Answer
$(-\infty,-1]\cup[3,\infty)$
Work Step by Step
The solutions of $|u|\gt c $ are the numbers that satisfy $ u\lt -c $ or $ u\gt c.$
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$ 3|x-1|+2\geq 8\qquad $ ... $/-2$
$ 3|x-1| \geq 6\qquad $ ... $/\div 3$
$|x-1| \geq 2$
$|x-1| \geq 2\Rightarrow \left\{\begin{array}{ll}
x-1\leq -2 & \\
or & \\
x-1 \geq 2 &
\end{array}\right.$
$ \begin{array}{lllll}
x-1\leq -2 & /+1 & ...or... & x-1 \geq 2 & /+1\\
x\leq-1 & & & x\geq 3 & \\
x\in(-\infty,-1] & & or & x\in[3,\infty) & \\
& & & &
\end{array} $
Solution set: $(-\infty,-1]\cup[3,\infty)$
