Answer
The solution set is $\color{blue}{\left\{1, 3\right\}}$.
Work Step by Step
RECALL:
$|x| =a \longrightarrow x = a \text{ or } x = -a$
Thus, using the rule above gives:
$|x-2| = 1 \longrightarrow x-2=1 \text{ or } x-2=-1$
Solve each equation to obtain:
$x-2=1
\\x=1+2
\\x=3$
or
$x-2=-1
\\x=-1+2
\\x=1$
Therefore, the solution set is $\color{blue}{\left\{1, 3\right\}}$.
