# Absolute Value Equations

The absolute value of a number is defined by

Suppose you want to solve an absolute value equation of the form

Replace with , then take cases.

Example. Solve .

Replace with :

The solutions are and .

It's often easier to put the " " on the side with the number rather than the side with the variable, this way:

This is the approach I'll use in the examples that follow.

Example. Solve .

Remove the absolute values and put a " " on the "2". Then take cases and solve.

The solutions are and .

Example. Solve .

Remove the absolute values and put a " " on the "6". Then take cases and solve.

The solutions are and .

Example. Solve .

Remove the absolute values and put a " " on the "5". Then take cases and solve.

The solutions are and .

Example. Solve .

If you work this problem like the others, you'll get two answers, but they won't be right.

The equation says an absolute value ( ) is negative (-8). Since an absolute value can't be negative, the equation has no solutions.

*Example. Solve .

Since I have two absolute value expressions, I'll go back to my original procedure: Remove the absolute values from an expression and put " " on it. Doing so, I get

Now I have 4 cases:

Case 1.

Since this is a contradiction, this case doesn't give any solutions.

Case 2.

Case 3.

Case 4.

Since this is a contradiction, this case doesn't give any solutions.

The solutions are and .