A shipping company uses two box sizes: large (L) and small (S). The inequality 5L + 2S leq 36 gives...

1. TRANSLATE the problem information

• Given information:
  • Inequality: \(5\mathrm{L} + 2\mathrm{S} \leq 36\) (total empty-box weight constraint)
  • Ordered pair: \((\mathrm{L}, \mathrm{S}) = (4, 8)\)
  • \(\mathrm{L}\) = large boxes, \(\mathrm{S}\) = small boxes

• What this tells us: We have 4 large boxes and 8 small boxes to test in the weight constraint

2. INFER our approach

• We need to substitute \(\mathrm{L} = 4\) and \(\mathrm{S} = 8\) into the inequality
• Then check if the result satisfies the \(\leq 36\) condition
• Finally, match our interpretation with the answer choices

3. SIMPLIFY by substituting values

• Substitute into \(5\mathrm{L} + 2\mathrm{S}\):
  \(5(4) + 2(8) = 20 + 16 = 36\)

4. INFER the inequality relationship

• We found that \(5\mathrm{L} + 2\mathrm{S} = 36\)
• Since \(36 \leq 36\) is true, the ordered pair satisfies the constraint
• This means the total weight "does not exceed" 36 pounds

5. TRANSLATE back to find the correct choice

• We need: 4 large boxes, 8 small boxes, does not exceed 36 pounds
• Checking choices: Only (B) matches both the correct order and inequality direction

Answer: B

Why Students Usually Falter on This Problem

Most Common Error Path:

TRANSLATE confusion: Students may confuse the order in the ordered pair \((\mathrm{L}, \mathrm{S}) = (4, 8)\), thinking it means 8 large boxes and 4 small boxes instead of 4 large and 8 small.

When they substitute \(5(8) + 2(4) = 40 + 8 = 48\), they get \(48 \leq 36\), which is false. This leads them to think the weight exceeds 36 pounds and they may select Choice (C) (8 large, 4 small, at least 36 pounds).

Second Most Common Error:

TRANSLATE misunderstanding of inequality direction: Students correctly identify 4 large and 8 small boxes, correctly calculate 36, but misinterpret what \(\leq\) means in context. They think "\(36 \leq 36\)" means the weight is "at least" 36 pounds instead of "does not exceed" 36 pounds.

This may lead them to select Choice (D) (4 large, 8 small, at least 36 pounds).

The Bottom Line:

This problem tests careful reading of mathematical notation - both ordered pair interpretation and inequality direction. Success requires precise TRANSLATION of symbolic math back to real-world language.