\(\frac{2}{\mathrm{x}-2} + \frac{3}{\mathrm{x}+5} = \frac{\mathrm{rx}+\mathrm{t}}{(\mathrm{x}-2)(\mathrm{x}+5)}\) The equation above is true for all ...

1. INFER the solution strategy

• Given: \(\frac{2}{\mathrm{x}-2} + \frac{3}{\mathrm{x}+5} = \frac{\mathrm{rx}+\mathrm{t}}{(\mathrm{x}-2)(\mathrm{x}+5)}\)
• Key insight: The right side already shows us what the common denominator should be
• Strategy: Convert both fractions on the left to have denominator \((\mathrm{x}-2)(\mathrm{x}+5)\)

2. SIMPLIFY the first fraction

• Convert \(\frac{2}{\mathrm{x}-2}\) to have denominator \((\mathrm{x}-2)(\mathrm{x}+5)\):
• Multiply by \(\frac{\mathrm{x}+5}{\mathrm{x}+5}\): \(\frac{2}{\mathrm{x}-2} \times \frac{\mathrm{x}+5}{\mathrm{x}+5} = \frac{2(\mathrm{x}+5)}{(\mathrm{x}-2)(\mathrm{x}+5)}\)
• Expand numerator: \(2(\mathrm{x}+5) = 2\mathrm{x} + 10\)
• Result: \(\frac{2\mathrm{x}+10}{(\mathrm{x}-2)(\mathrm{x}+5)}\)

3. SIMPLIFY the second fraction

• Convert \(\frac{3}{\mathrm{x}+5}\) to have denominator \((\mathrm{x}-2)(\mathrm{x}+5)\):
• Multiply by \(\frac{\mathrm{x}-2}{\mathrm{x}-2}\): \(\frac{3}{\mathrm{x}+5} \times \frac{\mathrm{x}-2}{\mathrm{x}-2} = \frac{3(\mathrm{x}-2)}{(\mathrm{x}-2)(\mathrm{x}+5)}\)
• Expand numerator: \(3(\mathrm{x}-2) = 3\mathrm{x} - 6\)
• Result: \(\frac{3\mathrm{x}-6}{(\mathrm{x}-2)(\mathrm{x}+5)}\)

4. SIMPLIFY by adding the fractions

• Now we have: \(\frac{2\mathrm{x}+10}{(\mathrm{x}-2)(\mathrm{x}+5)} + \frac{3\mathrm{x}-6}{(\mathrm{x}-2)(\mathrm{x}+5)}\)
• Add numerators: \((2\mathrm{x}+10) + (3\mathrm{x}-6) = 2\mathrm{x} + 10 + 3\mathrm{x} - 6 = 5\mathrm{x} + 4\)
• Combined result: \(\frac{5\mathrm{x}+4}{(\mathrm{x}-2)(\mathrm{x}+5)}\)

5. INFER the values by comparing coefficients

• We have: \(\frac{5\mathrm{x}+4}{(\mathrm{x}-2)(\mathrm{x}+5)} = \frac{\mathrm{rx}+\mathrm{t}}{(\mathrm{x}-2)(\mathrm{x}+5)}\)
• Since denominators are identical, numerators must be equal: \(5\mathrm{x} + 4 = \mathrm{rx} + \mathrm{t}\)
• Comparing coefficients:
  • Coefficient of x: \(\mathrm{r} = 5\)
  • Constant term: \(\mathrm{t} = 4\)
• Calculate: \(\mathrm{rt} = 5 \times 4 = 20\)

Answer: C. 20

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make arithmetic errors when expanding the numerators or combining like terms.

For example, they might incorrectly expand \(2(\mathrm{x}+5)\) as \(2\mathrm{x} + 5\) instead of \(2\mathrm{x} + 10\), or calculate \(3(\mathrm{x}-2)\) as \(3\mathrm{x} + 6\) instead of \(3\mathrm{x} - 6\). These errors lead to wrong coefficients when comparing with \(\mathrm{rx} + \mathrm{t}\). A student who gets \(\mathrm{r} = 3\) and \(\mathrm{t} = 8\) would calculate \(\mathrm{rt} = 24\), leading them to guess or select an incorrect answer.

Second Most Common Error:

Poor INFER reasoning about common denominators: Students attempt to add the fractions without recognizing they need a common denominator first.

They might try to directly add numerators and denominators separately, getting something like \(\frac{2+3}{\mathrm{x}-2+\mathrm{x}+5} = \frac{5}{2\mathrm{x}+3}\), which doesn't match the required form. This leads to confusion about how to proceed and often results in random answer selection.

The Bottom Line:

This problem tests whether students can systematically combine rational expressions - a skill that requires both strategic thinking about common denominators and careful algebraic manipulation. Success depends on methodical execution rather than shortcuts.