# SOLVING EQUATIONS WITH ABSOLUTE VALUES ON BOTH SIDES

The picture shown below explains, how to solve the equations in which we have absolute value sign on both sides.

## Practice Problems

Problem 1 :

Solve for x :

|x - 3| = |3x + 2|

Solution :

Based on the idea given above, we have

x - 3  =  (3x + 2)

x - 3  =  3x + 2

- 5  =  2x

- 5/2  =  x

(or)

x - 3  =  - (3x + 2)

x - 3  =  - 3x - 2

- 1  =  - 4x

1/4  =  x

Justify and evaluation :

Plug x = -5/2 and x = 1/4 in the given absolute value equation.

|-5/2 - 3| = |3(-5/2) + 2|

|-11/2|  =  |-15/2 + 2|

|-11/2|  =  |-11/2|

11/2  =  11/2

|1/4 - 3| = |3(1/4) + 2|

|-11/4|  =  |3/4 + 2|

|11/4|  =  |11/4|

11/4  =  11/4

Substituting x = -5/2 and x = 1/4 into the original equation results in true statements.

Both the answers x = -5/2 and x = 1/4 are correct and acceptable.

Problem 2 :

Solve for x :

|x - 7|  =  |2x - 2|

Solution :

Based on the idea given above, we have

x - 7  =  2x - 2

x - 7  =  2x - 2

- 5  =  x

(or)

x - 7  =  - (2x - 2)

x - 7  =  -2x + 2

3x  =  9

x  =  3

Justify and evaluation :

Plug x = -5 and x = 3 in the given absolute value equation.

x  =  -5

|-5 - 7|  =  |2(-5) - 2|

|-12|  =  |-10 - 2|

|-12|  =  |-12|

12  =  12

x  =  3

|3 - 7|  =  |2(3) - 2|

|-4|  =  |6 - 2|

|-4|  =  |4|

4  =  4

Substituting x = -5 and x = 3 into the original equation results in true statements.

Both the answers x = -5 and x = 3 are correct and acceptable.

Problem 3 :

Solve for z :

|2z + 5|  =  |2z - 1|

Solution :

Based on the idea given above, we have

2z + 5  =  2z - 1

5  =  - 1

5  =  -1

The above statement is false.

No solution here.

(or)

2z + 5  =  - (2z - 1)

2z + 5  =  -2z + 1

4  =  -4z

-1  =  z

Justify and evaluation :

Plug z = -1 in the given absolute value equation.

|2(-1) + 5|  =  |2(-1) - 1|

|-2 + 5|  =  |-2 - 1|

|3|  =  |-3|

3  =  3

Substituting z = -1 into the original equation results in true statement.

So, the answer z  = -1 is correct and acceptable.

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