A thermostat maintains a room at a set temperature of 68°F with a tolerance of at most 1.2°F from the...

1. TRANSLATE the problem information

• Given information:
  • Set temperature: 68°F
  • Tolerance: at most 1.2°F from set temperature
  • Condition: \(|\mathrm{T} - 68| \leq 1.2\)
• What this tells us: The actual temperature can be up to 1.2 degrees above or below 68°F

2. SIMPLIFY the absolute value inequality

• Convert \(|\mathrm{T} - 68| \leq 1.2\) to compound inequality:
  \(-1.2 \leq \mathrm{T} - 68 \leq 1.2\)
• Add 68 to all parts:
  \(68 - 1.2 \leq \mathrm{T} \leq 68 + 1.2\)
  \(66.8 \leq \mathrm{T} \leq 69.2\)

3. APPLY CONSTRAINTS to check each answer choice

• Test each temperature against our interval \([66.8, 69.2]\):
  • A) 66.5°F: Since \(66.5 \lt 66.8\), this is too cold
  • B) 66.7°F: Since \(66.7 \lt 66.8\), this is too cold
  • C) 69.1°F: Since \(66.8 \leq 69.1 \leq 69.2\), this works ✓
  • D) 69.3°F: Since \(69.3 \gt 69.2\), this is too hot

Answer: C) 69.1°F

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may misinterpret "tolerance of at most 1.2°F" and think it means the temperature can only vary by exactly 1.2°F, leading them to only consider 66.8°F and 69.2°F as valid temperatures. They might then select the closest answer choice without understanding that any temperature within the range is acceptable.

This leads to confusion when none of the choices match exactly 66.8 or 69.2, causing them to guess randomly.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up \(|\mathrm{T} - 68| \leq 1.2\) but make arithmetic errors when converting to the interval. For example, they might calculate \(68 - 1.2 = 66.6\) instead of 66.8, leading them to incorrectly accept choice B) 66.7°F as valid.

This may lead them to select Choice B (66.7°F).

The Bottom Line:

This problem tests whether students can translate real-world tolerance language into mathematical inequalities and then systematically apply the constraints. The key insight is recognizing that "at most 1.2°F from 68°F" creates an acceptable range, not just specific endpoint values.