Let's examine Mr. Knotts's work step-by-step to verify whether it’s correct.
We start with:
[tex]\[ \frac{x}{x^2 - 1} - \frac{1}{x - 1} \][/tex]
### Step 1
[tex]\[ \frac{x}{(x+1)(x-1)} - \frac{1}{x-1} \][/tex]
This factorization of [tex]\( x^2 - 1 \)[/tex] as [tex]\((x + 1)(x - 1)\)[/tex] is correct.
### Step 2
[tex]\[ \frac{x}{(x+1)(x-1)} - \frac{1(x + 1)}{(x+1)(x-1)} \][/tex]
Mr. Knotts found a common denominator, which is [tex]\((x + 1)(x - 1)\)[/tex]. He then adjusted the second fraction to have this common denominator. To do this properly, he multiplied the numerator and the denominator of the second term by [tex]\( x + 1 \)[/tex]. This is correct.
### Step 3
[tex]\[ \frac{x - (x + 1)}{(x+1)(x-1)} \][/tex]
However, there is a mistake in this step. The term should be:
[tex]\[ \frac{x - (x + 1)}{(x + 1)(x - 1)} = \frac{x - x - 1}{(x + 1)(x - 1)} \][/tex]
The correction should lead to:
[tex]\[ \frac{-1}{(x + 1)(x - 1)} \][/tex]
### Step 4
Mr. Knotts's work shows:
[tex]\[ \frac{1}{(x+1)(x-1)} \][/tex]
However, the correct simplified form is:
[tex]\[ \frac{-1}{(x+1)(x-1)} \][/tex]
Thus, upon review, it appears Mr. Knotts made a mistake in the sign while simplifying the numerator in Step 3. Therefore, the correct form should have been [tex]\( \frac{-1}{(x+1)(x-1)} \)[/tex].
### Conclusion
The correct statement is that Mr. Knotts should have obtained:
[tex]\[ \frac{-1}{(x + 1)(x - 1)} \][/tex]
Hence, his final step should reflect this negative sign. Therefore, Mr. Knotts’s work in Step 3 and Step 4 is incorrect due to the sign error in simplifying the numerator.
