1. **Problem statement:** We have a triangle with two sides of lengths 17 and 19. We want to find the largest possible whole-number length for the third side. 2. **Formula and rule:** For any triangle with sides $a$, $b$, and $c$, the triangle inequality must hold: $$a + b > c, \quad b + c > a, \quad c + a > b$$ 3. Since we want the largest possible whole-number length for the third side, call it $x$. The two known sides are 17 and 19. 4. Apply the triangle inequality involving $x$: - $17 + 19 > x \implies 36 > x$ - $17 + x > 19 \implies x > 2$ - $19 + x > 17 \implies x > -2$ (always true since $x$ is positive) 5. From these inequalities, the third side $x$ must satisfy: $$2 < x < 36$$ 6. The largest whole number less than 36 is 35. **Final answer:** The largest possible whole-number length for the third side is $\boxed{35}$.