Question:x/(x-1) = (2x+1)/(x+1)Which of the following is a solution to the equation above?

1. TRANSLATE the problem information

• Given equation: \(\frac{\mathrm{x}}{\mathrm{x}-1} = \frac{2\mathrm{x}+1}{\mathrm{x}+1}\)
• Need to find which answer choice satisfies this equation

2. INFER the solution approach

• Since we have a rational equation (fractions equal to each other), cross-multiplication will eliminate the denominators
• This should lead to a polynomial equation we can solve

3. SIMPLIFY by cross-multiplying

• Cross-multiply: \(\mathrm{x}(\mathrm{x}+1) = (2\mathrm{x}+1)(\mathrm{x}-1)\)
• Left side: \(\mathrm{x}(\mathrm{x}+1) = \mathrm{x}^2 + \mathrm{x}\)
• Right side: \((2\mathrm{x}+1)(\mathrm{x}-1) = 2\mathrm{x}^2 - 2\mathrm{x} + \mathrm{x} - 1 = 2\mathrm{x}^2 - \mathrm{x} - 1\)

4. SIMPLIFY to standard quadratic form

• Set expressions equal: \(\mathrm{x}^2 + \mathrm{x} = 2\mathrm{x}^2 - \mathrm{x} - 1\)
• Move all terms to right side: \(0 = 2\mathrm{x}^2 - \mathrm{x} - 1 - \mathrm{x}^2 - \mathrm{x}\)
• Combine like terms: \(0 = \mathrm{x}^2 - 2\mathrm{x} - 1\)

5. SIMPLIFY using the quadratic formula

• For \(\mathrm{x}^2 - 2\mathrm{x} - 1 = 0\), we have \(\mathrm{a} = 1\), \(\mathrm{b} = -2\), \(\mathrm{c} = -1\)
• \(\mathrm{x} = \frac{2 ± \sqrt{4 + 4}}{2}\)
  \(= \frac{2 ± \sqrt{8}}{2}\)
  \(= \frac{2 ± 2\sqrt{2}}{2}\)
  \(= 1 ± \sqrt{2}\)
• Solutions: \(\mathrm{x} = 1 + \sqrt{2}\) and \(\mathrm{x} = 1 - \sqrt{2}\)

6. APPLY CONSTRAINTS to verify domain

• Original equation has denominators \((\mathrm{x}-1)\) and \((\mathrm{x}+1)\)
• Domain restrictions: \(\mathrm{x} ≠ 1\) and \(\mathrm{x} ≠ -1\)
• Both \(1 + \sqrt{2}\) and \(1 - \sqrt{2}\) satisfy these restrictions ✓

Answer: A (\(1 - \sqrt{2}\))

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Incorrectly expanding \((2\mathrm{x}+1)(\mathrm{x}-1)\)

Students often make distribution errors when expanding \((2\mathrm{x}+1)(\mathrm{x}-1)\), such as:

• Forgetting the middle terms: getting \(2\mathrm{x}^2 - 1\) instead of \(2\mathrm{x}^2 - \mathrm{x} - 1\)
• Sign errors: getting \(2\mathrm{x}^2 + \mathrm{x} - 1\) instead of \(2\mathrm{x}^2 - \mathrm{x} - 1\)
• Incomplete distribution: only multiplying \(2\mathrm{x}\) by \((\mathrm{x}-1)\) and forgetting to multiply \(1\) by \((\mathrm{x}-1)\)

This leads to an incorrect quadratic equation, producing wrong solutions that don't match any answer choice, causing confusion and guessing.

Second Most Common Error:

Insufficient APPLY CONSTRAINTS reasoning: Not checking domain restrictions

Some students solve the quadratic correctly but fail to verify that their solutions don't make denominators zero. While both solutions are actually valid in this problem, students who don't perform this check may second-guess themselves or incorrectly eliminate valid solutions in similar problems.

The Bottom Line:

This problem tests whether students can systematically work through rational equation algebra without making expansion errors, while also remembering to verify that solutions are within the original equation's domain.