Solve absolute value equations'

1
Section 4.3

• Solve absolute value equations.
• Use boundary points to solve absolute value
  inequalities.
• Use equivalent compound inequalities to solve
  absolute value inequalities.
• Solve problems using absolute value inequalities.

2
Absolute Value

• The absolute value of a, written a, is the
  distance from 0 to a.

3
Absolute Value

• The absolute value of a, written a, is the
  distance from 0 to a.
• 2 2

4
Absolute Value

• The absolute value of a, written a, is the
  distance from 0 to a.
• 2 2
• -2 2

5
Absolute Value

• The absolute value of a, written a, is the
  distance from 0 to a.
• 2 2
• -2 2
• x 2
• Then x 2 or x -2

6
Rewriting an Absolute Value Equation without
absolute value bars

• If c is a positive real number and X represents
  any algebraic expression, then X c is
  equivalent to X c or X -c.

7
Solve 2x 3 11

• 2x 3 11 original equation

8
Solve 2x 3 11

• 2x 3 11 original equation
• 2x 3 11 or 2x 3
  -11

9
Solve 2x 3 11

• 2x 3 11 original equation
• 2x 3 11 or 2x 3
  -11
• 2x 14

10
Solve 2x 3 11

• 2x 3 11 original equation
• 2x 3 11 or 2x 3
  -11
• 2x 14
• x 7

11
Solve 2x 3 11

• 2x 3 11 original equation
• 2x 3 11 or 2x 3
  -11
• 2x 14 2x
  -8
• x 7

12
Solve 2x 3 11

• 2x 3 11 original equation
• 2x 3 11 or 2x 3
  -11
• 2x 14 2x
  -8
• x 7
  x -4

13
Solve 2x 3 11

• 2x 3 11 original equation
• 2x 3 11 or 2x 3
  -11
• 2x 14 2x
  -8
• x 7
  x -4
• Check2(7) 3 11 yes

14
Solve 2x 3 11

• 2x 3 11 original equation
• 2x 3 11 or 2x 3
  -11
• 2x 14 2x
  -8
• x 7
  x -4
• Check2(7) 3 11 yes 2(-4) 3 11 yes

15
Solve 51 4x 15 0
16
Solve 51 4x 15 0

• 51 4x 15 0 original equation

17
Solve 51 4x 15 0

• 51 4x 15 0 original equation
• 51 4x 15 add 15 to both
  sides

18
Solve 51 4x 15 0

• 51 4x 15 0 original equation
• 51 4x 15 add 15 to both
  sides
• 1 4x 3 divide both sides
  by 5

19
Solve 51 4x 15 0

• 51 4x 15 0 original equation
• 51 4x 15 add 15 to both
  sides
• 1 4x 3 divide both sides
  by 5
• 1 4x 3 or 1 4x
  -3

20
Solve 51 4x 15 0

• 51 4x 15 0 original equation
• 51 4x 15 add 15 to both
  sides
• 1 4x 3 divide both sides
  by 5
• 1 4x 3 or 1 4x
  -3
• 4x 2

21
Solve 51 4x 15 0

• 51 4x 15 0 original equation
• 51 4x 15 add 15 to both
  sides
• 1 4x 3 divide both sides
  by 5
• 1 4x 3 or 1 4x
  -3
• 4x 2
• x (1/2)

22
Solve 51 4x 15 0

• 51 4x 15 0 original equation
• 51 4x 15 add 15 to both
  sides
• 1 4x 3 divide both sides
  by 5
• 1 4x 3 or 1 4x
  -3
• 4x 2 4x
  -4
• x (1/2)

23
Solve 51 4x 15 0

• 51 4x 15 0 original equation
• 51 4x 15 add 15 to both
  sides
• 1 4x 3 divide both sides
  by 5
• 1 4x 3 or 1 4x
  -3
• 4x 2 4x
  -4
• x (1/2)
  x 1

24
Solve 51 4x 15 0

• 51 4x 15 0 original equation
• 51 4x 15 add 15 to both
  sides
• 1 4x 3 divide both sides
  by 5
• 1 4x 3 or 1 4x
  -3
• 4x 2 4x
  -4
• x (1/2)
  x 1
• Verify your solutions

25
Equations with 2 absolute values

• If X1 X2, then X1 X2 or X1 -X2

26
Solve 3x 1 x 5
27
Solve 3x 1 x 5

• 3x 1 x 5 original
  equation

28
Solve 3x 1 x 5

• 3x 1 x 5 original
  equation
• 3x 1 x 5 or 3x 1 (
  x 5)

