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