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

1. INFER the approach needed

• Given: \(\frac{3}{3\mathrm{x} - 1} - \frac{1}{\mathrm{x} + 2}\)
• Key insight: To subtract rational expressions, we need a common denominator
• The common denominator will be the product of both denominators: \((3\mathrm{x} - 1)(\mathrm{x} + 2)\)

2. SIMPLIFY by converting to common denominators

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

3. SIMPLIFY by subtracting the numerators

• Combine: \(\frac{3(\mathrm{x} + 2) - 1(3\mathrm{x} - 1)}{(3\mathrm{x} - 1)(\mathrm{x} + 2)}\)
• Expand the numerator:
  • \(3(\mathrm{x} + 2) = 3\mathrm{x} + 6\)
  • \(1(3\mathrm{x} - 1) = 3\mathrm{x} - 1\)
• Substitute: \(\frac{3\mathrm{x} + 6 - (3\mathrm{x} - 1)}{(3\mathrm{x} - 1)(\mathrm{x} + 2)}\)

4. SIMPLIFY the final expression

• Distribute the negative sign: \(3\mathrm{x} + 6 - 3\mathrm{x} + 1\)
• Combine like terms: \(3\mathrm{x} - 3\mathrm{x} + 6 + 1 = 7\)
• Final result: \(\frac{7}{(3\mathrm{x} - 1)(\mathrm{x} + 2)}\)

Answer: C. \(\frac{7}{(\mathrm{x} + 2)(3\mathrm{x} - 1)}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Students make sign errors when subtracting the second numerator. They may write \([3\mathrm{x} + 6 - 3\mathrm{x} - 1]\) instead of \([3\mathrm{x} + 6 - 3\mathrm{x} + 1]\), forgetting that subtracting \((3\mathrm{x} - 1)\) means subtracting both terms, changing the -1 to +1.

This leads to a final numerator of 5 instead of 7, causing them to select Choice B (\(\frac{5}{(\mathrm{x} + 2)(3\mathrm{x} - 1)}\)).

Second Most Common Error:

Poor INFER reasoning: Students attempt to subtract the fractions without finding a common denominator first, trying to subtract numerators and denominators separately: \(\frac{3}{3\mathrm{x} - 1} - \frac{1}{\mathrm{x} + 2} = \frac{3-1}{(3\mathrm{x}-1)-(\mathrm{x}+2)}\).

This leads to confusion and an expression that doesn't match any of the given choices, causing them to get stuck and guess randomly.

The Bottom Line:

This problem tests your ability to systematically handle rational expression operations. Success depends on recognizing the need for a common denominator and carefully managing signs during the subtraction process.