Question:5/(x+3) - 2/(x-3) = (mx+k)/(x^2-9)The equation above is true for all values of x where x neq ±3, and m...

1. INFER the key structural insight

• Given equation: \(\frac{5}{\mathrm{x}+3} - \frac{2}{\mathrm{x}-3} = \frac{\mathrm{mx}+\mathrm{k}}{\mathrm{x}^2-9}\)
• Key insight: The right side denominator \(\mathrm{x}^2 - 9\) is a difference of squares
• This means: \(\mathrm{x}^2 - 9 = (\mathrm{x}+3)(\mathrm{x}-3)\)
• Strategy: Combine the left side fractions using this common denominator

2. SIMPLIFY the left side by finding common denominators

• Rewrite each fraction with denominator \((\mathrm{x}+3)(\mathrm{x}-3)\):
  • \(\frac{5}{\mathrm{x}+3} = \frac{5(\mathrm{x}-3)}{(\mathrm{x}+3)(\mathrm{x}-3)}\)
  • \(\frac{2}{\mathrm{x}-3} = \frac{2(\mathrm{x}+3)}{(\mathrm{x}+3)(\mathrm{x}-3)}\)

3. SIMPLIFY by combining the fractions

• Subtract the fractions:
  \(\frac{5(\mathrm{x}-3) - 2(\mathrm{x}+3)}{(\mathrm{x}+3)(\mathrm{x}-3)}\)

4. SIMPLIFY the numerator through distribution

• Distribute carefully:
  \(5(\mathrm{x}-3) - 2(\mathrm{x}+3) = 5\mathrm{x} - 15 - 2\mathrm{x} - 6\)
• Combine like terms:
  \((5\mathrm{x} - 2\mathrm{x}) + (-15 - 6) = 3\mathrm{x} - 21\)
• Result:
  \(\frac{3\mathrm{x} - 21}{\mathrm{x}^2 - 9}\)

5. INFER the coefficient values

• Now we have:
  \(\frac{3\mathrm{x} - 21}{\mathrm{x}^2 - 9} = \frac{\mathrm{mx} + \mathrm{k}}{\mathrm{x}^2 - 9}\)
• Since denominators are identical, numerators must be equal:
  \(3\mathrm{x} - 21 = \mathrm{mx} + \mathrm{k}\)
• Compare coefficients: \(\mathrm{m} = 3\) (coefficient of x), \(\mathrm{k} = -21\) (constant term)

6. Calculate the final answer

• \(\mathrm{k} - \mathrm{m} = -21 - 3 = -24\)

Answer: A) -24

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make sign errors when distributing the subtraction, writing \(5(\mathrm{x}-3) - 2(\mathrm{x}+3)\) as \(5\mathrm{x} - 15 - 2\mathrm{x} + 6\) instead of \(5\mathrm{x} - 15 - 2\mathrm{x} - 6\).

This leads to getting \(\frac{3\mathrm{x} - 9}{\mathrm{x}^2 - 9}\) instead of \(\frac{3\mathrm{x} - 21}{\mathrm{x}^2 - 9}\), giving them \(\mathrm{k} = -9\) instead of \(\mathrm{k} = -21\). With \(\mathrm{m} = 3\), they calculate \(\mathrm{k} - \mathrm{m} = -9 - 3 = -12\).
This may lead them to select Choice C (-12).

Second Most Common Error:

Poor INFER reasoning: Students don't recognize that \(\mathrm{x}^2 - 9\) factors as \((\mathrm{x}+3)(\mathrm{x}-3)\), so they struggle to find a systematic way to combine the left-side fractions.

Without this insight, they might attempt cross-multiplication or other inefficient methods, leading to algebraic mistakes or getting stuck entirely. This leads to confusion and guessing.

The Bottom Line:

This problem tests whether students can recognize structural patterns in algebraic expressions and execute multi-step rational expression operations without sign errors. The key breakthrough is seeing the factored form connection between the denominators.