a triangle has side lengths of 1 over x-1, -1 over x^2-1, and 1 over x^2-2x-3 meters. find a simplified expression for the perimeter of the triangle in terms of x. (perimeter is sum of all side lengths)

Answer 1

To find the perimeter of the triangle, we need to add up all the side lengths.

First, let's simplify each side length:
Side 1: 1/(x-1)
Side 2: -1/(x^2-1) = -1/((x-1)(x+1))
Side 3: 1/(x^2-2x-3) = 1/((x+1)(x-3))

Now, let's add up all the side lengths to find the perimeter:
Perimeter = Side 1 + Side 2 + Side 3
= 1/(x-1) + (-1/((x-1)(x+1))) + 1/((x+1)(x-3))

To simplify this expression, we can find a common denominator and add up the fractions:

Perimeter = ((x+1)(x-3) + (-1)(x-1) + (x-1))/((x+1)(x-1)(x-3))
= (x^2 - 2x - 3 - x + 1 + x - 1)/((x+1)(x-1)(x-3))
= (x^2 - 2x - 3)/((x+1)(x-1)(x-3))

Therefore, the simplified expression for the perimeter of the triangle in terms of x is:
Perimeter = (x^2 - 2x - 3)/((x+1)(x-1)(x-3)) meters.