If 4x + 2 = 12, what is the value of 16x + 8?

1. TRANSLATE the problem information

• Given: \(4x + 2 = 12\)
• Find: \(16x + 8\)

2. INFER the strategic approach

• Look for a relationship between what we know \((4x + 2)\) and what we want \((16x + 8)\)
• Notice that \(16x + 8\) can be written as \(4(4x + 2)\)
• This means we can substitute directly without solving for x!

3. SIMPLIFY using the pattern

• Since \(16x + 8 = 4(4x + 2)\) and we know \(4x + 2 = 12\):
• \(16x + 8 = 4(12) = 48\)

Answer: B. 48

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize the factoring pattern and instead solve the long way by finding x first.

They solve \(4x + 2 = 12\) to get \(x = 2.5\), then calculate \(16x + 8 = 16(2.5) + 8\). While this approach works, it creates more opportunities for arithmetic errors during the multiplication and addition steps. Some students make calculation mistakes in this process, potentially leading to incorrect answer choices.

Second Most Common Error:

Inadequate SIMPLIFY execution: Students recognize they need to solve \(4x + 2 = 12\) but make algebraic errors in the process.

For example, they might forget to subtract 2 from both sides and think \(4x = 12\), leading to \(x = 3\). Then \(16x + 8 = 16(3) + 8 = 56\). This may lead them to select Choice C (56).

The Bottom Line:

This problem rewards pattern recognition over brute-force calculation. The key insight is seeing that the expression we want is exactly 4 times the expression we're given, making the solution immediate once this connection is made.