To ensure the quality of a chemical reaction, a laboratory's climate control system is designed to maintain a target temperature...

1. TRANSLATE the problem information

• Given information:
  • Target temperature: 22.0°C
  • Maximum deviation: 1.5°C above or below the target
  • Need to find: inequality for acceptable temperatures t

• What this tells us: The actual temperature can be 1.5 degrees higher OR 1.5 degrees lower than 22.0°C

2. INFER the approach

• Since we can go both above AND below the target, we need to find both the minimum and maximum acceptable temperatures
• This will create a range with two boundaries, requiring a compound inequality

3. Calculate the temperature boundaries

• Minimum temperature: \(\mathrm{22.0 - 1.5 = 20.5°C}\)
• Maximum temperature: \(\mathrm{22.0 + 1.5 = 23.5°C}\)

4. APPLY CONSTRAINTS to form the inequality

• The temperature must be at least 20.5°C: \(\mathrm{t \geq 20.5}\)
• The temperature must be at most 23.5°C: \(\mathrm{t \leq 23.5}\)
• Combining both conditions: \(\mathrm{20.5 \leq t \leq 23.5}\)

Answer: D (\(\mathrm{20.5 \leq t \leq 23.5}\))

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may misinterpret "maximum deviation of 1.5°C above or below" and think it means the temperature can only be 1.5°C total, rather than understanding it means 1.5°C in each direction from the target.

This leads them to think the range is just from 0 to 1.5, causing them to select Choice A (\(\mathrm{t \leq 1.5}\)).

Second Most Common Error:

Inadequate APPLY CONSTRAINTS reasoning: Students correctly calculate either the minimum (20.5°C) or maximum (23.5°C) but fail to recognize they need BOTH bounds for a complete solution.

They might focus only on "temperature can't exceed 23.5" and select Choice C (\(\mathrm{t \leq 23.5}\)), or only on "temperature can't go below 20.5" and think about \(\mathrm{t \geq 20.5}\), then get confused when that's not an option and select Choice B (\(\mathrm{t \leq 20.5}\)).

The Bottom Line:

This problem tests whether students can translate a real-world constraint (deviation from target) into mathematical language while recognizing that ranges require two boundaries, not just one.