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.
|
