1. **State the problem:** Diego walks between 9 and 12 miles per week and works out at the gym between 4.5 and 6 hours per week. We need to write a system of inequalities to represent this situation. 2. **Define variables:** Let $x$ be the miles Diego walks per week. Let $y$ be the hours Diego works out at the gym per week. 3. **Write inequalities:** Since Diego walks at least 9 miles but no more than 12 miles: $$9 \leq x \leq 12$$ Since Diego works out at least 4.5 hours but no more than 6 hours: $$4.5 \leq y \leq 6$$ 4. **Express as a system:** $$\begin{cases} 9 \leq x \leq 12 \\ 4.5 \leq y \leq 6 \end{cases}$$ 5. **Explanation:** This system restricts $x$ and $y$ to intervals representing Diego's weekly walking and gym hours. **Final answer:** $$\boxed{\begin{cases} 9 \leq x \leq 12 \\ 4.5 \leq y \leq 6 \end{cases}}$$