1. **State the problem:** Solve the compound inequality $n + 2 \leq -5$ and $n + 6 \geq -6$. 2. **Understand the problem:** We need to find all values of $n$ that satisfy both inequalities simultaneously. 3. **Solve the first inequality:** $$n + 2 \leq -5$$ Subtract 2 from both sides: $$n + 2 - 2 \leq -5 - 2$$ $$n \leq -7$$ 4. **Solve the second inequality:** $$n + 6 \geq -6$$ Subtract 6 from both sides: $$n + 6 - 6 \geq -6 - 6$$ $$n \geq -12$$ 5. **Combine the inequalities:** We want values of $n$ such that: $$-12 \leq n \leq -7$$ 6. **Interpretation:** The solution is all numbers $n$ between $-12$ and $-7$, inclusive. **Final answer:** $$\boxed{-12 \leq n \leq -7}$$