1. **State the problem:** Solve the compound inequality $$\frac{1}{6} < \frac{2x - 13}{12} \leq \frac{2}{3}$$. 2. **Understand the inequality:** We want to find all values of $x$ such that the expression $\frac{2x - 13}{12}$ is greater than $\frac{1}{6}$ and less than or equal to $\frac{2}{3}$. 3. **Break the compound inequality into two parts:** $$\frac{1}{6} < \frac{2x - 13}{12}$$ and $$\frac{2x - 13}{12} \leq \frac{2}{3}$$ 4. **Solve the first inequality:** Multiply both sides by 12 (positive, so inequality direction stays the same): $$12 \times \frac{1}{6} < 12 \times \frac{2x - 13}{12}$$ $$2 < 2x - 13$$ Add 13 to both sides: $$2 + 13 < 2x - 13 + 13$$ $$15 < 2x$$ Divide both sides by 2: $$\frac{15}{2} < x$$ or $$x > 7.5$$ 5. **Solve the second inequality:** Multiply both sides by 12: $$12 \times \frac{2x - 13}{12} \leq 12 \times \frac{2}{3}$$ $$2x - 13 \leq 8$$ Add 13 to both sides: $$2x - 13 + 13 \leq 8 + 13$$ $$2x \leq 21$$ Divide both sides by 2: $$x \leq \frac{21}{2}$$ or $$x \leq 10.5$$ 6. **Combine the two results:** $$7.5 < x \leq 10.5$$ **Final answer:** $$\boxed{7.5 < x \leq 10.5}$$