Which expression is equivalent to 6/(2x-1) - 3/(x+2)?

1. INFER the solution strategy

• Given: \(\frac{6}{2x-1} - \frac{3}{x+2}\)
• Key insight: To subtract fractions, we need a common denominator
• The common denominator must contain both \((2x-1)\) and \((x+2)\) as factors
• Common denominator = \((2x-1)(x+2)\)

2. SIMPLIFY by rewriting each fraction with the common denominator

• For \(\frac{6}{2x-1}\): multiply top and bottom by \((x+2)\)
  • \(\frac{6}{2x-1} = \frac{6(x+2)}{(2x-1)(x+2)}\)
• For \(\frac{3}{x+2}\): multiply top and bottom by \((2x-1)\)
  • \(\frac{3}{x+2} = \frac{3(2x-1)}{(x+2)(2x-1)}\)

3. SIMPLIFY the subtraction

• Now we have: \(\frac{6(x+2) - 3(2x-1)}{(2x-1)(x+2)}\)
• Expand the numerator:
  • \(6(x+2) = 6x + 12\)
  • \(3(2x-1) = 6x - 3\)
• Subtract: \((6x + 12) - (6x - 3) = 6x + 12 - 6x + 3 = 15\)

4. Write the final answer

• \(\frac{15}{(2x-1)(x+2)} = \frac{15}{(x+2)(2x-1)}\)

Answer: C

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Sign errors when handling the subtraction, particularly with the negative sign in front of \(\frac{3}{x+2}\).

Students often write: \(6(x+2) - 3(2x-1) = 6x + 12 - 6x - 3 = 9\)

This gives them \(\frac{9}{(x+2)(2x-1)}\), which isn't among the choices, leading to confusion and guessing.

Second Most Common Error:

Poor INFER reasoning: Attempting to subtract by finding separate common denominators or trying to subtract numerators and denominators separately.

Some students write: \(\frac{6-3}{(2x-1)+(x+2)} = \frac{3}{3x+1}\), which leads them to select Choice B \(\frac{9}{x+1}\) after making additional arithmetic errors.

The Bottom Line:

This problem tests whether students can systematically apply the common denominator method while carefully tracking signs during polynomial expansion. The key is recognizing that rational expression subtraction follows the same rules as numerical fraction subtraction, just with more complex algebraic manipulation.