1. **State the problem:** We are given two sides of a triangle with lengths 4 inches and 7 inches, and we need to find the possible whole-number lengths of the third side. 2. **Recall the triangle inequality theorem:** 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 remaining side. This gives three inequalities: $$a + b > c$$ $$a + c > b$$ $$b + c > a$$ 3. **Apply the theorem to our sides:** Let the third side be $x$. Then: - $4 + 7 > x \implies 11 > x \implies x < 11$ - $4 + x > 7 \implies x > 3$ - $7 + x > 4 \implies x > -3$ (always true since $x$ is positive) 4. **Combine inequalities:** The third side $x$ must satisfy: $$3 < x < 11$$ 5. **Find whole-number lengths:** The possible whole numbers between 3 and 11 are: $$4,5,6,7,8,9,10$$ **Final answer:** 4,5,6,7,8,9,10