Question:The minimum value of x is 12 more than one-sixth of a number n. Which inequality shows the possible values...

1. TRANSLATE the problem information

• Given information:
  • "The minimum value of x is 12 more than one-sixth of a number n"
• What this tells us:
  • We need to convert this English phrase into mathematical notation
  • "One-sixth of a number n" = \(\frac{\mathrm{n}}{6}\)
  • "12 more than" means we add 12
  • So "12 more than one-sixth of n" = \(\frac{\mathrm{n}}{6} + 12\)

2. INFER what "minimum value" means

• Key insight: If something is the minimum value of x, then x can be equal to that value OR greater than that value
• This means we need the "greater than or equal to" symbol: ≥
• Therefore: \(\mathrm{x} \geq \frac{\mathrm{n}}{6} + 12\)

3. Match with answer choices

Looking at the options, our inequality \(\mathrm{x} \geq \frac{\mathrm{n}}{6} + 12\) matches choice (B).

Answer: B

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Confusing "minimum" with "maximum"

Students often think that if something is the "minimum value," then x must be less than or equal to that value. This incorrect reasoning leads them to choose \(\mathrm{x \leq \frac{n}{6} + 12}\) instead of \(\mathrm{x \geq \frac{n}{6} + 12}\).

This may lead them to select Choice A (\(\mathrm{x \leq \frac{n}{6} + 12}\))

Second Most Common Error:

Poor TRANSLATE execution: Misinterpreting "12 more than"

Some students get confused about the order of operations in the phrase "12 more than one-sixth of n" and incorrectly translate it as \(\frac{\mathrm{n}}{6} - 12\) instead of \(\frac{\mathrm{n}}{6} + 12\).

This may lead them to select Choice D (\(\mathrm{x \geq \frac{n}{6} - 12}\))

The Bottom Line:

This problem tests whether you can accurately translate English to math notation AND understand the logical meaning of "minimum value." The key breakthrough is recognizing that a minimum value establishes a lower bound, meaning the variable must be greater than or equal to that bound.