2x/(x-1) = 3 + x/(x-1) What is the solution to the equation above?...

1. INFER the solving strategy

• Given equation: \(\frac{2\mathrm{x}}{\mathrm{x}-1} = 3 + \frac{\mathrm{x}}{\mathrm{x}-1}\)
• Key insight: Both terms on the left have the same denominator \((\mathrm{x}-1)\), and one term on the right also has this denominator
• Strategy: Multiply both sides by \((\mathrm{x}-1)\) to eliminate all denominators

2. SIMPLIFY by multiplying both sides by (x-1)

• Left side: \(\frac{2\mathrm{x}}{\mathrm{x}-1} \times (\mathrm{x}-1) = 2\mathrm{x}\)
• Right side: \([3 + \frac{\mathrm{x}}{\mathrm{x}-1}] \times (\mathrm{x}-1) = 3(\mathrm{x}-1) + \mathrm{x}\)

The equation becomes: \(2\mathrm{x} = 3(\mathrm{x}-1) + \mathrm{x}\)

3. SIMPLIFY the right side using the distributive property

• \(3(\mathrm{x}-1) = 3\mathrm{x} - 3\)
• So: \(2\mathrm{x} = 3\mathrm{x} - 3 + \mathrm{x}\)
• Combine like terms: \(2\mathrm{x} = 4\mathrm{x} - 3\)

4. SIMPLIFY to isolate x

• Subtract 4x from both sides: \(2\mathrm{x} - 4\mathrm{x} = -3\)
• Combine: \(-2\mathrm{x} = -3\)
• Divide by -2: \(\mathrm{x} = \frac{3}{2}\)

Answer: D. 3/2

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students correctly multiply by \((\mathrm{x}-1)\) but make distribution errors when expanding \(3(\mathrm{x}-1)\). They might write \(3(\mathrm{x}-1) = 3\mathrm{x} - 1\) instead of \(3\mathrm{x} - 3\), or forget to distribute entirely.

Following this error path with \(3(\mathrm{x}-1) = 3\mathrm{x} - 1\):

• \(2\mathrm{x} = 3\mathrm{x} - 1 + \mathrm{x} = 4\mathrm{x} - 1\)
• \(2\mathrm{x} - 4\mathrm{x} = -1\)
• \(-2\mathrm{x} = -1\)
• \(\mathrm{x} = \frac{1}{2}\)

This may lead them to select Choice A (1/2)

Second Most Common Error:

Poor INFER reasoning: Students attempt to solve without eliminating denominators first, trying to work directly with the fractions and getting confused about how to combine terms with different denominators.

This leads to confusion and abandoning systematic solution, causing them to guess among the answer choices.

The Bottom Line:

This problem tests whether students can systematically eliminate denominators in rational equations and then execute multi-step algebraic simplification without arithmetic errors. The key is recognizing that multiplying by \((\mathrm{x}-1)\) cleans up the equation structure significantly.