Which expression is equivalent to 2/(x^2 - 1) - 1/(x - 1)?\(\frac{1}{(\mathrm{x} - 1)(\mathrm{x} + 1)}\)\(\frac{-1}{(\mathrm{x} - 1)(\mathrm{x} + 1)}\...

1. INFER the approach needed

• We have: \(\frac{2}{x^2 - 1} - \frac{1}{x - 1}\)
• To subtract fractions, we need a common denominator
• Key insight: \(x^2 - 1\) looks like a difference of squares that we can factor

2. SIMPLIFY by factoring the first denominator

• Factor \(x^2 - 1\) using difference of squares: \(x^2 - 1 = (x - 1)(x + 1)\)
• Our expression becomes: \(\frac{2}{(x - 1)(x + 1)} - \frac{1}{x - 1}\)

3. SIMPLIFY by finding the common denominator

• The LCD is \((x - 1)(x + 1)\)
• Rewrite the second fraction: \(\frac{1}{x - 1} = \frac{x + 1}{(x - 1)(x + 1)}\)
• Now we have: \(\frac{2}{(x - 1)(x + 1)} - \frac{x + 1}{(x - 1)(x + 1)}\)

4. SIMPLIFY by combining the numerators

• Subtract the numerators: \(\frac{2 - (x + 1)}{(x - 1)(x + 1)}\)
• Distribute the negative: \(\frac{2 - x - 1}{(x - 1)(x + 1)} = \frac{1 - x}{(x - 1)(x + 1)}\)

5. SIMPLIFY by factoring and canceling

• Factor out -1 from numerator: \((1 - x) = -(x - 1)\)
• Expression becomes: \(\frac{-(x - 1)}{(x - 1)(x + 1)}\)
• Cancel the common factor \((x - 1)\): \(\frac{-1}{x + 1}\)

Answer: D. \(\frac{-1}{x + 1}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skills when combining numerators: Students correctly get to \(\frac{2}{(x - 1)(x + 1)} - \frac{x + 1}{(x - 1)(x + 1)}\) but make errors when subtracting \((x + 1)\) from 2. They might forget to distribute the negative sign, getting \((2 + x + 1)\) instead of \((2 - x - 1)\), leading to \((3 + x)\) in the numerator instead of \((1 - x)\).

This completely changes the final answer and may lead them to select Choice C (\(\frac{1}{x + 1}\)) or cause confusion and guessing.

Second Most Common Error:

Missing conceptual knowledge about difference of squares: Students don't recognize that \(x^2 - 1 = (x - 1)(x + 1)\), so they struggle to find the common denominator or try to work with \(x^2 - 1\) directly, making the algebra much more complicated.

This leads to getting stuck early in the problem and abandoning systematic solution, resulting in guessing.

The Bottom Line:

This problem requires careful algebraic manipulation at every step. The key insight is recognizing the difference of squares factoring, but success depends on methodical execution of fraction operations and not making sign errors when combining terms.