1
00:00:05,000 --> 00:00:09,600
Okay, what does this have written on the board?

2
00:00:13,040 --> 00:00:14,560
It's an absolute-value inequality.

3
00:00:17,880 --> 00:00:19,440
And it has two variables.

4
00:00:19,640 --> 00:00:20,620
So what does that tell us?

5
00:00:20,620 --> 00:00:28,200
The solutions are ordered pairs, so we graph the solution set on a coordinate plane.

6
00:00:33,620 --> 00:00:34,880
This is an equation.

7
00:00:37,180 --> 00:00:39,720
What does the graph of this equation look like?

8
00:00:44,000 --> 00:00:45,400
We graphed these last week?

9
00:00:45,880 --> 00:00:46,780
Oh, oh, oh.

10
00:00:46,680 --> 00:00:47,720
Four to the right.

11
00:00:48,200 --> 00:00:49,620
It's a V looking thing.

12
00:00:49,700 --> 00:00:50,600
Oh yeah.

13
00:00:50,580 --> 00:00:52,220
Is it a V or an upside down V?

14
00:00:52,220 --> 00:00:53,680
No, just a V.

15
00:00:54,100 --> 00:00:55,280
A V because of what?

16
00:00:55,900 --> 00:00:57,360
Because the slope is positive.

17
00:00:57,720 --> 00:00:59,160
Positive coefficient there.

18
00:01:00,600 --> 00:01:01,500
OK.

19
00:01:03,060 --> 00:01:05,240
Let's graph this for a moment.

20
00:01:06,520 --> 00:01:08,300
We're going to use some transformations.

21
00:01:08,980 --> 00:01:10,460
Where's the vertex going to be?

22
00:01:13,180 --> 00:01:14,760
The vertex is at 3, negative 4.

23
00:01:15,940 --> 00:01:18,005
In vertex form, use the opposite of the number inside the absolute value and the

24
00:01:18,005 --> 00:01:18,755
number outside.

25
00:01:22,500 --> 00:01:28,120
The boundary is a V with slopes 2 and negative 2.

26
00:01:29,060 --> 00:01:37,360
Starting at the vertex, use rise 2 and run 1 on each arm.

27
00:01:37,360 --> 00:01:40,406
The boundary graph is part of the solution set because the inequality includes

28
00:01:40,406 --> 00:01:41,156
equality.

29
00:01:44,540 --> 00:01:49,420
Are all of those ordered pairs also solutions to the inequality?

30
00:01:51,280 --> 00:01:52,400
Definitely, because what?

31
00:01:52,820 --> 00:01:54,621
Every ordered pair on the solid boundary satisfies the equality part of the

32
00:01:54,621 --> 00:01:55,371
inequality.

33
00:01:57,660 --> 00:01:58,560
Okay.

34
00:01:58,240 --> 00:02:03,728
If the inequality were strictly less than, the boundary would be dashed instead of

35
00:02:03,728 --> 00:02:04,478
solid.

36
00:02:08,160 --> 00:02:12,540
Okay, so this is just some of the solutions

37
00:02:13,280 --> 00:02:15,400
There are many more solutions beyond the boundary.

38
00:02:17,360 --> 00:02:18,740
Shade one side of the boundary to show all remaining solutions.

39
00:02:27,280 --> 00:02:29,180
And how do I decide which side to shade?

40
00:02:30,300 --> 00:02:31,200
Use a test point to decide which side of the boundary to shade.

41
00:02:31,000 --> 00:02:31,900
Let's do a test point.

42
00:02:35,400 --> 00:02:36,944
The origin, zero comma zero, is a convenient test point when it is not on the

43
00:02:36,944 --> 00:02:37,694
boundary.

44
00:02:40,460 --> 00:02:44,840
Test zero less than or equal to 2 times the absolute value of zero minus 3, minus 4.

45
00:02:46,760 --> 00:02:48,000
The absolute value is 3, so the right side is 6 minus 4, which is 2.

46
00:02:50,020 --> 00:02:57,200
Zero is less than or equal to 2, so the origin is in the solution region.

47
00:02:57,200 --> 00:03:05,580
Shade the entire side of the V that contains the origin.

48
00:03:05,580 --> 00:03:12,080
Every point in the shaded green region is a solution to the two-variable

49
00:03:12,080 --> 00:03:13,080
absolute-value inequality.

50
00:03:14,740 --> 00:03:20,233
Graph the boundary with the same transformations used for an equation, then shade

51
00:03:20,233 --> 00:03:21,500
the additional solutions.

52
00:03:31,900 --> 00:03:35,180
Okay, so what am I going to draw first?

53
00:03:38,160 --> 00:03:46,418
The boundary equation is y equals negative one half times the absolute value of x

54
00:03:46,418 --> 00:03:48,620
plus 1, plus 1.

55
00:03:48,620 --> 00:03:55,454
Use a dashed boundary because the inequality is strict and does not include

56
00:03:55,454 --> 00:03:56,204
equality.

57
00:03:55,980 --> 00:04:01,020
Points on a dashed boundary are not solutions.

58
00:04:01,020 --> 00:04:12,400
The vertex is at negative 1, 1.

59
00:04:12,400 --> 00:04:28,440
The negative coefficient reflects the graph, so the boundary is an upside-down V.

60
00:04:28,880 --> 00:04:34,420
The two slopes are one half and negative one half.

61
00:04:34,420 --> 00:04:36,300
Now choose a test point; the origin works because it is not on the boundary.

62
00:04:38,320 --> 00:04:39,460
Can we choose zero, zero?

63
00:04:40,760 --> 00:04:42,620
Sure, as long as it's not on that line.

64
00:04:42,860 --> 00:04:44,880
If it's on the line, we've got to choose any other ordered pair.

65
00:04:50,020 --> 00:05:02,887
Test whether 0 is greater than negative one half times the absolute value of 0 plus

66
00:05:02,887 --> 00:05:05,300
1, plus 1.

67
00:05:07,520 --> 00:05:08,820
The right side is one half, so 0 is not greater than one half.

68
00:05:13,400 --> 00:05:23,020
The origin is not a solution, so shade the region on the other side of the dashed V.