Question

mr. knotts found the difference of the following expression. which statement is true about mr. knotts’s work?
\\(\frac{x}{x^2 - 1} - \frac{1}{x - 1}\\)
knotts’s work
step 1: \\(\frac{x}{(x + 1)(x - 1)} - \frac{1}{x - 1}\\)
step 2: \\(\frac{x}{(x + 1)(x - 1)} - \frac{1(x + 1)}{(x + 1)(x - 1)}\\)
step 3: \\(\frac{x - x + 1}{(x + 1)(x - 1)}\\)
step 4: \\(\frac{1}{(x + 1)(x - 1)}\\)
in step 3, he did not distribute the negative sign when he subtracted.
in step 4, he did not correctly subtract the numerator.
in step 1, he did not correctly factor the denominator.
in step 2, he did not use the correct common denominator.

Explanation:

Step1: Analyze Step 1

Factor \(x^2 - 1\) as \((x + 1)(x - 1)\), so Step 1 is correct.

Step2: Analyze Step 2

The common denominator for \(\frac{x}{(x + 1)(x - 1)}\) and \(\frac{1}{x - 1}\) is \((x + 1)(x - 1)\), so we multiply numerator and denominator of \(\frac{1}{x - 1}\) by \((x + 1)\), Step 2 is correct.

Step3: Analyze Step 3

When subtracting \(\frac{x}{(x + 1)(x - 1)}-\frac{x + 1}{(x + 1)(x - 1)}\), the numerator should be \(x-(x + 1)=x - x - 1=-1\), but Mr. Knotts did \(x - x + 1\), so he did not distribute the negative sign.

Step4: Analyze Step 4 (not needed as Step 3 error is found)

Answer:

In step 3, he did not distribute the negative sign when he subtracted.