A manufacturer produces ball bearings that have a target diameter of 12 millimeters. For a ball bearing to be considered...

1. TRANSLATE the problem information

• Given information:
  • Target diameter: 12 millimeters
  • Tolerance: 0.04 millimeters
  • Actual diameter: \(\mathrm{d}\) millimeters
  • Condition: \(\mathrm{d}\) must be "within a tolerance of 0.04 millimeters of the target diameter"

• What this tells us: The difference between actual and target diameter cannot exceed 0.04 mm in either direction

2. INFER the mathematical relationship

• "Within tolerance" means the distance between actual and target values must be ≤ tolerance
• Distance is always represented by absolute value since it's positive regardless of direction
• We need: \(\mathrm{|actual\ value - target\ value| \leq tolerance}\)

3. TRANSLATE into mathematical notation

• Absolute difference between \(\mathrm{d}\) and 12: \(\mathrm{|d - 12|}\)
• This difference must be ≤ 0.04: \(\mathrm{|d - 12| \leq 0.04}\)

4. Verify against answer choices

• (A) \(\mathrm{|d + 12| \leq 0.04}\): Wrong center (-12 instead of 12)
• (B) \(\mathrm{|d - 12| \geq 0.04}\): Represents unacceptable values
• (C) \(\mathrm{|d - 0.04| \leq 12}\): Tolerance and target swapped
• (D) \(\mathrm{|d - 12| \leq 0.04}\): Correct setup ✓

Answer: D

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students misinterpret "within tolerance" and set up the inequality backwards, thinking about what values are NOT acceptable rather than what values ARE acceptable.

They might reason: "If it's not acceptable when the difference is more than 0.04, then \(\mathrm{|d - 12| \geq 0.04}\)." This logic reversal leads them to select Choice B (\(\mathrm{|d - 12| \geq 0.04}\)).

Second Most Common Error:

Poor TRANSLATE reasoning: Students confuse which value should be the center of the absolute value expression and which should be the bound.

They might think the tolerance (0.04) is what we're measuring distance from, setting up \(\mathrm{|d - 0.04| \leq 12}\). This leads them to select Choice C (\(\mathrm{|d - 0.04| \leq 12}\)).

The Bottom Line:

The key insight is that "within tolerance" always means \(\mathrm{|actual - target| \leq tolerance}\). The target value (12) goes inside the absolute value bars, and the tolerance (0.04) becomes the upper bound.