1. Objectives
Solve equations in one variable that contain absolute-value expressions.

4. To solve absolute-value equations, perform inverse operations to isolate the absolute-value expression on one side of the equation. Then you must consider two cases.

5. Additional Example 1A: Solving Absolute-Value Equations
Solve the equation.
|x| = 12
Think: What numbers are 12 units from 0?
|x| = 12
12 units
10 8 6 4 2 4 6 8 10
12 2 12 •
Case 1
x = 12
Case 2
x = –12
Rewrite the equation as two cases.
The solutions are {12, –12}.

6. Additional Example 1B: Solving Absolute-Value Equations
Solve the equation.
3|x + 7| = 24
|x + 7| = 8
Case 1
x + 7 = 8
Case 2
x + 7 = –8
– 7 –7
– 7 – 7
x = 1
x = –15
The solutions are {1, –15}.

7. Check It Out! Example 1a
Solve the equation.
|x| – 3 = 4
|x| – 3 = 4
|x| = 7
Case 1
x = 7
Case 2
x = –7
The solutions are {7, –7}.

8. Check It Out! Example 1b
Solve the equation.
8 =|x  2.5|
8 =|x  2.5|
Case 1
8 = x  2.5
Case 2
 8 = x  2.5
10.5 = x
5.5 = x
The solutions are {10.5, –5.5}.

9. The table summarizes the steps for solving absolute-value equations.
Solving an Absolute-Value Equation
1. Use inverse operations to isolate the absolute-value expression.
2. Rewrite the resulting equation as two cases that do not involve absolute values.
3. Solve the equation in each of the two cases.

10. Not all absolute-value equations have two solutions
Not all absolute-value equations have two solutions. If the absolute-value expression equals 0, there is one solution. If an equation states that an absolute-value is negative, there are no solutions.

11. Additional Example 2A: Special Cases of Absolute-Value Equations
Solve the equation.
8 = |x + 2|  8
8 = |x + 2|  8
0 = |x + 2|
0 = x + 2
 2
2 = x
The solution is {2}.

12. Additional Example 2B: Special Cases of Absolute-Value Equations
Solve the equation.
3 + |x + 4| = 0
3 + |x + 4| = 0
 3
|x + 4| = 3
This equation has no solution.

13. Remember!
Absolute value must be nonnegative because it represents a distance.

14. Check It Out! Example 2a
Solve the equation.
2  |2x  5| = 7
2  |2x  5| = 7
 2
 |2x  5| = 5
|2x  5| = 5
This equation has no solution.

15. Check It Out! Example 2b
Solve the equation.
6 + |x  4| = 6
6 + |x  4| = 6
|x  4| = 0
x  4 = 0
x = 4

16. Lesson Quiz
Solve each equation.
1. 15 = |x| |x – 7| = 14
3. |x + 1|– 9 = – |5 + x| – 3 = –2
|x – 8| = 6
–15, 15
0, 14 –1
–6, –4
no solution
6. Inline skates typically have wheels with a diameter of 74 mm. The wheels are manufactured so that the diameters vary from this value by at most 0.1 mm. Write and solve an absolute-value equation to find the minimum and maximum diameters of the wheels.
|x – 74| = 0.1; 73.9 mm; 74.1 mm