Absolute Value Equations

Follow these steps to solve an absolute value equality which contains one absolute value:

Isolate the absolute value on one side of the equation.
Is the number on the other side of the equation negative? If you answered yes, then the equation has no solution. If you answered no, then go on to step 3.
Write two equations without absolute values. The first equation will set the quantity inside the bars equal to the number on the other side of the equal sign; the second equation will set the quantity inside the bars equal to the opposite of the number on the other side.
Solve the two equations.

Follow these steps to solve an absolute value equality which contains two absolute values (one on each side of the equation):

Write two equations without absolute values. The first equation will set the quantity inside the bars on the left side equal to the quantity inside the bars on the right side. The second equation will set the quantity inside the bars on the left side equal to the opposite of the quantity inside the bars on the right side.
Solve the two equations.

Let's look at some examples.

Example 1: Solve |2x - 1| + 3 = 6

Step 1: Isolate the absolute value	\|2x - 1\| + 3 = 6 \|2x - 1\| = 3
Step 2: Is the number on the other side of the equation negative?	No, it’s a positive number, 3, so continue on to step 3
Step 3: Write two equations without absolute value bars	2x - 1 = 3	2x - 1 = -3
Step 4: Solve both equations	2x - 1 = 3 2x = 4 x = 2	2x - 1 = -3 2x = -2 x = -1

Example 2: Solve |3x - 6| - 9 = -3

Step 1: Isolate the absolute value	\|3x - 6\| - 9 = -3 \|3x - 6\| = 6
Step 2: Is the number on the other side of the equation negative?	No, it’s a positive number, 6, so continue on to step 3
Step 3: Write two equations without absolute value bars	3x - 6 = 6	3x - 6 = -6
Step 4: Solve both equations	3x - 6 = 6 3x = 12 x = 4	3x - 6 = -6 3x = 0 x = 0

Example 3: Solve |5x + 4| + 10 = 2

Step 1: Isolate the absolute value	\|5x + 4\| + 10 = 2 \|5x + 4\| = -8
Step 2: Is the number on the other side of the equation negative?	Yes, it’s a negative number, -8. There is no solution to this problem.

Example 4: Solve |x - 7| = |2x - 2|

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

Example 5: Solve |x - 3| = |x + 2|

Step 1: Write two equations without absolute value bars	x - 3 = x + 2	x - 3 = -(x + 2)
Step 4: Solve both equations	x - 3 = x + 2 - 3 = -2 false statement No solution from this equation	x - 3 = -x - 2 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	x - 3 = 3 - x	x - 3 = -(3 - x)
Step 4: Solve both equations	x - 3 = 3 - x 2x - 3 = 3 2x = 6 x = 3	x - 3 = -(3 - x) 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