200 --> 00:04:41.769 So, just like the last few problems, 00:04:41.769 --> 00:04:47.640 4x -1 could be equal to 19. 00:04:47.640 --> 00:04:51.670 Or 4x -1 might evaluate to -19. 00:04:51.670 --> 00:04:53.130 Because then when you take the absolute value, 00:04:53.130 --> 00:04:54.800 you're going to get 19 again. 00:04:54.800 --> 00:04:59.100 Or 4x -1 could be equal to -19. 00:04:59.100 --> 00:05:00.970 Then you just solve these two equations. 00:05:00.970 --> 00:05:02.945 Add 1 to both sides of this equation-- 00:05:02.945 --> 00:05:04.274 we could do them simultaneous, even. 00:05:04.274 --> 00:05:08.510 Add 1 to both sides of this, you get 4x is equal to 20. 00:05:08.510 --> 00:05:11.005 Add 1 to both sides of this equation, 00:05:11.005 --> 00:05:15.340 you get 4x is equal to -18. 00:05:15.340 --> 00:05:20.210 Divide both sides of this by 4, you get x is equal to 5. 00:05:20.210 --> 00:05:23.920 Divide both sides of this by 4, you get x is equal to -18/4, 00:05:23.920 --> 00:05:31.770 which is equal to -9/2. 00:05:31.770 --> 00:05:35.730 So both of these x values satisfy the equation. 00:05:35.730 --> 00:05:36.587 Try it out. 00:05:36.587 --> 00:05:39.580 -9/2 x 4. 00:05:39.580 --> 00:05:41.570 This will become a -18. 00:05:41.570 --> 00:05:44.200 -18 minus 1 is -19. 00:05:44.200 --> 00:05:46.740 Take the absolute value, you get 19. 00:05:46.740 --> 00:05:49.920 You put a 5 here, 4 x 5 is 20. 00:05:49.920 --> 00:05:51.960 Minus 1 is +19. 00:05:51.960 --> 00:05:53.260 So you take the absolute value. 00:05:53.260 --> 00:05:55.920 Once again, you'll get a 19. 00:05:55.920 --> 00:05:58.580 Let's try to graph one of these, just for fun. 00:05:58.580 --> 00:05:59.283 So let's say 00:05:59.283 --> 00:06:04.990 I have y is equal to the absolute value of x +3. 00:06:04.990 --> 00:06:07.840 So this is a function, or a graph, 00:06:07.840 --> 00:06:09.410 with an absolute value in it. 00:06:09.410 --> 00:06:11.820 So let's think about two scenarios. 00:06:11.820 --> 00:06:13.136 There's one scenario 00:06:13.136 --> 00:06:16.430 where the thing inside of the absolute value is positive. 00:06:16.430 --> 00:06:18.873 So you have the scenario where x + 3 00:06:18.873 --> 00:06:23.420 I'll write it over here: x + 3 is > 0. 00:06:23.420 --> 00:06:29.370 And then you have the scenario where x +3 is < 0. 00:06:29.370 --> 00:06:32.658 When x +3 is > 0, 00:06:32.658 --> 00:06:36.490 this graph, or this line--or I guess we can't call it a line-- 00:06:36.490 --> 00:06:41.690 this function, is the same thing as y is equal to x +3. 00:06:41.690 --> 00:06:44.370 If this thing over here is > 0, 00:06:44.370 --> 00:06:46.750 then the absolute value sign is irrelevant. 00:06:46.750 --> 00:06:48.780 So then this thing is the same thing 00:06:48.780 --> 00:06:50.280 as y is equal to x +3. 00:06:50.280 --> 00:06:52.590 But when is x +3 > 0? 00:06:52.590 --> 00:06:56.366 Well, if you subtract 3 from both sides, 00:06:56.366 --> 00:06:59.910 you get x is > -3. 00:06:59.910 --> 00:07:02.249 So when x is > -3, 00:07:02.249 --> 00:07:08.460 this graph is going to look just like y is equal to x +3. 00:07:08.460 --> 00:07:11.500 Now, when x +3 is < 0. 00:07:11.500 --> 00:07:13.328 When the situation where this-- 00:07:13.328 --> 00:07:16.509 the inside of our absolute value sign--is negative, 00:07:16.509 --> 00:07:20.356 in that situation this equation is going to be 00:07:20.356 --> 00:07:26.250 y is equal to the negative of x +3. 00:07:26.250 --> 00:07:27.540 How can I say that? 00:07:27.540 --> 00:07:30.520 Well, look, if this is going to be a negative number, if x 00:07:30.520 --> 00:07:33.060 plus 3 is going to be a negative number-- that's what 00:07:33.060 --> 00:07:36.010 we're assuming here-- if it's going to be a negative number, 00:07:36.010 --> 00:07:38.090 then when you take the absolute value of a negative 00:07:38.090 --> 00:07:40.050 number, you're going to make it positive. 00:07:40.050 --> 00:07:43.280 That's just like multiplying it by negative 1. 00:07:43.280 --> 00:07:45.870 If you know you're taking the absolute value of a negative 00:07:45.870 --> 00:07:48.890 number, it's just like multiplying it by negative 1, 00:07:48.890 --> 00:07:51.010 because you're going to make it positive. 