1. **Problem statement:** We have a triangle with two sides of lengths 4 and 7. 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 4 and 7, so: - $4 + 7 > x \implies 11 > x$ - $4 + x > 7 \implies x > 3$ - $7 + x > 4 \implies x > -3$ (always true since $x$ is positive) 4. **Combine inequalities:** From above, $3 < x < 11$. 5. **Find largest whole number:** The largest whole number less than 11 and greater than 3 is 10. **Final answer:** The largest possible whole-number length for the third side is **10**.