3x + 2y leq 675For which of the following tables are all the values of x and their corresponding values...

1. TRANSLATE the problem information

• Given information:
  • Inequality: \(3\mathrm{x} + 2\mathrm{y} \leq 675\)
  • Four tables with different \((\mathrm{x},\mathrm{y})\) coordinate pairs
  • Need to find which table has ALL pairs satisfying the inequality

• What this tells us: We need to test every single pair in each table - if even one pair fails, that entire table is wrong.

2. INFER the approach

• Strategy: Substitute each \((\mathrm{x},\mathrm{y})\) pair into \(3\mathrm{x} + 2\mathrm{y}\) and check if the result is \(\leq 675\)
• Key insight: We can eliminate a table as soon as we find one pair that doesn't work
• We need to find the table where ALL pairs satisfy the inequality

3. SIMPLIFY by testing each table systematically

Choice A: Test \((200, 38)\)

• \(3(200) + 2(38) = 600 + 76 = 676\)
• Is \(676 \leq 675\)? No! → Choice A eliminated

Choice B: Test each pair

• \((200, 37)\): \(3(200) + 2(37) = 600 + 74 = 674 \leq 675\) ✓
• \((220, 8)\): \(3(220) + 2(8) = 660 + 16 = 676 \leq 675\)? No! → Choice B eliminated

Choice C: Test each pair

• \((220, 7)\): \(3(220) + 2(7) = 660 + 14 = 674 \leq 675\) ✓
• \((200, 38)\): \(3(200) + 2(38) = 600 + 76 = 676 \leq 675\)? No! → Choice C eliminated

Choice D: Test each pair

• \((220, 7)\): \(3(220) + 2(7) = 660 + 14 = 674 \leq 675\) ✓
• \((210, 22)\): \(3(210) + 2(22) = 630 + 44 = 674 \leq 675\) ✓
• \((200, 37)\): \(3(200) + 2(37) = 600 + 74 = 674 \leq 675\) ✓

4. APPLY CONSTRAINTS to confirm the answer

• All three pairs in Choice D satisfy the inequality
• No other choice has all pairs working

Answer: D

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students test just the first pair in each table and select the first table where that pair works, not realizing they need ALL pairs to work.

For example, they might test \((200, 37)\) from Choice B, see that \(674 \leq 675\), and immediately choose B without checking the other pairs in that table. This leads them to select Choice B instead of continuing to verify all pairs.

Second Most Common Error:

Poor SIMPLIFY execution: Students make arithmetic errors when calculating \(3\mathrm{x} + 2\mathrm{y}\), particularly with the multiplication or addition steps.

For instance, they might calculate \(3(220) + 2(7)\) as \(660 + 14 = 676\) instead of 674, leading to incorrect eliminations and potentially guessing among the remaining choices.

The Bottom Line:

This problem tests systematic verification skills - students must resist the urge to stop after finding one working pair and instead methodically check every single coordinate pair in each table.