A medical device maintains a patient's blood pH during a procedure. The acceptable pH is 7.40 with a tolerance of...

1. TRANSLATE the problem information

• Given information:
  • Target pH: 7.40
  • Tolerance: 0.05
  • Need absolute value inequality for acceptable pH values (p)

• What this tells us: The actual pH can be up to 0.05 units away from 7.40 in either direction

2. INFER the acceptable range

• With tolerance of 0.05, pH can vary from:
  • Lower bound: \(\mathrm{7.40 - 0.05 = 7.35}\)
  • Upper bound: \(\mathrm{7.40 + 0.05 = 7.45}\)
• Acceptable range: \(\mathrm{7.35 \leq p \leq 7.45}\)

3. INFER the absolute value form

• Key insight: The compound inequality \(\mathrm{7.35 \leq p \leq 7.45}\) can be rewritten as an absolute value inequality
• The range extends 0.05 units on each side of the center value 7.40
• This means: \(\mathrm{|p - 7.40| \leq 0.05}\) (distance from p to 7.40 is at most 0.05)

4. INFER which answer choice matches

• Look for inequality with:
  • Center point: 7.40 (the target value)
  • Bound: 0.05 (the tolerance)
• Choice (B): \(\mathrm{|p - 7.40| \leq 0.05}\) ✓

Answer: B

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students confuse which value should be the center of the absolute value inequality. They might think the center should be one of the boundary values (7.35 or 7.45) instead of the target value (7.40).

For example, seeing that the range goes from 7.35 to 7.45, they might incorrectly reason that 7.35 is somehow the "center" and write \(\mathrm{|p - 7.35| \leq 0.05}\). This completely misses that absolute value inequalities measure distance from a central point.

This may lead them to select Choice A (\(\mathrm{|p - 7.35| \leq 0.05}\)).

Second Most Common Error:

Poor TRANSLATE reasoning: Students might double the tolerance value, thinking that since tolerance goes ±0.05, the total "spread" is 0.10, and incorrectly use this as the bound.

They get the center correct (7.40) but use 0.10 instead of 0.05 as the tolerance in the inequality.

This may lead them to select Choice D (\(\mathrm{|p - 7.40| \leq 0.10}\)).

The Bottom Line:

This problem tests whether students understand that in tolerance situations, the absolute value inequality centers on the target value (not a boundary), and the bound equals the tolerance amount (not double the tolerance).