Triangle Side Length Analysis: Comparing a, a-2, and a+1

Given the values of the sides of a triangle, is it a triangle with different sides?

To solve this problem, we'll determine whether a triangle with side lengths $a$, $a-2$, and $a+1$ is scalene:

• Step 1: Verify the triangle inequality theorem.
  - Check $a + (a-2) > (a+1)$: $2a - 2 > a + 1$ simplifies to $a > 3$. - Check $(a-2) + (a+1) > a$: $(2a - 1) > a$ simplifies to $a > 1$. - Check $a + (a+1) > (a-2)$: $2a + 1 > a - 2$ simplifies to $a > -3/2$, which is always true for $a > 2$.
• Step 2: Check if all sides are different.
  - Compare $a \neq a-2$: True, always holds as $a > 2$.
  - Compare $a \neq a+1$: True, always holds.
  - Compare $a-2 \neq a+1$: True, simplifies to $a \neq 3$, which holds since $a > 3$.

All side lengths satisfy the triangle inequality and are different. Therefore, the triangle is scalene. The solution to the problem is "Yes," this is a triangle with different sides.