Scalene triangle side length

In a scalene triangle, no three sides are of the same length. In all triangles, for two given side lengths $a$ and $b$ , the third side length $c$ satisfies the inequality $\left | a - b \right | < c < a + b$ . Since the given triangle's side lengths are integral, the shortest side length possible for a triangle would be 34 (which is one more than the lower bound, which cannot be a side length since $c \neq \left | a - b \right |$ . However, if the third side length is 34, then the triangle would not be scalene. Therefore, the shortest possible length for the third side is 35.