Solving Absolute Value Equations
Solving an Equation of the Form  z = w
Here are several examples of solving equations of the form z = w.
Example 1
Solve: 4x  34 = 6x + 14
Solution
Replace 4x = 34 with z and 6x + 14 with w:
So: 
4x  34 = 6x + 14
z = w
z = w or z = w 
Substitute 4x  34 for z and 6x + 14 for w.
Now, solve for x.
So: 
4x  34
4x
2x
x 
= 6x + 14 = 6x + 48
= 48
= 24 
or
or
or
or
or 
4x  34 = 4x  34 =
4x =
10x =
x = 
(6x + 14) 6x  14
6x + 20
20
2 
Let's check the solutions:
Check x = 24 
Check x = 3 
4x  34
Is 4(24)  34
Is 130
Is 130 
= 6x + 14
= 6(24) + 14 ?
= 130 ?
= 130 ? Yes 
4x  34
Is 4(2)  34
Is 26
Is 26 
= 6x + 14 = 6(2) + 14
= 26 ?
= 26 ? Yes 
So, the solutions are x = 24 and x = 2.
Example 2
Solve: 3x  4 = 3x + 16
Solution
Replace 3x  4 with z and
3x + 16 with w:
So: 
3x  4 = 3x + 16 z = w
z = w or z = w 
Substitute 3x  4 for
z and 3x + 16 for w.
Now, solve for x.

3x + 4
3x
0

= 3x + 16 = 3x + 20
= 20

or
or
or

3x  4 =
3x  4 =
6x =
x =

(3x + 16) 3x  16
12
2

Since 0 = 20 is a contradiction, the left equation does not lead to a
solution.
Check x = 2 
3x  4
Is 3(2)  4
Is 10
Is 10 
= 3x + 16 = 3(2) + 16 ?
= 10 ?
= 10 ? Yes 
Thus, 2 is the only solution.