29
Solve 3x 1 x 5

• 3x 1 x 5 original
  equation
• 3x 1 x 5 or 3x 1 (
  x 5)
• 2x 1 5

30
Solve 3x 1 x 5

• 3x 1 x 5 original
  equation
• 3x 1 x 5 or 3x 1 (
  x 5)
• 2x 1 5
• 2x 6

31
Solve 3x 1 x 5

• 3x 1 x 5 original
  equation
• 3x 1 x 5 or 3x 1 (
  x 5)
• 2x 1 5
• 2x 6
• x 3

32
Solve 3x 1 x 5

• 3x 1 x 5 original
  equation
• 3x 1 x 5 or 3x 1 (
  x 5)
• 2x 1 5 3x 1
  x 5
• 2x 6
• x 3

33
Solve 3x 1 x 5

• 3x 1 x 5 original
  equation
• 3x 1 x 5 or 3x 1 (
  x 5)
• 2x 1 5 3x 1
  x 5
• 2x 6 4x 1
  -5
• x 3

34
Solve 3x 1 x 5

• 3x 1 x 5 original
  equation
• 3x 1 x 5 or 3x 1 (
  x 5)
• 2x 1 5 3x 1
  x 5
• 2x 6 4x 1
  -5
• x 3 4x
  -4

35
Solve 3x 1 x 5

• 3x 1 x 5 original
  equation
• 3x 1 x 5 or 3x 1 (
  x 5)
• 2x 1 5 3x 1
  x 5
• 2x 6 4x 1
  -5
• x 3 4x
  -4
• 
  x -1

36
Solve 3x 1 x 5

• 3x 1 x 5 original
  equation
• 3x 1 x 5 or 3x 1 (
  x 5)
• 2x 1 5 3x 1
  x 5
• 2x 6 4x 1
  -5
• x 3 4x
  -4
• 
  x -1
• Verify your answer

37
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39
Solve and graph x 4 lt 3
40
Solve and graph x 4 lt 3
Find the boundary points for x 4 3
41
Solve and graph x 4 lt 3
Find the boundary points for x 4 3 x 4
3 or x 4
-3
42
Solve and graph x 4 lt 3
Find the boundary points for x 4 3 x 4
3 or x 4
-3 x 7
43
Solve and graph x 4 lt 3
Find the boundary points for x 4 3 x 4
3 or x 4
-3 x 7
x 1
44
Solve and graph x 4 lt 3
Find the boundary points for x 4 3 x 4
3 or x 4
-3 x 7
x 1 Locate the boundary points on a
number line.
45
Solve and graph x 4 lt 3
Find the boundary points for x 4 3 x 4
3 or x 4
-3 x 7
x 1 Locate the boundary points on a
number line. Now choose test values in each
interval and test them by plugging them into the
original inequality.
46
Solve and graph x 4 lt 3
Find the boundary points for x 4 3 x 4
3 or x 4
-3 x 7
x 1 Locate the boundary points on a
number line. Now choose test values in each
interval and test them by plugging them into the
original inequality. Graph the solution set.
47
Solve and graph 2x 3 5
48
Solve and graph 2x 3 5
Find the boundary points for 2x 3 5
49
Solve and graph 2x 3 5
Find the boundary points for 2x 3 5 2x 3
5 or 2x
3 -5
50
Solve and graph 2x 3 5
Find the boundary points for 2x 3 5 2x 3
5 or 2x
3 -5 2x 2
51
Solve and graph 2x 3 5
Find the boundary points for 2x 3 5 2x 3
5 or 2x
3 -5 2x 2 x 1
52
Solve and graph 2x 3 5
Find the boundary points for 2x 3 5 2x 3
5 or 2x
3 -5 2x 2
2x -8 x 1
53
Solve and graph 2x 3 5
Find the boundary points for 2x 3 5 2x 3
5 or 2x
3 -5 2x 2
2x -8 x 1
x -4
54
Solve and graph 2x 3 5
Find the boundary points for 2x 3 5 2x 3
5 or 2x
3 -5 2x 2
2x -8 x 1
x -4 Locate the
boundary points on a number line.
55
Solve and graph 2x 3 5
Find the boundary points for 2x 3 5 2x 3
5 or 2x
3 -5 2x 2
2x -8 x 1
x -4 Locate the
boundary points on a number line. Now choose
test values in each interval and test them by
plugging them into the original inequality.
56
Solve and graph 2x 3 5
Find the boundary points for 2x 3 5 2x 3
5 or 2x
3 -5 2x 2
2x -8 x 1
x -4 Locate the
boundary points on a number line. Now choose
test values in each interval and test them by
plugging them into the original inequality. Graph
the solution set.
57
Purple Box on 268
58
Solve 2x 1 lt 5
59
Solve 2x 1 lt 5

