For a 3-week period in a town in Illinois, the lowest recorded temperature was 31°F and the highest recorded temperature...

1. TRANSLATE the problem information

• Given information:
  • Lowest recorded temperature: \(\mathrm{31°F}\)
  • Highest recorded temperature: \(\mathrm{67°F}\)
  • Need inequality for ANY recorded temperature \(\mathrm{t}\)
• This tells us we have established minimum and maximum bounds for the temperature range.

2. INFER what "any recorded temperature" means

• If \(\mathrm{31°F}\) was the LOWEST temperature, then no temperature could be below \(\mathrm{31°F}\)
• If \(\mathrm{67°F}\) was the HIGHEST temperature, then no temperature could be above \(\mathrm{67°F}\)
• Therefore, ANY temperature \(\mathrm{t}\) must be between these values, including the endpoints

3. TRANSLATE this reasoning into mathematical notation

• Temperature must be at least \(\mathrm{31°F}\): \(\mathrm{t \geq 31}\)
• Temperature must be at most \(\mathrm{67°F}\): \(\mathrm{t \leq 67}\)
• Combining both constraints: \(\mathrm{31 \leq t \leq 67}\)

4. APPLY CONSTRAINTS to eliminate incorrect choices

• Choice A (\(\mathrm{t \geq 98}\)): Impossible since maximum was only \(\mathrm{67°F}\)
• Choice B (\(\mathrm{t \geq 67}\)): Misses temperatures between \(\mathrm{31°F}\) and \(\mathrm{67°F}\)
• Choice D (\(\mathrm{t \leq 31}\)): Misses temperatures between \(\mathrm{31°F}\) and \(\mathrm{67°F}\)
• Choice C (\(\mathrm{31 \leq t \leq 67}\)): Captures the complete range

Answer: C. \(\mathrm{31 \leq t \leq 67}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students focus on only one boundary instead of recognizing that "any recorded temperature" requires BOTH minimum and maximum constraints.

For example, they might think "the highest was \(\mathrm{67°F}\), so \(\mathrm{t \leq 67}\)" or "the lowest was \(\mathrm{31°F}\), so \(\mathrm{t \geq 31}\)" but fail to combine both conditions. This partial reasoning may lead them to select Choice B (\(\mathrm{t \geq 67}\)) or Choice D (\(\mathrm{t \leq 31}\)).

Second Most Common Error:

Poor TRANSLATE reasoning: Students misinterpret what the temperature bounds mean in context.

They might think the inequality should describe temperatures OUTSIDE the recorded range, or confuse which direction the inequality symbols should face. This confusion about the relationship between recorded extremes and possible values leads to selecting Choice A (\(\mathrm{t \geq 98}\)) or guessing randomly.

The Bottom Line:

This problem tests whether students understand that a range is defined by BOTH its minimum and maximum values, and that any value within that range must satisfy both boundary conditions simultaneously. The key insight is translating "between the lowest and highest" into a compound inequality.