## Elementary and Intermediate Algebra: Concepts & Applications (6th Edition)

$\bf{\text{Solution Outline:}}$ To solve the given equation, $|y-2|=|2-y| ,$ use the definition of an absolute value equality. Then use the properties of equality to isolate the variable. $\bf{\text{Solution Details:}}$ Since $|x|=|y|$ implies $x=y \text{ or } x=-y,$ the equation above is equivalent to \begin{array}{l}\require{cancel} y-2=2-y \\\\\text{OR}\\\\ y-2=-(2-y) .\end{array} Solving each equation results to \begin{array}{l}\require{cancel} y-2=2-y \\\\ y+y=2+2 \\\\ 2y=4 \\\\ y=\dfrac{4}{2} \\\\ y=2 \\\\\text{OR}\\\\ y-2=-(2-y) \\\\ y-2=-2+y \\\\ y-y=-2+2 \\\\ 0=0 \text{ (TRUE)} .\end{array} Since the solution above ended with a TRUE statement, then the solution set is the set of all real numbers.