If 3x + 2 = 8, what is thevalue of 9x + 6?

1. INFER the solution strategy

Given: \(\mathrm{3x + 2 = 8}\), find \(\mathrm{9x + 6}\)

You have two strategic options:

• Option A: Solve for x first, then substitute into \(\mathrm{9x + 6}\)
• Option B: Notice that \(\mathrm{9x + 6}\) looks like it might relate to \(\mathrm{3x + 2}\)

Let's explore both approaches:

2. SIMPLIFY using Option A (Direct method)

• Solve \(\mathrm{3x + 2 = 8}\) for x:
  • Subtract 2 from both sides: \(\mathrm{3x = 6}\)
  • Divide both sides by 3: \(\mathrm{x = 2}\)

• Substitute \(\mathrm{x = 2}\) into \(\mathrm{9x + 6}\):
  \(\mathrm{9(2) + 6 = 18 + 6 = 24}\)

3. SIMPLIFY using Option B (Pattern recognition method)

• INFER the relationship: Notice that \(\mathrm{9x + 6 = 3(3x + 2)}\)
• Since \(\mathrm{3x + 2 = 8}\), we have: \(\mathrm{9x + 6 = 3(8) = 24}\)

Answer: 24

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students solve for x correctly (\(\mathrm{x = 2}\)) but make arithmetic errors when calculating \(\mathrm{9x + 6}\).

For example, they might compute \(\mathrm{9×2 + 6 = 18 + 6 = 23}\) instead of 24, or calculate \(\mathrm{9×2 = 16}\) instead of 18. These careless arithmetic mistakes lead to incorrect final answers even when the approach is sound.

Second Most Common Error:

Poor INFER reasoning: Students don't recognize either solution strategy and get overwhelmed by seeing a different expression (\(\mathrm{9x + 6}\)) than what they solved for.

They might think they need additional information or get confused about how to use the equation \(\mathrm{3x + 2 = 8}\) to find something that looks completely different. This leads to confusion and guessing.

The Bottom Line:

This problem tests whether students can connect given information to what they're asked to find. The key insight is recognizing that you can either solve systematically for the variable first, or identify patterns that allow for more direct calculation.