\(3(2\mathrm{x} - 6) - 11 = 4(\mathrm{x} - 3) + 6\) If x is the solution to the equation above,...

1. TRANSLATE the problem information

• Given equation: \(\mathrm{3(2x - 6) - 11 = 4(x - 3) + 6}\)
• Find: the value of \(\mathrm{x - 3}\) (not x itself)

2. INFER the strategic approach

• Since we need \(\mathrm{x - 3}\), look for patterns involving \(\mathrm{(x - 3)}\)
• Notice that \(\mathrm{2x - 6}\) can be factored as \(\mathrm{2(x - 3)}\)
• This means we can work with \(\mathrm{(x - 3)}\) as a single unit instead of solving for x first

3. SIMPLIFY by substituting the factored form

• Rewrite \(\mathrm{2x - 6}\) as \(\mathrm{2(x - 3)}\):
  \(\mathrm{3[2(x - 3)] - 11 = 4(x - 3) + 6}\)
• SIMPLIFY the left side:
  \(\mathrm{6(x - 3) - 11 = 4(x - 3) + 6}\)

4. SIMPLIFY by treating \(\mathrm{(x - 3)}\) as a single variable

• Let \(\mathrm{y = (x - 3)}\), so our equation becomes:
  \(\mathrm{6y - 11 = 4y + 6}\)
• SIMPLIFY by collecting like terms:
  \(\mathrm{6y - 4y = 6 + 11}\)
  \(\mathrm{2y = 17}\)
  \(\mathrm{y = \frac{17}{2}}\)

5. INFER the final answer

• Since \(\mathrm{y = (x - 3)}\), we have \(\mathrm{x - 3 = \frac{17}{2}}\)

Answer: B. \(\mathrm{\frac{17}{2}}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize the strategic opportunity to work with \(\mathrm{(x - 3)}\) as a unit. Instead, they immediately distribute everything and solve for x first.

They would get: \(\mathrm{6x - 18 - 11 = 4x - 12 + 6}\), leading to \(\mathrm{6x - 29 = 4x - 6}\), then \(\mathrm{2x = 23}\), so \(\mathrm{x = \frac{23}{2}}\). Then they calculate \(\mathrm{x - 3 = \frac{23}{2} - 3 = \frac{23}{2} - \frac{6}{2} = \frac{17}{2}}\). While this reaches the correct answer, it's more work and creates more opportunities for arithmetic errors.

However, some students stop at \(\mathrm{x = \frac{23}{2}}\) and forget the problem asks for \(\mathrm{x - 3}\), leading them to select Choice A (\(\mathrm{\frac{23}{2}}\)).

Second Most Common Error:

Poor SIMPLIFY execution: Students make arithmetic errors during the algebraic manipulations, particularly when combining constants or handling signs.

For example, they might incorrectly combine \(\mathrm{-18 - 11}\) as \(\mathrm{-28}\) instead of \(\mathrm{-29}\), or make sign errors when moving terms across the equals sign. This leads to wrong values for x, causing them to get stuck and guess among the remaining choices.

The Bottom Line:

This problem rewards strategic thinking - recognizing that factoring \(\mathrm{2x - 6}\) as \(\mathrm{2(x - 3)}\) creates a more direct path to the answer. Students who jump straight to distribution miss this insight and either work harder than necessary or forget what the question actually asks for.