00:07:51.010 --> 00:07:53.870 And this is going to be the situation. 00:07:53.870 --> 00:07:55.840 x plus 3 is less than 0. 00:07:55.840 --> 00:07:59.850 If we subtract 3 from both sides, when x is less than 00:07:59.850 --> 00:08:01.280 negative 3. 00:08:01.280 --> 00:08:03.920 So when x is less than negative 3, the graph will 00:08:03.920 --> 00:08:05.040 look like this. 00:08:05.040 --> 00:08:08.280 When x is greater than negative 3, the graph will 00:08:08.280 --> 00:08:09.600 look like that. 00:08:09.600 --> 00:08:11.300 So let's see what that would make the 00:08:11.300 --> 00:08:13.670 entire graph look like. 00:08:13.670 --> 00:08:21.520 Let me draw my axes. 00:08:21.520 --> 00:08:26.070 That's my x-axis, that's my y-axis. 00:08:26.070 --> 00:08:29.090 So just let me multiply this out, just so we have it in mx 00:08:29.090 --> 00:08:29.870 plus b form. 00:08:29.870 --> 00:08:36.070 So this is equal to negative x minus 3. 00:08:36.070 --> 00:08:37.409 So let's just figure out what this graph would 00:08:37.409 --> 00:08:38.620 look like in general. 00:08:38.620 --> 00:08:42.020 Negative x minus 3. 00:08:42.020 --> 00:08:47.380 The y-intercept is negative 3, so 1, 2, 3. 00:08:47.380 --> 00:08:51.060 And negative x means it slopes downward, has a 00:08:51.060 --> 00:08:52.290 downward slope of 1. 00:08:52.290 --> 00:08:53.540 So it would look like this. 00:08:56.840 --> 00:09:02.830 The x-intercept would be at x is equal to--. 00:09:02.830 --> 00:09:07.740 So if you say y is equal to 0, that would happen when x is 00:09:07.740 --> 00:09:08.575 equal to negative 3. 00:09:08.575 --> 00:09:10.380 So it's going to go through that line, 00:09:10.380 --> 00:09:11.920 that point right there. 00:09:11.920 --> 00:09:14.190 And the graph, if we didn't have this constraint right 00:09:14.190 --> 00:09:15.600 here, would look something like this. 00:09:19.890 --> 00:09:22.760 That's if we didn't constrain it to a certain interval on 00:09:22.760 --> 00:09:23.880 the x-axis. 00:09:23.880 --> 00:09:27.080 Now this graph, what does it look like? 00:09:27.080 --> 00:09:27.480 Let's see. 00:09:27.480 --> 00:09:31.810 It has its y-intercept at positive 3. 00:09:31.810 --> 00:09:33.230 Just like that. 00:09:33.230 --> 00:09:35.260 And where's its x-intecept? 00:09:35.260 --> 00:09:37.970 When y is equal to 0, x is negative 3. 00:09:37.970 --> 00:09:39.760 So it also goes through that point right there, and it has 00:09:39.760 --> 00:09:40.620 a slope of 1. 00:09:40.620 --> 00:09:43.710 So it would look something like this. 00:09:43.710 --> 00:09:45.330 That's what this graph looks like. 00:09:45.330 --> 00:09:48.100 Now, what we figured out is that this absolute value 00:09:48.100 --> 00:09:52.030 function, it looks like this purple graph when x is less 00:09:52.030 --> 00:09:53.830 than negative 3. 00:09:53.830 --> 00:09:57.070 So when x is less than negative 3-- that's x is equal 00:09:57.070 --> 00:09:59.593 to negative 3 right there-- when x is less than negative 00:09:59.593 --> 00:10:03.170 3, it looks like this purple graph. 00:10:03.170 --> 00:10:04.570 Right there. 00:10:04.570 --> 00:10:07.390 So that's when x is less than negative 3. 00:10:07.390 --> 00:10:10.830 But when x is greater than negative 3, it looks like the 00:10:10.830 --> 00:10:12.160 green graph. 00:10:12.160 --> 00:10:14.640 It looks like that. 00:10:14.640 --> 00:10:17.480 So this graph looks like this strange v. 00:10:17.480 --> 00:10:21.430 When x is greater than negative 3, this is positive. 00:10:21.430 --> 00:10:24.950 So we have the graph of-- we have a positive slope. 00:10:24.950 --> 00:10:28.270 But then when x is less than negative 3, we're essentially 00:10:28.270 --> 00:10:30.550 taking the negative of the function, if you want to view 00:10:30.550 --> 00:10:32.280 it that way, and so we have this negative slope. 00:10:32.280 --> 00:10:35.060 So you kind of have this v-shaped function, this 00:10:35.060 --> 00:10:38.250 v-shaped graph, which is indicative of an absolute 00:10:38.250 --> 00:10:39.950 value function.