The minimum value of x is 12 less than 6 times another number n. Which inequality shows the possible values...

1. TRANSLATE the problem information

• Given information:
  • The minimum value of \(\mathrm{x}\) is 12 less than 6 times another number \(\mathrm{n}\)

• What this tells us:
  • We need to express "6 times n" as \(\mathrm{6n}\)
  • We need to express "12 less than 6n" as \(\mathrm{6n - 12}\)
  • This value \(\mathrm{(6n - 12)}\) represents the smallest possible value for \(\mathrm{x}\)

2. INFER what minimum value means for inequalities

• Key insight: If \(\mathrm{6n - 12}\) is the minimum value of \(\mathrm{x}\), then:
  • \(\mathrm{x}\) can equal \(\mathrm{6n - 12}\) (\(\mathrm{x}\) reaches its minimum)
  • \(\mathrm{x}\) can be greater than \(\mathrm{6n - 12}\) (\(\mathrm{x}\) exceeds its minimum)
  • \(\mathrm{x}\) cannot be less than \(\mathrm{6n - 12}\) (this would violate the minimum)

• Therefore: \(\mathrm{x \geq 6n - 12}\)

3. Match with answer choices

Looking at the options, choice B shows \(\mathrm{x \geq 6n - 12}\), which matches our reasoning.

Answer: B

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students incorrectly translate "12 less than 6n" as "\(\mathrm{12 - 6n}\)" instead of "\(\mathrm{6n - 12}\)"

The phrase "12 less than [something]" means "[something] - 12", not "12 - [something]". This translation error leads students to think the minimum value is \(\mathrm{12 - 6n}\).

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

Second Most Common Error:

Poor INFER reasoning: Students correctly translate to get \(\mathrm{6n - 12}\) but misunderstand what "minimum value" means for inequalities

They might think: "If \(\mathrm{6n - 12}\) is the minimum, then \(\mathrm{x}\) must be less than or equal to this value" – confusing minimum with maximum.

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

The Bottom Line:

This problem tests two critical skills working together: precise language translation and understanding the logical relationship between minimum values and inequality directions. Success requires both accurate translation and conceptual clarity about what "minimum" means in mathematical contexts.