Answer
$ x\in [-5,-1] \cup[3,7].$
Work Step by Step
$$2\leq |x-1|\leq 6\Longrightarrow 2\leq x-1\leq 6, \quad -6\leq x-1\leq -2\\ \Longrightarrow 2+1\leq x\leq 6+1, \quad 1-6\leq x\leq 1-2\\
\Longrightarrow 3\leq x\leq 7, \quad -5\leq x\leq -1.$$
So, $ x\in [-5,-1] \cup[3,7].$
