1. **Problem statement:** We have a triangle with two sides of lengths 17 and 4. We want to find the largest possible whole-number length for the third side. 2. **Triangle inequality rule:** For any triangle with sides $a$, $b$, and $c$, the sum of the lengths of any two sides must be greater than the length of the third side. This gives us three inequalities: $$a + b > c$$ $$a + c > b$$ $$b + c > a$$ 3. **Apply the rule:** Let the third side be $x$. Given sides are 17 and 4, so: - $17 + 4 > x \implies 21 > x$ - $17 + x > 4 \implies x > 4 - 17 \implies x > -13$ (always true since side lengths are positive) - $4 + x > 17 \implies x > 13$ 4. **Combine inequalities:** From above, $x$ must satisfy: $$13 < x < 21$$ 5. **Find largest whole number:** The largest whole number less than 21 and greater than 13 is 20. **Final answer:** The largest possible whole-number length for the third side is **20**.