1. **State the problem:** We have two inequalities involving an unknown value $x$: - $3 + x \leq 5$ - $8 + x \geq 5$ We want to find the range of values of $x$ that satisfy both inequalities simultaneously. 2. **Solve the first inequality:** $$3 + x \leq 5$$ Subtract 3 from both sides: $$\cancel{3} + x - \cancel{3} \leq 5 - 3$$ $$x \leq 2$$ 3. **Solve the second inequality:** $$8 + x \geq 5$$ Subtract 8 from both sides: $$\cancel{8} + x - \cancel{8} \geq 5 - 8$$ $$x \geq -3$$ 4. **Combine both inequalities:** $$-3 \leq x \leq 2$$ This means $x$ must be greater than or equal to $-3$ and less than or equal to $2$ to satisfy both inequalities. **Final answer:** $$\boxed{-3 \leq x \leq 2}$$