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Concept

# Absolute Value Equation

An absolute value equation is an equation that contains the absolute value of a variable expression. $\left| 2x-8 \right|=5$ Equations of the type $|x|=a,$ where $a\geq0,$ can be solved by thinking of the absolute value of a number $a$ as the distance between $a$ and $0$ on the number line. For example, $|x|=4$ are all values of $x$ which are $4$ units away from $0.$

As there are two points on the number line that fulfill this requirement, there are two solutions to the equation $|x|=4,$ namely $x=4$ and $x=\text{-} 4.$

Concept

## Number of Solutions

The absolute value equation $|x|=4$ has two solutions. Equations of the same type, $|x|=a,$ will have $0,$ $1,$ or $2$ solutions, depending on $a.$ More complex absolute value equations may have more than $2$ solutions.

Equation Number of solutions
$|x|=\text{-} 4$ $0$
$|x|=0$ $1$
$|x|=4$ $2$
$\left|x^2-4 \right|=2$ $4$
Concept

## Solving Absolute Value Equations

When solving absolute value equations algebraically, it is necessary to take into consideration that the absolute value of both a negative number and a positive yields a positive. One possible strategy is to, when removing the absolute value, split the equation into two cases: one case where the expression inside the absolute value is negative and one where it is positive.

Absolute value equations can also be solved graphically or numerically.