If 1 - 3/(x - 3) = (-4)/(2x - 6), what is the solution to the equation?

1. TRANSLATE the problem information

• Given equation: \(1 - \frac{3}{\mathrm{x} - 3} = \frac{-4}{2\mathrm{x} - 6}\)
• Need to find the value of x that makes this equation true

2. INFER the solution strategy

• Notice that the right side denominator \(2\mathrm{x} - 6\) can be factored as \(2(\mathrm{x} - 3)\)
• This means both fractions will have denominators involving \((\mathrm{x} - 3)\)
• Strategy: Use the LCD method to eliminate fractions, then solve the resulting linear equation

3. SIMPLIFY by rewriting with factored denominator

• Rewrite: \(1 - \frac{3}{\mathrm{x} - 3} = \frac{-4}{2(\mathrm{x} - 3)}\)
• Now we can see the LCD clearly: \(2(\mathrm{x} - 3)\)

4. SIMPLIFY by eliminating fractions

• Multiply every term by the LCD: \(2(\mathrm{x} - 3)\)
• \(2(\mathrm{x} - 3) \times 1 - 2(\mathrm{x} - 3) \times \frac{3}{\mathrm{x} - 3} = 2(\mathrm{x} - 3) \times \frac{-4}{2(\mathrm{x} - 3)}\)
• Cancel common factors: \(2(\mathrm{x} - 3) - 2 \times 3 = -4\)

5. SIMPLIFY the resulting linear equation

• \(2\mathrm{x} - 6 - 6 = -4\)
• \(2\mathrm{x} - 12 = -4\)
• \(2\mathrm{x} = 8\)
• \(\mathrm{x} = 4\)

Answer: B. 4

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize that \(2\mathrm{x} - 6 = 2(\mathrm{x} - 3)\), making it difficult to find a common approach to eliminate fractions. Without this key insight, they may attempt to cross-multiply incorrectly or get stuck trying to combine fractions with different denominators.

This leads to confusion and guessing among the answer choices.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up the LCD method but make arithmetic errors when:

• Distributing the \(2(\mathrm{x} - 3)\) across terms
• Combining like terms (\(-6 - 6 = -12\))
• Solving the final equation (\(2\mathrm{x} - 12 = -4\) becomes \(2\mathrm{x} = 8\))

Common arithmetic mistakes include:

• Getting \(2\mathrm{x} = 4\) instead of \(2\mathrm{x} = 8\) → leads to Choice A (2)
• Writing \(2\mathrm{x} - 12 = 4\) instead of \(2\mathrm{x} - 12 = -4\) → leads to Choice D (8)
• Making errors in the final steps → leads to Choice C (5)

The Bottom Line:

Success on this problem requires recognizing the factoring opportunity that creates a path to eliminate fractions, combined with careful arithmetic execution throughout multiple algebraic steps.