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