1. Solve a simple absolute value equation
EXAMPLE 1
Solve a simple absolute value equation
Solve |x – 5| = 7. Graph the solution.
SOLUTION
| x – 5 | = 7
Write original equation.
x – 5 = – 7 or x – 5 = 7
Write equivalent equations.
x = 5 – 7 or x = 5 + 7
Solve for x.
x = –2 or x = 12
Simplify.

2. EXAMPLE 1
Solve a simple absolute value equation
ANSWER
The solutions are –2 and 12. These are the values of x that are 7 units away from 5 on a number line. The graph is shown below.

3. Solve an absolute value equation
EXAMPLE 2
Solve an absolute value equation
Solve |5x – 10 | = 45.
SOLUTION
| 5x – 10 | = 45
Write original equation.
5x – 10 = 45 or 5x – 10 = –45
Expression can equal 45 or –45 .
5x = 55 or x = –35
Add 10 to each side.
x = 11 or x = –7
Divide each side by 5.

4. Solve an absolute value equation
EXAMPLE 2
Solve an absolute value equation
ANSWER
The solutions are 11 and –7. Check these in the original equation.
Check:
| 5x – 10 | = 45
| 5x – 10 | = 45
| 5(11) – 10 | = 45 ?
| 5(–7) – 10 | = 45 ?
|45| = 45 ?
| – 45| = 45 ?
45 = 45
45 = 45

5. Check for extraneous solutions
EXAMPLE 3
Check for extraneous solutions
Solve |2x + 12 | = 4x. Check for extraneous solutions.
SOLUTION
| 2x + 12 | = 4x
Write original equation.
2x = 4x or 2x = – 4x
Expression can equal 4x or – 4 x
12 = 2x or 12 = –6x
Add –2x to each side.
6 = x or –2 = x
Solve for x.

6. Check for extraneous solutions
EXAMPLE 3
Check for extraneous solutions
Check the apparent solutions to see if either is extraneous.
CHECK
| 2x + 12 | = 4x
| 2x + 12 | = 4x
| 2(6) +12 | = 4(6) ?
| 2(–2) +12 | = 4(–2) ?
|24| = 24 ?
|8| = – 8 ?
24 = 24
8 = –8
The solution is 6. Reject –2 because it is an extraneous solution.
ANSWER

7. Solve the equation. Check for extraneous solutions.
GUIDED PRACTICE
for Examples 1, 2 and 3
Solve the equation. Check for extraneous solutions.
1. | x | = 5
The solutions are –5 and 5. These are the values of x that are 5 units away from 0 on a number line. The graph is shown below.
ANSWER
– 3
– 4
– 2
– 1 1 2 3 4 5 6 7
– 5
– 6
– 7

8. Solve the equation. Check for extraneous solutions.
GUIDED PRACTICE
for Examples 1, 2 and 3
Solve the equation. Check for extraneous solutions.
2. |x – 3| = 10
The solutions are –7 and 13. These are the values of x that are 10 units away from 3 on a number line. The graph is shown below.
ANSWER
– 3
– 4
– 2
– 1 1 2 3 4 5 6 7
– 5
– 6
– 7 8 9 10 11 12 13

9. GUIDED PRACTICE
for Examples 1, 2 and 3
Solve the equation. Check for extraneous solutions.
3. |x + 2| = 7
The solutions are –9 and 5. These are the values of x that are 7 units away from – 2 on a number line.
ANSWER

10. GUIDED PRACTICE
for Examples 1, 2 and 3
Solve the equation. Check for extraneous solutions.
4. |3x – 2| = 13
ANSWER
The solutions are 5 and

11. GUIDED PRACTICE
for Examples 1, 2 and 3
Solve the equation. Check for extraneous solutions.
5. |2x + 5| = 3x
The solution of is 5. Reject 1 because it is an extraneous solution.
ANSWER

12. GUIDED PRACTICE
for Examples 1, 2 and 3
Solve the equation. Check for extraneous solutions.
6. |4x – 1| = 2x + 9
ANSWER
The solutions are – and 5. 3 1