Which of the following ordered pairs \((\mathrm{x}, \mathrm{y})\) makes the expression 12 - 3x - 2y positive? \((3, 1)\) \((2,...

1. TRANSLATE the problem requirement

• Given information:
  • Expression: \(12 - 3\mathrm{x} - 2\mathrm{y}\)
  • Condition: Must be positive
  • Three ordered pairs to test: (3,1), (2,3), (4,1)

• Mathematical translation: \(12 - 3\mathrm{x} - 2\mathrm{y} \gt 0\)

2. INFER a strategic rearrangement

• The inequality \(12 - 3\mathrm{x} - 2\mathrm{y} \gt 0\) can be rearranged for easier evaluation
• Moving terms: \(12 \gt 3\mathrm{x} + 2\mathrm{y}\)
• Equivalent form: \(3\mathrm{x} + 2\mathrm{y} \lt 12\)

This format allows us to substitute each pair and compare a single result to 12.

3. SIMPLIFY by testing each ordered pair systematically

Testing I. (3, 1):

• \(3\mathrm{x} + 2\mathrm{y} = 3(3) + 2(1)\)
• \(= 9 + 2\)
• \(= 11\)
• Is \(11 \lt 12\)? Yes ✓

Testing II. (2, 3):

• \(3\mathrm{x} + 2\mathrm{y} = 3(2) + 2(3)\)
• \(= 6 + 6\)
• \(= 12\)
• Is \(12 \lt 12\)? No ✗ (equal, not less than)

Testing III. (4, 1):

• \(3\mathrm{x} + 2\mathrm{y} = 3(4) + 2(1)\)
• \(= 12 + 2\)
• \(= 14\)
• Is \(14 \lt 12\)? No ✗ (greater than 12)

4. APPLY CONSTRAINTS to select the final answer

• Only pair I satisfies the strict inequality
• Answer choice A corresponds to "I only"

Answer: A

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may misinterpret "positive" as meaning \(\geq 0\) instead of \(\gt 0\), or they might incorrectly set up the original inequality.

This conceptual confusion leads them to accept pair II since \(12 - 3(2) - 2(3) = 0\), thinking "zero or positive" satisfies the condition. This may lead them to select Choice C (I and II only).

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up \(3\mathrm{x} + 2\mathrm{y} \lt 12\) but make arithmetic errors when substituting values, particularly with the order of operations or sign errors.

Common calculation mistakes include computing \(3(2) + 2(3)\) as 12 instead of \(6 + 6 = 12\), or similar computational errors that lead to incorrect comparisons. This leads to confusion and guessing among the remaining choices.

The Bottom Line:

This problem tests precision in both language interpretation (what "positive" means mathematically) and systematic algebraic evaluation. Students must maintain accuracy through multiple substitutions while respecting the strict inequality requirement.