Which expression is equivalent to 4/(4x-5) - 1/(x+1)?

1. INFER the solution strategy

• Given: \(\frac{4}{4x-5} - \frac{1}{x+1}\)
• Key insight: To subtract fractions, we need a common denominator
• The common denominator must be \((4x-5)(x+1)\) since these are the two different denominators

2. SIMPLIFY by rewriting each fraction with the common denominator

• First fraction: \(\frac{4}{4x-5} = \frac{4(x+1)}{(4x-5)(x+1)}\)
• Second fraction: \(\frac{1}{x+1} = \frac{4x-5}{(x+1)(4x-5)}\)

3. SIMPLIFY the subtraction

• Now we have: \(\frac{4(x+1)}{(4x-5)(x+1)} - \frac{4x-5}{(x+1)(4x-5)}\)
• Since denominators are the same: \(\frac{4(x+1) - (4x-5)}{(4x-5)(x+1)}\)

4. SIMPLIFY the numerator

• Expand: \(4(x+1) = 4x + 4\)
• Handle the subtraction: \(4(x+1) - (4x-5) = 4x + 4 - 4x + 5\)
• Combine like terms: \(4x - 4x + 4 + 5 = 9\)

5. Write final answer

• \(\frac{9}{(4x-5)(x+1)} = \frac{9}{(x+1)(4x-5)}\)

Answer: D. \(\frac{9}{(x+1)(4x-5)}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make sign errors when subtracting \((4x-5)\)

When subtracting \(\frac{1}{x+1}\), students must subtract the entire expression \((4x-5)\) in the numerator. Many students write:

\(4(x+1) - (4x-5) = 4x + 4 - 4x - 5 = -1\)

This incorrect handling of the negative sign leads to a final answer of \(\frac{-1}{(x+1)(4x-5)}\).
This may lead them to select Choice C (\(\frac{-1}{(x+1)(4x-5)}\))

Second Most Common Error:

Poor INFER reasoning: Students attempt to subtract denominators directly

Some students think they can subtract fractions by subtracting numerators and denominators separately:

\(\frac{4}{4x-5} - \frac{1}{x+1} = \frac{4-1}{(4x-5)-(x+1)} = \frac{3}{3x-6}\)

This fundamental misunderstanding of fraction operations may lead them to select Choice B (\(\frac{3}{3x-6}\))

The Bottom Line:

This problem requires solid understanding of rational expression operations AND careful algebraic manipulation. The combination of conceptual reasoning (finding common denominators) and precise execution (handling signs in subtraction) creates multiple opportunities for error.