Definition of Absolute Value

Formally, the absolute value can be defined as a piece-wise function. $$\big| x \big| = \left\{ \begin{array}{rl} x & \mbox{if } x \geq 0 \\ -x & \mbox{if } x < 0 \end{array} \right.$$

One way of interpreting the absolute value is to think of $|x|$ as the distance from the real number $x$ to the origin $0$. For example, $\vert -2 \vert =2$ and $|2|=2$ since $-2$ and $2$ are both two units away from the origin.

Absolute Value Equations

The general procedure to solve an equation involving a single absolute value expression is as follows:

1. Isolate the absolute value expression on one side of the equation.
2. Consider both the postive and negative cases of the absolute value in order to obtain all possible solutions of the equation.
3. Check for extraneous solutions by making a substitution into the original equation to determine if your possible solutions are valid.