Question

Mr. Knotts found the difference of the following expression. Which statement is true about Mr. Knotts’s work?

StartFraction x Over x squared minus 1 EndFraction minus StartFraction 1 Over x minus 1 EndFraction

Step 1: StartFraction x Over (x + 1) (x minus 1) EndFraction minus StartFraction 1 Over x minus 1 EndFraction

Step 2: StartFraction x Over (x + 1) (x minus 1) EndFraction minus StartFraction 1 (x + 1) Over (x + 1) (x minus 1) EndFraction

Step 3: StartFraction x minus x + 1 Over (x + 1) (x minus 1) EndFraction

Step 4: StartFraction 1 Over (x + 1) (x minus 1) EndFraction
In step 1, he did not correctly factor the denominator.
In step 2, he did not use the correct common denominator.
In step 3, he did not distribute the negative sign when he subtracted.
In step 4, he did not correctly subtract the numerator.

Answer 1

The true statement is: "In step 3, he did not distribute the negative sign when he subtracted."

Reason: With common denominator (x+1)(x-1),
x/(x^2−1) − 1/(x−1) = x/((x+1)(x−1)) − (x+1)/((x+1)(x−1))
= (x − (x+1))/((x+1)(x−1)) = (x − x − 1)/((x+1)(x−1)) = −1/((x+1)(x−1)).

He wrote x − x + 1 instead of x − x − 1, so Step 3 is the error.