Follow these steps to solve an absolute value equality which contains one absolute value:
Follow these steps to solve an absolute value equality
which contains two absolute values (one on each side of the equation):
Let's look at some examples.
Example 1: Solve 2x  1 + 3 = 6
Step 1: Isolate the absolute value 
2x  1 = 3 

Step 2: Is the number on the other side of the equation negative? 


Step 3: Write two equations without absolute value bars 


Step 4: Solve both equations 
2x = 4 x = 2 
2x = 2 x = 1 
Example 2: Solve 3x  6  9 = 3
Step 1: Isolate the absolute value 
3x  6 = 6 

Step 2: Is the number on the other side of the equation negative? 


Step 3: Write two equations without absolute value bars 


Step 4: Solve both equations 
3x = 12 x = 4 
3x = 0 x = 0 
Example 3: Solve 5x + 4 + 10 = 2
Step 1: Isolate the absolute value 
5x + 4 = 8 
Step 2: Is the number on the other side of the equation negative? 

Example 4: Solve x  7 = 2x  2
Step 1: Write two equations without absolute value bars 


Step 4: Solve both equations 
x  7 = 2 x = 5 x = 5 
3x  7= 2 3x = 9 x = 3 
Example 5: Solve x  3 = x + 2
Step 1: Write two equations without absolute value bars 


Step 4: Solve both equations 
 3 = 2 false statement No solution from this equation 
2x  3= 2 2x = 1 x = 1/2 
So the only solution to this problem is x = 1/2
Example 6: Solve x  3 = 3  x
Step 1: Write two equations without absolute value bars 


Step 4: Solve both equations 
2x  3 = 3 2x = 6 x = 3 
x  3= 3 + x 3 = 3 All real numbers are solutions to this equation 
Since 3 is included in the set of real numbers, we will just say that the solution to this equation is All Real Numbers