#### Answer

set of all real numbers

#### Work Step by Step

$\bf{\text{Solution Outline:}}$
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.