For x neq -4, what is the solution to the equation 2 - (x-1)/(x+4) = 3/(x+4)?

1. TRANSLATE the problem information

• Given equation: \(2 - \frac{(\mathrm{x}-1)}{(\mathrm{x}+4)} = \frac{3}{(\mathrm{x}+4)}\)
• Domain restriction: \(\mathrm{x} \neq -4\) (to avoid division by zero)
• Goal: Find the value of x

2. INFER the solution strategy

• Notice both fractions have the same denominator \((\mathrm{x}+4)\)
• Strategy: Multiply both sides by \((\mathrm{x}+4)\) to clear all denominators
• This will convert the rational equation into a simpler linear equation

3. SIMPLIFY by clearing denominators

• Multiply both sides by \((\mathrm{x}+4)\):
  \(2(\mathrm{x}+4) - (\mathrm{x}-1) = 3\)

4. SIMPLIFY through algebraic expansion

• Distribute on the left side:
  \(2(\mathrm{x}+4) = 2\mathrm{x} + 8\)
  \(-(\mathrm{x}-1) = -\mathrm{x} + 1\) (watch the sign!)
• Combined: \(2\mathrm{x} + 8 - \mathrm{x} + 1 = 3\)

5. SIMPLIFY by combining like terms

• Combine x terms: \(2\mathrm{x} - \mathrm{x} = \mathrm{x}\)
• Combine constants: \(8 + 1 = 9\)
• Result: \(\mathrm{x} + 9 = 3\)

6. SIMPLIFY to isolate x

• Subtract 9 from both sides: \(\mathrm{x} = 3 - 9 = -6\)

7. APPLY CONSTRAINTS to verify solution validity

• Check domain: Is \(-6 \neq -4\)? Yes ✓
• The solution is valid

Answer: (A) -6

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Sign error when distributing \(-(\mathrm{x}-1)\)

Students often write \(-(\mathrm{x}-1) = -\mathrm{x} - 1\) instead of \(-\mathrm{x} + 1\), forgetting that the negative sign applies to the entire expression (x-1). This gives them:
\(2\mathrm{x} + 8 - \mathrm{x} - 1 = 3\)
\(\mathrm{x} + 7 = 3\)
\(\mathrm{x} = -4\)

However, \(\mathrm{x} = -4\) violates the domain restriction, which might confuse them further and lead to guessing among the remaining choices.

Second Most Common Error:

Poor INFER reasoning: Not recognizing the denominator-clearing strategy

Some students try to solve by moving terms around without clearing denominators first, leading to increasingly complex fractions. They might attempt to subtract \(\frac{3}{(\mathrm{x}+4)}\) from 2, creating messy work like:
\(2 - \frac{3}{(\mathrm{x}+4)} = \frac{(\mathrm{x}-1)}{(\mathrm{x}+4)}\)

This approach quickly becomes unwieldy and often leads them to abandon systematic solution and guess.

The Bottom Line:

This problem requires careful attention to signs during algebraic manipulation. The key insight is recognizing that multiplying by the common denominator transforms a potentially complex rational equation into a straightforward linear equation.