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

1. INFER the approach needed

• We have two fractions with different denominators that need to be subtracted
• Strategy: Find a common denominator, then combine the fractions

2. INFER what common denominator to use

• The denominators are \(\mathrm{(4x-5)}\) and \(\mathrm{(x+1)}\)
• Since these have no common factors, use their product: \(\mathrm{(4x-5)(x+1)}\)

3. SIMPLIFY by converting each fraction

• First fraction: \(\mathrm{\frac{4}{4x-5} = \frac{4(x+1)}{(4x-5)(x+1)}}\)
• SIMPLIFY the numerator: \(\mathrm{4(x+1) = 4x + 4}\)
• So: \(\mathrm{\frac{4}{4x-5} = \frac{4x+4}{(4x-5)(x+1)}}\)

• Second fraction: \(\mathrm{\frac{1}{x+1} = \frac{1(4x-5)}{(x+1)(4x-5)}}\)
• SIMPLIFY the numerator: \(\mathrm{1(4x-5) = 4x-5}\)
• So: \(\mathrm{\frac{1}{x+1} = \frac{4x-5}{(4x-5)(x+1)}}\)

4. SIMPLIFY by subtracting the fractions

• Now subtract: \(\mathrm{\frac{4x+4}{(4x-5)(x+1)} - \frac{4x-5}{(4x-5)(x+1)}}\)
• Combine numerators: \(\mathrm{\frac{(4x+4) - (4x-5)}{(4x-5)(x+1)}}\)
• SIMPLIFY the numerator: \(\mathrm{(4x+4) - (4x-5) = 4x + 4 - 4x + 5 = 9}\)
• Final result: \(\mathrm{\frac{9}{(4x-5)(x+1)} = \frac{9}{(x+1)(4x-5)}}\)

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

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students correctly set up the common denominator but make arithmetic errors when expanding or combining terms in the numerator.

For example, when computing \(\mathrm{(4x+4) - (4x-5)}\), they might get:

• \(\mathrm{4x + 4 - 4x - 5 = -1}\) (forgetting to distribute the negative sign)
• Or \(\mathrm{4x + 4 - 4x + 5 = 4 + 5 = 9}\), but then write it as \(\mathrm{4x + 9}\)

This may lead them to select Choice C (\(\mathrm{\frac{-1}{(x+1)(4x-5)}}\)) or get confused and guess.

Second Most Common Error:

Poor INFER reasoning about common denominators: Students might try to find a common denominator by just adding the denominators: \(\mathrm{(4x-5) + (x+1) = 5x-4}\), or attempt some other incorrect approach.

This leads to completely wrong expressions that don't match any answer choice, causing them to get stuck and randomly select an answer.

The Bottom Line:

This problem tests whether students can systematically execute the multi-step process of combining rational expressions, with particular emphasis on careful algebraic manipulation and sign handling.