1. SOLVING ABSOLUTE-VALUE EQUATIONS
You can solve some absolute-value equations using mental math. For instance, you learned that the equation | x | 8 has two solutions: 8 and 8.
To solve absolute-value equations, you can use the fact that the expression inside the absolute value symbols can be either positive or negative.

2. The equation has two solutions: 7 and –3.
Solving an Absolute-Value Equation
Solve | x  2 |  5
Solve | x  2 |  5
The expression x  2 can be equal to 5 or 5.
SOLUTION
The expression x  2 can be equal to 5 or 5.
x  2 IS POSITIVE
| x  2 |  5
x  2  5
x  7
x  2 IS POSITIVE
| x  2 |  5
x  2  5
x  7
x  3
x  2 IS NEGATIVE
| x  2 |  5
x  2  5
x  2 IS POSITIVE
x  2  5
x  2 IS POSITIVE
x  2 IS NEGATIVE
x  2 IS NEGATIVE
x  2  5
| x  2 |  5
| x  2 |  5
x  2  5
x  2  5
x  7
x  3
The equation has two solutions: 7 and –3.
CHECK
| 7  2 |  | 5 |  5
| 3  2 |  | 5 |  5

3. 2x  7 IS POSITIVE | 2x  7 |  5  4 | 2x  7 |  9 2x  7  +9
Solving an Absolute-Value Equation
Solve | 2x  7 |  5  4
Solve | 2x  7 |  5  4
Isolate the absolute value expression on one side of the equation.
SOLUTION
Isolate the absolute value expression on one side of the equation.
2x  7 IS POSITIVE
| 2x  7 |  5  4
| 2x  7 |  9
2x  7  +9
2x  16
x  8
2x  7 IS POSITIVE
| 2x  7 |  5  4
| 2x  7 |  9
2x  7  +9
2x  16
x  8
2x  7 IS NEGATIVE
| 2x  7 |  5  4
| 2x  7 |  9
2x  7  9
2x  2
x  1
2x  7 IS POSITIVE
2x  7  +9
2x  7 IS POSITIVE
2x  7 IS NEGATIVE
2x  7 IS NEGATIVE
2x  7  9
| 2x  7 |  5  4
| 2x  7 |  5  4
| 2x  7 |  9
| 2x  7 |  9
2x  7  +9
2x  7  9
2x  16
2x  2
x  8
x  1
TWO SOLUTIONS
x  8
x  1

4. Solving an Absolute-Value Equation
Recall that | x | is the distance between x and 0. If | x |  8, then any number between 8 and 8 is a solution of the inequality.
Recall that x is the distance between x and 0. If x  8, then any number between 8 and 8 is a solution of the inequality.
Recall that | x | is the distance between x and 0. If | x |  8, then any number between 8 and 8 is a solution of the inequality.
 8  7  6  5  4  3  2 
You can use the following properties to solve absolute-value inequalities and equations.

5. SOLVING ABSOLUTE-VALUE EQUATIONS AND INEQUALITIES
SOLVING ABSOLUTE-VALUE INEQUALITIES
SOLVING ABSOLUTE-VALUE EQUATIONS AND INEQUALITIES
means
| a x  b |  c
means
a x  b  c and a x  b   c.
means
| a x  b |  c
When an absolute value is less than a number, the
means
a x  b  c and a x  b   c.
inequalities are connected by and. When an absolute
means
| a x  b |  c
value is greater than a number, the inequalities are
means
a x  b  c or a x  b   c.
connected by or.
means
| a x  b |  c
means
a x  b  c or a x  b   c.
means
| a x  b |  c
means
a x  b  c or a x  b   c.

6. The solution is all real numbers greater than 1 and less than 7.
Solving an Absolute-Value Inequality
Solve | x  4 | < 3
x  4 IS POSITIVE
x  4 IS NEGATIVE
| x  4 |  3
| x  4 |  3
x  4  3
x  4  3 
x  7
x  1
Reverse inequality symbol.
The solution is all real numbers greater than 1 and less than 7.
This can be written as 1  x  7.

7. Solve | 2x  1 | 3  6 and graph the solution.
Solving an Absolute-Value Inequality
| 2x  1 |  3  6
| 2x  1 |  9
2x  1  +9
x  4
2x  8
| 2x  1 | 3  6
2x  1  9
2x  10
x  5
2x + 1 IS POSITIVE
2x + 1 IS NEGATIVE
Solve | 2x  1 | 3  6 and graph the solution.
2x + 1 IS POSITIVE
2x + 1 IS NEGATIVE
| 2x  1 |  3  6
| 2x  1 | 3  6
| 2x  1 |  9
| 2x  1 |  9
2x  1  +9
2x  1  9 
2x  8
2x  10
The solution is all real numbers greater than or equal to 4 or less than or equal to  5. This can be written as the compound inequality x   5 or x  4.
x  4
x  5
Reverse inequality symbol.
 5 4.
 6  5  4  3  2 

8. Writing an Absolute-Value Inequality
You work in the quality control department of a manufacturing company. The diameter of a drill bit must be between 0.62 and 0.63 inch.
Write an absolute-value inequality
to represent this requirement. a.
Does a bit with a diameter of 0.623
inch meet the requirement? b.

9. Let d represent the diameter (in inches) of the drill bit.
Writing an Absolute-Value Inequality
The diameter of a drill bit must be between 0.62 and 0.63 inch.
a. Write an absolute-value inequality to represent this requirement.
Let d represent the diameter (in inches) of the drill bit.
Write a compound inequality.
0.62  d  0.63
Find the halfway point.
0.625
Subtract from each part of the compound inequality.
0.62   d   0.63  0.625
0.005  d   0.005
Rewrite as an absolute-value inequality.
| d  |  0.005
This inequality can be read as “the actual diameter must differ from inch by no more than inch.”

10. Because 0.002  0.005, the bit does meet the requirement.
Writing an Absolute-Value Inequality
The diameter of a drill bit must be between 0.62 and 0.63 inch.
b. Does a bit with a diameter of meet the requirement?
| d  |  0.005
|  |  0.005
| 0.002 |  0.005
 0.005
Because  , the bit does meet the requirement.