• 2x 1 lt 5 means -5 lt 2x 1 lt 5

60
Solve 2x 1 lt 5

• 2x 1 lt 5 means -5 lt 2x 1 lt 5
• -6 lt 2x
  lt 4

61
Solve 2x 1 lt 5

• 2x 1 lt 5 means -5 lt 2x 1 lt 5
• -6 lt 2x
  lt 4
• -3 lt x
  lt 2

62
Solve 2x 1 lt 5

• 2x 1 lt 5 means -5 lt 2x 1 lt 5
• -6 lt 2x
  lt 4
• -3 lt x
  lt 2
• Solution is (-3,2).

63
Solve 2x 1 5
2x 1 5
64
Solve 2x 1 5
2x 1 5 means 2x 1 -5 or
2x 1 5
65
Solve 2x 1 5
2x 1 5 means 2x 1 -5 or
2x 1 5 2x -6
66
Solve 2x 1 5
2x 1 5 means 2x 1 -5 or
2x 1 5 2x -6 x -3
67
Solve 2x 1 5
2x 1 5 means 2x 1 -5 or
2x 1 5 2x -6
2x 4 x -3
68
Solve 2x 1 5
2x 1 5 means 2x 1 -5 or
2x 1 5 2x -6
2x 4 x -3
x 2
69
Solve 2x 1 5
2x 1 5 means 2x 1 -5 or
2x 1 5 2x -6
2x 4 x -3
x 2 The
solution is xx -3 or x 2.
70
Solve 2x 1 5
2x 1 5 means 2x 1 -5 or
2x 1 5 2x -6
2x 4 x -3
x 2 The
solution is xx -3 or x 2. The solution is
-8,-3 U 2,8.
71
Solve -23x 5 7 -13
72
Solve -23x 5 7 -13

• -23x 5 7 -13 original
  inequality

73
Solve -23x 5 7 -13

• -23x 5 7 -13 original
  inequality
• -23x 5 -20 subtract 7 from both
  sides

74
Solve -23x 5 7 -13

• -23x 5 7 -13 original
  inequality
• -23x 5 -20 subtract 7 from both
  sides
• 3x 5 10 divide both sides by -2

75
Solve -23x 5 7 -13

• -23x 5 7 -13 original
  inequality
• -23x 5 -20 subtract 7 from both
  sides
• 3x 5 10 divide both sides by
  -2
• -10 3x 5 10 interpretation

76
Solve -23x 5 7 -13

• -23x 5 7 -13 original
  inequality
• -23x 5 -20 subtract 7 from both
  sides
• 3x 5 10 divide both sides by
  -2
• -10 3x 5 10 interpretation
• -15 3x 5 subtract 5 from all
  parts

77
Solve -23x 5 7 -13

• -23x 5 7 -13 original
  inequality
• -23x 5 -20 subtract 7 from both
  sides
• 3x 5 10 divide both sides by
  -2
• -10 3x 5 10 interpretation
• -15 3x 5 subtract 5 from all
  parts
• -5 x (5/3) divide all parts by 3

78
Solve -23x 5 7 -13

• -23x 5 7 -13 original
  inequality
• -23x 5 -20 subtract 7 from both
  sides
• 3x 5 10 divide both sides by
  -2
• -10 3x 5 10 interpretation
• -15 3x 5 subtract 5 from all
  parts
• -5 x (5/3) divide all parts by 3
• The solution set is x-5 x (5/3)

79
Solve -23x 5 7 -13

• -23x 5 7 -13 original
  inequality
• -23x 5 -20 subtract 7 from both
  sides
• 3x 5 10 divide both sides by
  -2
• -10 3x 5 10 interpretation
• -15 3x 5 subtract 5 from all
  parts
• -5 x (5/3) divide all parts by 3
• The solution set is x-5 x (5/3) or
• -5,5/3.

80
Solve -23x 5 7 -13

• -23x 5 7 -13 original
  inequality
• -23x 5 -20 subtract 7 from both
  sides
• 3x 5 10 divide both sides by
  -2
• -10 3x 5 10 interpretation
• -15 3x 5 subtract 5 from all
  parts
• -5 x (5/3) divide all parts by 3
• The solution set is x-5 x (5/3) or
• -5,5/3.
• Now graph the solution.

81
Homework

• 4-3
• 1,3,5,9,25,33,37,41,45,49,57,67,71