Absolute Value Functions,

1
Chapter 5 Functions
5.5
Absolute Value Functions, Equations, and
Inequalities
5.5.1
MATHPOWERTM 11, WESTERN EDITION
2
Absolute Value Equations
The absolute value of any number is its distance
from the origin on a number line.
6 6
-6 6
Given this definition for absolute value, there
would be two cases to consider when solving an
absolute value equation, the positive case and
the negative case.
Solve x - 2 6
From the definition of absolute value x - 2
6 means x - 2 6 or -(x - 2) 6
-(x - 2) 6 -x 2 6 -x 4
x -4
x - 2 6 x 8
The solution is x 8 or x -4.
Absolute value equations must be checked because
of the possibility of extraneous roots.
Check
- 4 - 2 6 - 6 6 6
6
8 - 2 6 6 6 6 6
5.5.2
3
Solving an Absolute Value Equation
There are 4 possible cases
Solve x - 3 - 3x 7 0
, - -, -, -
x - 3 3x 7
Two of these give the same results.
-(x - 3) 3x 7 -x 3 3x 7 -4x
4 x -1
x - 3 -(3x 7) x - 3 -3x - 7 4x -4
x -1
-(x - 3) -(3x 7) -x 3 -3x - 7
2x -10 x -5
x - 3 3x 7 -2x 10 x - 5
Check
-1 - 3 - 3(-1) 7 0
-4 - 4 0
4 - 4 0 0
0
-5 - 3 - 3(-5) 7 0
-8 - 8 0 8
- 8 0 0 0
Therefore, the solution is x -5 or x -1.
5.5.3
4
Solving Absolute Value Equations by Graphing
x - 3 - 3x 7 0
x - 3 3x 7
y 3x 7
y x - 3
y 3x 7
y x - 3
The solution will be where the two graphs
intersect.
Therefore, x -1 and x -5.
- 5
- 1
5.5.4
5
Solving Absolute Value Equations by Graphing
Solve graphically.
a) x - 2 6
y x - 2
The solution to the equation will be the
intersection of the two graphs y x - 2
and y 6.
y 6
-4
8
Therefore, the solution is x -4 and x 8.
5.5.5
6
Solving Absolute Value Equations by Graphing
b) x - 3 - 3x 7 0
The solution will be the x-intercepts.
Therefore, the solution is x -5 or x -1.
5.5.6
7
Solving Absolute Value Equations by Graphing
c) x - 2 2x - 1
y 2 x -1
y x - 2
There is only one solution at x 1.
5.5.7
8
Solving Absolute Value Equations by Graphing
d) x - 1 x - 4 7
The solution will be the intersection of the two
graphs y x - 1 x - 4
and y 7
y x - 1 x - 4
y 7
The solution is x -1 and x 6.
5.5.8
9
Solving Absolute Value Equations Algebraically
x - 5 2
(x - 5) 2 x - 5 2 x 7
-(x - 5) 2 -x 5 2 x 3
Check
x - 5 2 7 - 5 2 2 2
2 2
x - 5 2 3 - 5 2 -2 2
2 2
5.5.9
10
Solving Absolute Value Equations
Algebraically
a) x 1 x - 1
No solution
-(x 1) x - 1 -x - 1 x - 1 -2x
0 x 0
(x 1) x - 1 x 1 x - 1
1 ? -1
b) x - 2 - 2x 11 0
x - 2 2x 11
-(x - 2) (2x 11) -x 2 2x 11
-3x 9 x -3
(x - 2) (2x 11) x - 2 2x 11
-x 13 x -13
5.5.10
11
Solving Absolute Value Equations contd
Graphically
a) x 1 x - 1
5.5.11
12
Solving Absolute Value Equations contd
Graphically
b) x - 2 - 2x 11 0
x -13 x -3
5.5.12
13
Solving Absolute Value Equations, Algebraically

and Graphically
x - 3 x - 8 17
- (x - 3) -(x - 8) 17 -x 3 - x
8 17 -2x 11 17
-2x 6
x -3
(x - 3) (x - 8) 17 2x - 11 17
2x 28
x 14
-(x - 3) (x - 8) 17 -x 3 x - 8
17 -5 ? 17
(x - 3) -(x - 8) 17 x - 3 - x 8
17 5 ? 17
5.5.13
14
Solving Absolute Value Equations, Algebraically
and Graphically
contd
x - 3 x - 8 17
x -3 or x 14
5.5.14
15
Solving Absolute Value Inequalities by Graphing
Solve for x 2x - 1 lt 5
The graph of y 2x - 1 is on or below the
graph of y 5 when -2 lt x and x lt 3.
y 2x - 1
y 5
-2 x and x 3
Therefore, the solution is -2 lt x and x
lt 3.
-2
3
5.5.15
16
Solving Absolute Value Inequalities by Graphing
x 2 lt -x 6
The graph of y x 2 is below the graph
of y 2x 6 when x lt 2.
y -x 6
y x 2
Therefore, the solution is x lt 2.
x lt 2
5.5.16
17
Solving Absolute Value Inequalities by Graphing
x - 1 x 3 gt 6
The graph of y x - 1 x 3 is above
the graph of y 6 when -4 gt x or x gt 2.
y x - 1 x 3
x gt 2
-4 gt x
y 6
Therefore, the solution is -4 gt x or x gt 2.
5.5.17
18
Solving Absolute Value Inequalities
Solve 3x - 2 lt 7.
(3x - 2) lt 7 3x - 2 lt 7 3x lt 9
x lt 3
-(3x - 2) lt 7 -3x 2 lt 7 -3x lt 5
5.5.18
19
Solving Absolute Value Inequalities
Solve x - 1 x - 3 2.
-(x - 1) -(x - 3) 2 -x 1 - x
3 2 -2x -2
x 1
(x - 1) (x - 3) 2 2x - 4 2
2x 6
x 3
5.5.19
20
Assignment
Suggested Questions
49, 53, 57, 58, 63, 65, 67, 68, 73, 79, 87,
101, 103, 110, 113
Pages 296-298 1, 3, 5, 13, 15, 21, 27, 33, 35,
36, 37, 38, 39, 41, 44
5.5.20