\(\frac{1}{3}(\mathrm{x} + 6) - \frac{1}{2}(\mathrm{x} + 6) = -8\) What value of x is the solution to the given equation?...

1. INFER the most efficient approach

• Looking at \(\frac{1}{3}(x + 6) - \frac{1}{2}(x + 6) = -8\), notice that both terms contain \((x + 6)\)
• Instead of distributing first, we can factor out the common term \((x + 6)\)
• This will make the problem much simpler to solve

2. SIMPLIFY by factoring out the common term

• Factor out \((x + 6)\): \((x + 6)(\frac{1}{3} - \frac{1}{2}) = -8\)
• Now we need to calculate the coefficient: \(\frac{1}{3} - \frac{1}{2}\)

3. SIMPLIFY the fraction arithmetic

• Find common denominator: \(\frac{1}{3} - \frac{1}{2} = \frac{2}{6} - \frac{3}{6} = -\frac{1}{6}\)
• Our equation becomes: \((x + 6)(-\frac{1}{6}) = -8\)

4. SIMPLIFY to isolate (x + 6)

• Multiply both sides by -6: \((x + 6) = -8 × (-6) = 48\)
• So: \(x + 6 = 48\)

5. SIMPLIFY to find x

• Subtract 6 from both sides: \(x = 48 - 6 = 42\)

Answer: 42

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Not recognizing the factoring opportunity and instead distributing each fraction first.

Students might distribute to get: \(\frac{1}{3}x + 2 - \frac{1}{2}x - 3 = -8\), then combine like terms to get \(-\frac{1}{6}x - 1 = -8\). While this approach works, it's more complex and creates more opportunities for arithmetic errors.

This path doesn't lead to a wrong answer if executed correctly, but increases the chance of calculation mistakes.

Second Most Common Error:

Poor SIMPLIFY execution: Making arithmetic errors when calculating \(\frac{1}{3} - \frac{1}{2}\).

Students might incorrectly compute \(\frac{1}{3} - \frac{1}{2}\) as \(\frac{1}{6}\) (forgetting the negative sign) or get confused with the fraction arithmetic. With the wrong coefficient, they would get \((x + 6)(\frac{1}{6}) = -8\), leading to \(x + 6 = -48\), and finally \(x = -54\) instead of the correct answer.

The Bottom Line:

This problem rewards students who can recognize patterns and factor strategically. The key insight is seeing \((x + 6)\) as a common factor rather than rushing to distribute.