Answer
$[-\displaystyle \frac{19}{3},7]$
Work Step by Step
The solutions of $|u| \lt c $ are the numbers that satisfy $-c \lt u \lt c.$
The idea is to isolate $x$ in the middle.
Adding, subtracting or multiplying/dividing with a positive number preserves order (the inequality symbols remain).
Multiplying/dividing with a negative number inverts the order (the inequality symbols changes).
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$|3(x-1)+2|\leq 20\qquad $... is equivalent to ...
$-20\leq 3(x-1)+2\leq 20 \qquad $ ... simplify the middle
$-20\leq 3x-1\leq 20\qquad $ ... add $1$ to all parts
$-19\leq 3x\leq 21 \qquad $ ... divide with $3$
$-\displaystyle \frac{19}{3}\leq x\leq 7$
Interval borders are included in the interval.
The solution set is $[-\displaystyle \frac{19}{3},7]$
