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, or 144

2. INFER the most efficient approach

• Key insight: Notice that \(6\mathrm{x} + 6 = 6(\mathrm{x} + 1)\)
• This means we can work directly with \((\mathrm{x} + 1)\) rather than solving for x first
• Since we know \(3(\mathrm{x} + 1) = 24\), we can find \(6(\mathrm{x} + 1)\) without finding x individually

3. SIMPLIFY to find (x + 1)

• From \(3(\mathrm{x} + 1) = 24\)
• Divide both sides by 3: \((\mathrm{x} + 1) = 8\)

4. SIMPLIFY to find the target expression

• Since \(6\mathrm{x} + 6 = 6(\mathrm{x} + 1)\), and \((\mathrm{x} + 1) = 8\)
• Therefore: \(6(\mathrm{x} + 1) = 6(8) = 48\)

Answer: B (48)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize the factoring pattern \(6\mathrm{x} + 6 = 6(\mathrm{x} + 1)\) and instead use the longer method of solving for x first, creating more opportunities for arithmetic errors.

When solving \(3(\mathrm{x} + 1) = 24\)
\(\mathrm{x} + 1 = 8\)
\(\mathrm{x} = 7\)
then calculating \(6\mathrm{x} + 6 = 6(7) + 6\), students often make calculation errors like \(6(7) = 36\) instead of 42, or they calculate \(6(7) = 42\) correctly but forget to add 6, giving them just 42.

This may lead them to select Choice A (42).

Second Most Common Error:

Poor SIMPLIFY execution: Students solve for x correctly \((\mathrm{x} = 7)\) but make a basic arithmetic error when substituting, such as calculating \(6 \times 7 = 36\) instead of 42.

Following this error: \(6\mathrm{x} + 6 = 36 + 6 = 42\), leading them to the wrong answer.

This may lead them to select Choice A (42).

The Bottom Line:

This problem rewards students who can recognize algebraic patterns and factor expressions strategically. The shortcut method using \(6(\mathrm{x} + 1) = 2 \times 3(\mathrm{x} + 1)\) eliminates multiple calculation steps and reduces error opportunities.