A number w is at least 3 more than twice the value of z. If the value of z is...

1. TRANSLATE the problem information

• Given information:
  • "w is at least 3 more than twice the value of z"
  • \(\mathrm{z = 5}\)
  • Need to find the smallest possible value of w

• The key phrase "at least" means "greater than or equal to" (\(\geq\))
• "3 more than twice z" means \(\mathrm{2z + 3}\)
• So: \(\mathrm{w \geq 2z + 3}\)

2. SIMPLIFY by substituting the known value

• Replace z with 5 in the inequality:

\(\mathrm{w \geq 2(5) + 3}\)

\(\mathrm{w \geq 10 + 3}\)

\(\mathrm{w \geq 13}\)

3. APPLY CONSTRAINTS to find the answer

• The inequality \(\mathrm{w \geq 13}\) means w can be 13, 14, 15, 16, and so on
• Since we want the smallest possible value, \(\mathrm{w = 13}\)

Answer: 13

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students confuse "at least" with "exactly" or "at most"

Some students might interpret "at least 3 more than twice z" as meaning \(\mathrm{w = 2z + 3}\) (using equality instead of inequality). With \(\mathrm{z = 5}\), they would still get \(\mathrm{w = 13}\), but they'd miss the conceptual understanding that w could be larger. While this leads to the correct numerical answer by coincidence, it shows fundamental misunderstanding of inequality language.

Second Most Common Error:

Incomplete APPLY CONSTRAINTS reasoning: Students solve correctly to \(\mathrm{w \geq 13}\) but don't identify the smallest value

Students might perform all the algebra correctly and arrive at \(\mathrm{w \geq 13}\), but when asked for the "smallest possible value," they might write "\(\mathrm{w \geq 13}\)" as their final answer instead of recognizing that this means \(\mathrm{w = 13}\). This leads to confusion about what the question is actually asking for.

The Bottom Line:

This problem tests whether students can accurately translate inequality language and then extract specific information from the resulting mathematical relationship. The translation step is crucial—misunderstanding "at least" changes the entire problem structure.