Answer
$\left\{x\ |\ -6 \lt x \lt 0 \right\}$ or $\left(-6,0\right)$
Work Step by Step
$1 \lt 1-\displaystyle \frac{1}{2}x \lt 4 \qquad$ ... add $-1$
$0 \lt -\displaystyle \frac{1}{2}x \lt 3 \qquad$ ... multiply with $-2 $ (negative)
... the inequalities change direction ...
$0 \gt x \gt -6 $
or
$-6 \lt x \lt 0$
Solution set: $\left\{x\ |\ -6 \lt x \lt 0 \right\}$ or $\left(-6,0\right)$
