During a laboratory experiment, the temperature was maintained between 68 degrees Fahrenheit and 78 degrees Fahrenheit. Which inequality best represen...

1. TRANSLATE the problem information

• Given information:
  • Temperature maintained "between 68°F and 78°F"
  • Need to represent this as inequality with variable \(\mathrm{t}\)

• What this tells us: The word "between" indicates a range with both minimum and maximum values

2. INFER what "between" means mathematically

• "Between 68 and 78" means:
  • Temperature cannot go below 68°F: \(\mathrm{t \geq 68}\)
  • Temperature cannot go above 78°F: \(\mathrm{t \leq 78}\)
  • Both conditions must be true at the same time

• This requires a compound inequality that shows both bounds

3. TRANSLATE into mathematical notation

• Combine both conditions: \(\mathrm{68 \leq t \leq 78}\)
• This reads as "68 is less than or equal to t, and t is less than or equal to 78"

4. APPLY CONSTRAINTS to eliminate wrong choices

• Check each option:
  • (A) \(\mathrm{t \leq 10}\): Too low, doesn't match the problem
  • (B) \(\mathrm{t \leq 68}\): Only shows one bound, missing the "at least 68" requirement
  • (C) \(\mathrm{t \leq 78}\): Only shows one bound, missing the "at least 68" requirement
  • (D) \(\mathrm{68 \leq t \leq 78}\): Shows both bounds correctly

Answer: D

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students often misinterpret "between" as meaning only one boundary instead of both boundaries.

They might think "between 68 and 78" means just "less than or equal to 78" or just "greater than or equal to 68," not realizing they need both conditions simultaneously. This incomplete translation leads them to focus on only one part of the range.

This may lead them to select Choice B (\(\mathrm{t \leq 68}\)) or Choice C (\(\mathrm{t \leq 78}\)).

Second Most Common Error:

Poor INFER reasoning: Students may recognize they need both bounds but get confused about which inequality symbols to use or how to write compound inequalities.

They might struggle with whether to use ≤ vs < or how to properly structure the compound inequality notation, leading to uncertainty about the correct mathematical representation.

This leads to confusion and guessing among the remaining choices.

The Bottom Line:

The key challenge is recognizing that "between" in mathematical contexts means you need BOTH a lower bound AND an upper bound, not just one constraint. Students must translate this everyday language into precise mathematical notation that captures both boundaries simultaneously.