An absolute value equation is an equation that involves the absolute value of a variable expression. Equations of the form $|x|=a,$ where $a$ is a real number greater than zero, can be solved by looking for the numbers $x$ whose distance from $0$ in the number line equals $a.$ For example, the solutions of $|x|=4$ are all values of $x$ that are $4$ units away from $0.$

Since 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 $x=\text{-}4.$ However, solving an absolute value equation, in general, might require a more elaborate and structured approach.

Number of Solutions

Simple absolute value equations of the form $|x|=a,$ can have no, one, or two solutions, depending on the value of $a.$ However, more complex absolute value equations may have more than two solutions.

Equation	Number of Solutions	Solution(s)
$|x|=\text{-}4$	Zero	No solution
$|x|=0$	One	$0$
$|x|=4$	Two	$\text{-}4,4$
$\left|x^2-4\right|=2$	Four	$\text{-}\sqrt{2},\sqrt{2},\text{-}\sqrt{6},\sqrt{6}$