Question:If \(3(\mathrm{x} + 1) = 24\), what is the value of 6x + 6?424854144

1. TRANSLATE the problem information

• Given: \(3(\mathrm{x} + 1) = 24\)
• Find: The value of \(6\mathrm{x} + 6\)
• Available answer choices: 42, 48, 54, 144

2. INFER the most efficient approach

There are actually two smart ways to tackle this problem:

• Standard approach: Solve for x first, then substitute
• Pattern recognition: Notice the relationship between the expressions

Let's explore both methods so you can choose your preferred strategy.

3. SIMPLIFY using Method 1 (Standard approach)

• Start with: \(3(\mathrm{x} + 1) = 24\)
• Divide both sides by 3: \(\mathrm{x} + 1 = 8\)
• Subtract 1 from both sides: \(\mathrm{x} = 7\)
• Now substitute \(\mathrm{x} = 7\) into \(6\mathrm{x} + 6\):
  \(6(7) + 6 = 42 + 6 = 48\)

4. INFER the pattern for Method 2 (Shortcut)

• Notice that \(6\mathrm{x} + 6\) can be factored: \(6\mathrm{x} + 6 = 6(\mathrm{x} + 1)\)
• We already know that \(3(\mathrm{x} + 1) = 24\)
• To get \(6(\mathrm{x} + 1)\), we need to double \(3(\mathrm{x} + 1)\)
• Therefore: \(6(\mathrm{x} + 1) = 2 \times 3(\mathrm{x} + 1) = 2 \times 24 = 48\)

Answer: B (48)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER reasoning: Students solve correctly for \(\mathrm{x} = 7\), but then substitute into the wrong expression or make calculation errors.

For example, they might calculate \(6\mathrm{x}\) instead of \(6\mathrm{x} + 6\), getting \(6(7) = 42\), and select Choice A (42). Or they might incorrectly substitute and calculate something like \(6(7 + 1) = 48\) but then add 6 again, getting 54.

Second Most Common Error:

Missing the pattern connection: Students don't recognize that \(6\mathrm{x} + 6 = 6(\mathrm{x} + 1)\) and miss the elegant shortcut completely.

This leads to getting overwhelmed by what seems like a complex multi-step problem, causing confusion and potentially guessing among the answer choices.

The Bottom Line:

This problem tests whether students can either execute a straightforward substitution accurately OR recognize algebraic patterns that lead to more efficient solutions. The key insight is seeing the structural relationship between the given equation and the target expression.