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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.

 
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