For all real numbers x where x neq -4 and x neq 1/6, the expression 6/(6x - 1) - 1/(x...

1. INFER the approach needed

• Given: \(\frac{6}{6x - 1} - \frac{1}{x + 4}\) needs to be written as \(\frac{k}{(x + 4)(6x - 1)}\)
• Key insight: To combine fractions, we need a common denominator
• The target form tells us the common denominator should be \((x + 4)(6x - 1)\)

2. SIMPLIFY by converting to common denominator

• First fraction: \(\frac{6}{6x - 1} = \frac{6(x + 4)}{(6x - 1)(x + 4)}\)
• Second fraction: \(\frac{1}{x + 4} = \frac{1(6x - 1)}{(x + 4)(6x - 1)}\)

3. SIMPLIFY by combining the fractions

• Combined expression: \(\frac{6(x + 4) - 1(6x - 1)}{(x + 4)(6x - 1)}\)
• Focus on the numerator: \(6(x + 4) - 1(6x - 1)\)

4. SIMPLIFY the numerator

• Expand: \(6(x + 4) = 6x + 24\)
• Expand: \(-1(6x - 1) = -6x + 1\)
• Combine: \(6x + 24 - 6x + 1 = 25\)

5. INFER the final answer

• The expression equals \(\frac{25}{(x + 4)(6x - 1)}\)
• Therefore, \(k = 25\)

Answer: D (25)

Why Students Usually Falter on This Problem

Most Common Error Path:

Poor SIMPLIFY execution: Students make sign errors when handling the subtraction of the second fraction. They might incorrectly write the numerator as \(6(x + 4) - (6x - 1)\) instead of \(6(x + 4) - 1(6x - 1)\), forgetting that the entire second fraction is being subtracted. This leads to \(6x + 24 - 6x + 1 = 25\) becoming \(6x + 24 - 6x - 1 = 23\).

This may lead them to select Choice B (23).

Second Most Common Error:

Weak INFER skill: Students don't recognize that they need to find a common denominator or they choose the wrong common denominator. They might try to subtract the fractions incorrectly by working with the denominators separately, leading to confused calculations and abandoning the systematic approach.

This leads to confusion and guessing among the remaining answer choices.

The Bottom Line:

This problem tests whether students can systematically combine algebraic fractions while maintaining careful attention to signs and algebraic manipulation. The key challenge is recognizing the strategy (common denominators) and executing it without sign errors.