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 .

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Copyright 2011 by Bruce Ikenaga