Question

mr. knolls found the difference of the following expression. which statement is true about mr. knolls’s work?
\\(\frac{x}{x^2 - 1} - \frac{1}{x - 1}\\)

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)}

save and exit next submit
math.com and cogent

Explanation:

Step1: Analyze Step 3

To subtract the fractions, we have \(\frac{x}{(x + 1)(x - 1)}-\frac{x + 1}{(x + 1)(x - 1)}\) (from Step 2). The numerator should be \(x-(x + 1)\), not \(x - x+1\).

Step2: Expand the Correct Numerator

Expanding \(x-(x + 1)\) gives \(x - x-1=-1\), so the correct numerator after subtraction is \(-1\), not \(1\) as in Step 3 and Step 4. So there is a mistake in Step 3 (and thus Step 4) due to an error in distributing the negative sign when subtracting the numerators.

Answer:

There is an error in Step 3 (and subsequent Step 4) because when subtracting \(\frac{x}{(x + 1)(x - 1)}-\frac{x + 1}{(x + 1)(x - 1)}\), the numerator should be \(x-(x + 1)=x - x-1=-1\), not \(x - x + 1 = 1\). So the work has a mistake in the subtraction of the numerators (Step 3).