Answer
$(-7,9)$
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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$|\displaystyle \frac{3(x-1)}{4}| \lt 6\qquad $... is equivalent to ...
$-6 \lt \displaystyle \frac{3(x-1)}{4} \lt 6\qquad $ ... multiply all parts with $4$
$-24 \lt 3x-3 \lt 24\qquad $ ... add $3$ to all parts
$-21 \lt 3x \lt 27\qquad $ ... divide with $3$
$-7 \lt x \lt 9$
Interval borders are excluded from the interval.
The solution set is $(-7,9)$ .
