A car-share service records the duration t, in minutes, of each rental trip. During a promotional week, the duration of...

1. TRANSLATE the problem conditions

• Given information:
  • Trip durations were "at least 12 minutes"
  • Trip durations were "at most 45 minutes"

• What this tells us:
  • "At least 12" means \(\mathrm{t \geq 12}\)
  • "At most 45" means \(\mathrm{t \leq 45}\)

2. INFER how to combine the conditions

• Both conditions must be true for every trip
• We need a compound inequality that shows t satisfies both constraints simultaneously
• This gives us: \(\mathrm{12 \leq t \leq 45}\)

3. APPLY CONSTRAINTS to verify the answer

• Check that our inequality includes both endpoints (12 and 45 are allowed)
• Verify it excludes values outside the range

Answer: C (\(\mathrm{12 \leq t \leq 45}\))

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students confuse the direction of inequality symbols, particularly with "at least" and "at most." They might think "at least 12" means \(\mathrm{t \leq 12}\) because they focus on the number 12 rather than understanding that "at least" means "12 or more."

This may lead them to select Choice A (\(\mathrm{t \leq 12}\)) by incorrectly translating "at least 12."

Second Most Common Error:

Poor TRANSLATE reasoning: Students correctly identify the inequality directions but use strict inequalities (< and >) instead of inclusive inequalities (≤ and ≥). They don't recognize that "at least" and "at most" include the boundary values.

This may lead them to select Choice B (\(\mathrm{12 \lt t \lt 45}\)) by using strict inequalities when inclusive ones are needed.

The Bottom Line:

This problem tests precise translation of common English phrases into mathematical notation. The key is remembering that "at least" means "greater than or equal to" and "at most" means "less than or equal to," and both conditions must be satisfied together.