Triangle Side Analysis: Determining if X+1 Creates a Scalene Triangle

To determine if the triangle with sides $X$, $X+1$, $X+1$ is a scalene triangle, we need to confirm that all sides are of different lengths.

Using the triangle inequality theorem:

• $X + (X + 1) > (X + 1) \rightarrow 2X + 1 > X + 1$, which simplifies to $X > 0$.
• $X + (X + 1) > (X + 1) \rightarrow 2X + 1 > X + 1$, which also simplifies to $X > 0$.
• $(X + 1) + (X + 1) > X \rightarrow 2X + 2 > X$, which simplifies to $X > -2$.

All these conditions are satisfied if $X$ is a positive number. Next, check if all sides differ:

• $X$ and $X+1$ are different, by definition.
• However, the two longer sides are both $X+1$, meaning not all sides are different.

The triangle is thus not scalene, as it does not have all sides of different lengths.

Therefore, the correct answer is No.