Which expression is equivalent to 3/(x-5) - (2x+1)/(x^2-25)?

1. TRANSLATE the problem information

• Given: \(\frac{3}{\mathrm{x}-5} - \frac{2\mathrm{x}+1}{\mathrm{x}^2-25}\)
• Goal: Find an equivalent single rational expression

2. INFER the solution strategy

• To subtract rational expressions, we need a common denominator
• Notice that \(\mathrm{x}^2-25\) looks like a difference of squares: \(\mathrm{x}^2-25 = (\mathrm{x}-5)(\mathrm{x}+5)\)
• This means the LCD is \(\mathrm{x}^2-25\)

3. SIMPLIFY by converting to the common denominator

• The second fraction already has denominator \(\mathrm{x}^2-25\)
• Convert the first fraction: \(\frac{3}{\mathrm{x}-5} \times \frac{\mathrm{x}+5}{\mathrm{x}+5} = \frac{3\mathrm{x}+15}{\mathrm{x}^2-25}\)

4. SIMPLIFY by combining the fractions

• Now we have: \(\frac{3\mathrm{x}+15}{\mathrm{x}^2-25} - \frac{2\mathrm{x}+1}{\mathrm{x}^2-25}\)
• Combine numerators: \(\frac{3\mathrm{x}+15 - (2\mathrm{x}+1)}{\mathrm{x}^2-25}\)
• Distribute the negative: \(\frac{3\mathrm{x}+15 - 2\mathrm{x} - 1}{\mathrm{x}^2-25}\)
• Combine like terms: \(\frac{\mathrm{x}+14}{\mathrm{x}^2-25}\)

Answer: B. \(\frac{\mathrm{x}+14}{\mathrm{x}^2-25}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students forget to distribute the negative sign to both terms in \((2\mathrm{x}+1)\)

They write: \(3\mathrm{x}+15 - 2\mathrm{x}+1\) instead of \(3\mathrm{x}+15 - 2\mathrm{x} - 1\)
This gives them \(\frac{2\mathrm{x}+16}{\mathrm{x}^2-25}\)
This may lead them to select Choice C \((\frac{2\mathrm{x}+16}{\mathrm{x}^2-25})\)

Second Most Common Error:

Missing INFER insight: Students don't recognize that \(\mathrm{x}^2-25\) factors as \((\mathrm{x}-5)(\mathrm{x}+5)\)

Without this insight, they struggle to find the common denominator and may attempt incorrect algebraic manipulations or give up and guess randomly.

The Bottom Line:

This problem tests both pattern recognition (spotting the difference of squares) and careful algebraic execution (proper sign distribution). Success requires methodical attention to detail in multi-step simplification.