1. **Problem statement:** We have an acute triangle with two known sides measuring 5 inches and 8 inches. We want to find the greatest possible whole-number length of the unknown longest side. 2. **Triangle inequality rule:** For any triangle with sides $a$, $b$, and $c$, the sum of any two sides must be greater than the third side: $$a + b > c, \quad a + c > b, \quad b + c > a$$ 3. **Acute triangle condition:** For sides $a$, $b$, and $c$ with $c$ as the longest side, the triangle is acute if: $$a^2 + b^2 > c^2$$ 4. **Apply to our problem:** Let the unknown longest side be $x$. Given sides are 5 and 8, so: - Triangle inequalities: $$5 + 8 > x \Rightarrow 13 > x$$ $$5 + x > 8 \Rightarrow x > 3$$ $$8 + x > 5 \Rightarrow x > -3$$ (always true) - Acute condition: $$5^2 + 8^2 > x^2 \Rightarrow 25 + 64 > x^2 \Rightarrow 89 > x^2$$ 5. **Find the range for $x$:** - From triangle inequality: $x < 13$ - From acute condition: $x^2 < 89 \Rightarrow x < \sqrt{89} \approx 9.43$ 6. **Greatest whole number length:** Since $x$ must be less than both 13 and approximately 9.43, the greatest whole number less than 9.43 is 9. **Final answer:** The greatest possible whole-number length of the unknown side is **9 inches**.