1. 5.5 Solve Absolute Value Equations
You will solve absolute value equations.
Essential Question: How do you solve absolute value equations?
You will learn how to answer this question by rewriting absolute value equations as two equations and then solving the equations.

2. EXAMPLE 1
Solve an absolute value equation
Solve x = 7.
SOLUTION
The distance between x and 0 is 7. So, x = 7 or x = –7.
ANSWER
The solutions are 7 and –7.

3. EXAMPLE 1
GUIDED PRACTICE
for Example 1
Solve (a) |x| = 3 and (b) |x| = 15.
ANSWER
3, –3 a.
15, –15 b.

4. Solve an absolute value equation
EXAMPLE 2
Solve an absolute value equation
x – 3 = 8.
Solve
SOLUTION
Rewrite the absolute value equation as two equations. Then solve each equation separately.
x – 3 = 8
Write original equation.
x – 3 = 8 or x – 3 = –8
Rewrite as two equations.
x = 11 or x = –5
Add 3 to each side.
ANSWER
The solutions are 11 and –5. Check your solutions.

5. Solve an absolute value equation
EXAMPLE 2
Solve an absolute value equation
CHECK
|x – 3| = |x – 3| = 8
Write original inequality.
|11 – 3| = |–5 – 3| = 8 ?
Substitute for x.
| 8| = |–8| = 8 ?
Subtract.
8 = = 8
Simplify. The solution checks.
When can you perform operations on an absolute value expression?
Why is the expression x - 3 inside the absolute value symbol equal to 8 or -8?

6. Rewrite an absolute value equation
EXAMPLE 3
Rewrite an absolute value equation
3 2x – 7 – 5 = 4.
Solve
SOLUTION
First, rewrite the equation in the form ax + b = c.
3 2x – 7 – 5 = 4
Write original equation.
3 2x – 7 = 9
Add 5 to each side.
2x – 7 = 3
Divide each side by 3.

7. Rewrite an absolute value equation
EXAMPLE 3
Rewrite an absolute value equation
Next, solve the absolute value equation.
2x – 7 = 3
Write absolute value equation.
2x – 7 = or 2x – 7 = –3
Rewrite as two equations.
2x = 10 or x = 4
Add 7 to each side.
x = or x = 2
Divide each side by 2.
ANSWER
The solutions are 5 and 2.

8. GUIDED PRACTICE
for Examples 2 and 3
Solve the equation.
r – 7 = 9 2.
16, –2
ANSWER

9. GUIDED PRACTICE
for Examples 2 and 3
Solve the equation.
2 s = 18.9 3.
7.4, –7.4
ANSWER

10. GUIDED PRACTICE
for Examples 2 and 3
Solve the equation.
4 t + 9 – 5 = 19 4.
–3, –15
ANSWER

11. Decide if an equation has no solutions
EXAMPLE 4
Decide if an equation has no solutions
3x = –2, if possible.
Solve
3x = –2
Write original equation.
3x + 5 = –8
Subtract 6 from each side.
ANSWER
The absolute value of a number is never negative. So,
there are no solutions.
• Why is the absolute value of a number never negative?

12. EXAMPLE 5
Use absolute deviation
BASKETBALLS
Before the start of a professional basketball game, a basketball must be inflated to an air pressure of 8 pounds per square inch (psi) with an absolute error of 0.5 psi. (Absolute error is the absolute deviation of a measured value from an accepted value.) Find the minimum and maximum acceptable air pressures for the basketball.

13. EXAMPLE 5
Use absolute deviation
SOLUTION
Let p be the air pressure (in psi) of a basketball. Write a verbal model. Then write and solve an absolute value equation.
0.5 = p – 8

14. Use absolute deviation
EXAMPLE 5
Use absolute deviation p – 8
0.5 =
Write original equation. p 8
0.5 = – or
–0.5 =
Rewrite as two equations. p
8.5 = or
7.5 =
Add 8 to each side.
ANSWER
The minimum and maximum acceptable pressures are
7.5 psi and 8.5 psi.

15. GUIDED PRACTICE
for Examples 4 and 5
Solve the equation, if possible
m – = 2
ANSWER
no solution

16. GUIDED PRACTICE
for Examples 4 and 5
Solve the equation, if possible
–3 n +2 –7 = –10
ANSWER
1, 3

17. GUIDED PRACTICE
for Examples 4 and 5
7. The absolute deviation of x from 7.6 is 5.2. What are the values of x that satisfy this requirement?
12.8, 2.4
ANSWER

18. 5.5 Solve Absolute Value Equations
You will solve absolute value equations.
Essential Question: How do you solve absolute value equations?
You can rewrite an absolute
value equation as two equations and then solve each equation separately to solve the absolute
value equation.
• Some absolute value equations have no solution.
Rewrite an absolute value equation in the form |ax + b | = c, if necessary, and then rewrite it as two equations ax + b = c and
ax + b = -c. Solve each equation
separately.

19. Daily Homework Quiz
Solve the equation, if possible.
| x – 4 | = 13
ANSWER
–9, 17
2. | x + 2 | + 7 = 3
ANSWER
no solutions

20. Daily Homework Quiz
Solve the equation. If possible.
| 2x – 6 | + 4 = 20
ANSWER
–5, 11
4. –2 | x – 5 | + 7 = 12
ANSWER
no solutions

21. Daily Homework Quiz
A pattern for a 26-inch skirt for an absolute deviation of 1.5 inches. Find the minimum and maximum skirt lengths that can be made from the pattern. 5.
ANSWER
minimum : 24.5 in.; maximum 27.5 in.