1. **State the problem:** Solve the compound inequality $$0 \leq 3x - 5 \leq 7$$. 2. **Understand the compound inequality:** This means both inequalities must be true simultaneously: $$0 \leq 3x - 5$$ and $$3x - 5 \leq 7$$. 3. **Solve the first inequality:** $$0 \leq 3x - 5$$ Add 5 to both sides: $$0 + 5 \leq 3x - 5 + 5$$ $$5 \leq 3x$$ Divide both sides by 3: $$\frac{5}{\cancel{3}} \leq \frac{3x}{\cancel{3}}$$ $$\frac{5}{3} \leq x$$ 4. **Solve the second inequality:** $$3x - 5 \leq 7$$ Add 5 to both sides: $$3x - 5 + 5 \leq 7 + 5$$ $$3x \leq 12$$ Divide both sides by 3: $$\frac{3x}{\cancel{3}} \leq \frac{12}{\cancel{3}}$$ $$x \leq 4$$ 5. **Combine the results:** $$\frac{5}{3} \leq x \leq 4$$ **Final answer:** The solution to the compound inequality is $$x \in \left[\frac{5}{3}, 4\right]$$.