During the afternoon, the temperature inside a greenhouse stayed between 68 degrees Fahrenheit and 72 degrees Fahrenheit, inclusive. Which inequality...

1. TRANSLATE the problem information

• Given information:
  • Temperature stayed "between \(\mathrm{68°F}\) and \(\mathrm{72°F}\), inclusive"
  • Need to find which inequality represents this situation
• The word "inclusive" tells us that \(\mathrm{68°F}\) and \(\mathrm{72°F}\) are both allowed temperatures

2. INFER what "inclusive" means mathematically

• "Inclusive" means the endpoints are part of the solution
• This requires \(\leq\) symbols (not \(\lt\) symbols) to include 68 and 72
• The temperature can be exactly 68, exactly 72, or anything in between

3. TRANSLATE the range into inequality notation

• "Between 68 and 72, inclusive" becomes: \(\mathrm{68 \leq T \leq 72}\)
• This compound inequality captures both requirements:
  • Temperature must be at least 68: \(\mathrm{T \geq 68}\)
  • Temperature must be at most 72: \(\mathrm{T \leq 72}\)

4. APPLY CONSTRAINTS to evaluate each choice

• Check each option against our range requirement:
  • Choice A (\(\mathrm{T \lt 68}\)): Says temperature is below 68 - wrong direction entirely
  • Choice B (\(\mathrm{T \leq 68}\)): Only covers temperatures up to 68, ignores 69-72 range
  • Choice C (\(\mathrm{T \leq 72}\)): Only covers temperatures up to 72, ignores the minimum of 68
  • Choice D (\(\mathrm{68 \leq T \leq 72}\)): Captures the complete range requirement

Answer: D

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students misinterpret "between X and Y" as meaning only the values strictly between the endpoints, not including X and Y themselves.

They might think "between 68 and 72" means temperatures like 69, 70, 71 but not exactly 68 or 72. This leads them to look for inequalities with \(\lt\) symbols instead of \(\leq\) symbols, or they focus on just one boundary. This may lead them to select Choice A (\(\mathrm{T \lt 68}\)) if they completely misunderstand the range, or get confused and guess among the partial constraints.

Second Most Common Error:

Inadequate APPLY CONSTRAINTS reasoning: Students correctly understand the individual parts but fail to recognize that both the lower AND upper bounds must be satisfied simultaneously.

They might select Choice B (\(\mathrm{T \leq 68}\)) or Choice C (\(\mathrm{T \leq 72}\)) because each captures part of the requirement, not realizing that the complete solution needs both constraints working together in a compound inequality.

The Bottom Line:

This problem tests whether students can accurately translate everyday language about ranges into precise mathematical notation, particularly understanding what "inclusive" means and how to express it with the correct inequality symbols.