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EXAMPLE 1
Solve a simple absolute value equation
Solve |x – 5| = 7. Graph the solution.
SOLUTION
|x– 5|=7
x– 5=–7
Write original equation.
or x – 5 = 7
Write equivalent equations.
x = 5 – 7 or
x=5+7
Solve for x.
x = –2
x = 12
Simplify.
or
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.
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
5x = –35
x = 11 or
x = –7
Add 10 to each side.
Divide each side by 5.
EXAMPLE 2
Solve an absolute value equation
ANSWER
The solutions are 11 and –7. Check these in the
original equation.
Check:
| 5x – 10 | = 45
| 5(11) – 10 | =? 45
|45| =? 45
| 5x – 10 | = 45
| 5(–7) – 10 | =? 45
| – 45| =? 45
45 = 45
45 = 45
EXAMPLE 3
Check for extraneous solutions
Solve |2x + 12 | = 4x. Check for extraneous solutions.
SOLUTION
| 2x + 12 | = 4x
Write original equation.
2x + 12 = 4x or 2x + 12 = – 4x Expression can equal 4x or – 4 x
12 = 2x or 12 = –6x
6=x
or –2 = x
Add –2x to each side.
Solve for x.
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
24 = 24
|8| =? – 8
8 = –8
ANSWER
The solution is 6. Reject –2 because it is an
extraneous solution.
GUIDED PRACTICE
for Examples 1, 2 and 3
Solve the equation. Check for extraneous solutions.
1. | x | = 5
ANSWER
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.
5
–7 –6 –5 –4 –3 –2 –1
5
0
1
2
3
4
5
6
7
GUIDED PRACTICE
for Examples 1, 2 and 3
Solve the equation. Check for extraneous solutions.
2. |x – 3| = 10
ANSWER
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.
10
–7 –6–5–4 –3 –2–1 0 1 2
10
3 4
5
6 7 8 9 10 11 12 13
GUIDED PRACTICE
for Examples 1, 2 and 3
Solve the equation. Check for extraneous solutions.
3. |x + 2| = 7
ANSWER
The solutions are –9 and 5. These are the values of
x that are 7 units away from – 2 on a number line.
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
.
GUIDED PRACTICE
for Examples 1, 2 and 3
Solve the equation. Check for extraneous solutions.
5.
|2x + 5| = 3x
ANSWER
The solution of is 5. Reject 1 because it is an
extraneous solution.
GUIDED PRACTICE
for Examples 1, 2 and 3
Solve the equation. Check for extraneous solutions.
6.
|4x – 1| = 2x + 9
ANSWER
The solutions are –1 1 and 5.
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