Absolute Value Equations

Equations Involving Absolute Values

To solve an equation that involves an absolute value, turn the equation into an "Or" statement.

Example 

Solve 

        |x|  =  2 


Solution

        x  =  2 or x  =  -2

Example

         |x + 1|  =  2


Solution

        x + 1  =  2    or    x + 1  =  -2

        x  =  1     or     x  =  -3                  Subtracting 1 from both sides

Example

        |3x - 4|  =  5 


Solution

We write

        3x - 4  =  5     or     3x - 4  =  -5

        3x  =  9     or     3x  =  -1                    Adding 4 to both sides

        x  =  3     or     x  =  -1/3                     Dividing by 3

Example

        | 4x + 7|  =  -3

Solution

This equation has no solution, since an absolute value cannot be negative.

Example

        | 2x - 6|  =  0

Solution

Since positive and negative 0 mean the same thing, we only need one equation

        2x - 6  =  0

        2x  =  6

        x  =  3

Exercises

Solve the following.  Put your mouse on the yellow rectangle to check your solution.

1. |x|  =  6       
   
   

2. |x - 3|  =  6   
   
   

3. |x|  =  -4       
   
   

4. |2x + 1|  =  7   

Equations That have Absolute Value Signs on Both Sides

If we have absolute value signs on both sides of the equation, we can play the same game with two choices as follows.

Example:  

Solve 

        |3x + 4|  =  | 2x - 3|

Solution:

We can write

        3x + 4  =  2x - 3      or     3x + 4  =  -(2x - 3)

        3x + 4  =  2x - 3     or     3x + 4  =  -2x + 3          Distributing the (-)

        3x  =  2x - 7     or     3x  =  -2x - 1                        Subtracting 4

        x  =  -7      or     5x  =  -1                                      Subtracting (Adding)  the 2x 

        x  =  -7     or      or x   =  -1/5                                    Dividing by 5

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