Which expression is equivalent to 3/(x - 2) + 2/(3x + 1) for x neq 2 and x neq -1/3?

1. INFER the approach needed

• We need to add two rational expressions with different denominators
• To add fractions, we need a common denominator
• Since \((\mathrm{x} - 2)\) and \((3\mathrm{x} + 1)\) are distinct linear expressions, their least common denominator is their product: \((\mathrm{x} - 2)(3\mathrm{x} + 1)\)

2. SIMPLIFY by converting each fraction to the common denominator

• First fraction: \(\frac{3}{\mathrm{x} - 2} = \frac{3(3\mathrm{x} + 1)}{(\mathrm{x} - 2)(3\mathrm{x} + 1)}\)
• Second fraction: \(\frac{2}{3\mathrm{x} + 1} = \frac{2(\mathrm{x} - 2)}{(\mathrm{x} - 2)(3\mathrm{x} + 1)}\)

3. SIMPLIFY by expanding the numerators

• First numerator: \(3(3\mathrm{x} + 1) = 9\mathrm{x} + 3\)
• Second numerator: \(2(\mathrm{x} - 2) = 2\mathrm{x} - 4\)
• Combined expression: \(\frac{9\mathrm{x} + 3 + 2\mathrm{x} - 4}{(\mathrm{x} - 2)(3\mathrm{x} + 1)}\)

4. SIMPLIFY by combining like terms in the numerator

• Combine x terms: \(9\mathrm{x} + 2\mathrm{x} = 11\mathrm{x}\)
• Combine constants: \(3 + (-4) = -1\)
• Final result: \(\frac{11\mathrm{x} - 1}{(\mathrm{x} - 2)(3\mathrm{x} + 1)}\)

Answer: C) \(\frac{11\mathrm{x} - 1}{(\mathrm{x} - 2)(3\mathrm{x} + 1)}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make arithmetic errors when distributing or combining like terms in the numerator.

For example, they might incorrectly calculate \(3(3\mathrm{x} + 1)\) as \(9\mathrm{x} + 1\) instead of \(9\mathrm{x} + 3\), or make sign errors when combining \(3 + (-4)\). These calculation mistakes lead to wrong coefficients in the final numerator.

This may lead them to select Choice B \((9\mathrm{x} - 4)\) if they make multiple arithmetic errors, or get confused and guess among the remaining choices.

Second Most Common Error:

Poor INFER reasoning: Students attempt to add the fractions by simply adding numerators and denominators separately, not recognizing they need a common denominator.

They might try \(\frac{3}{\mathrm{x}-2} + \frac{2}{3\mathrm{x}+1} = \frac{3+2}{(\mathrm{x}-2)+(3\mathrm{x}+1)} = \frac{5}{4\mathrm{x}-1}\), completely bypassing the proper method for adding rational expressions.

This leads them to select Choice A \((\frac{5}{4\mathrm{x}-1})\).

The Bottom Line:

This problem tests whether students understand the fundamental principle of rational expression addition and can execute multi-step algebraic manipulation accurately. The arithmetic complexity creates multiple opportunities for errors even when the conceptual approach is correct.