Answer
$\displaystyle \left\{x\ |\ -\frac{1}{3} \lt x \leq \frac{1}{3} \right\}$ or $\displaystyle \left(-\frac{1}{3}, \ \frac{1}{3} \right]$
Work Step by Step
$\displaystyle \frac{1}{3} \lt \frac{x+1}{2} \leq \frac{2}{3} \qquad$... multiply with $6$, the LCD
$2 \lt 3(x+1) \leq 4$
$ 2 \lt 3x+3 \leq 4\qquad$ ... add $-3$
$-1 \lt 3x \leq 1 \qquad$... multiply with $\displaystyle \frac{1}{3}$
$-\displaystyle \frac{1}{3} \lt x \leq \frac{1}{3} $
Solution set: $\displaystyle \left\{x\ |\ -\frac{1}{3} \lt x \leq \frac{1}{3} \right\}$ or $\displaystyle \left(-\frac{1}{3}, \ \frac{1}{3} \right]$
