Consider the inequality 2x + 5y geq 29. Which table lists only ordered pairs (x, y) that are solutions to...

1. TRANSLATE the problem requirements

• Given: Inequality \(\mathrm{2x + 5y \geq 29}\)
• Need to find: Which table contains ONLY ordered pairs that satisfy this inequality
• Key word "only" means ALL pairs in the correct table must work

2. CONSIDER ALL CASES for systematic checking

• Must test every single ordered pair in each table
• Cannot stop after finding one that works - need to verify all three pairs per table
• One failing pair eliminates that entire table

3. TRANSLATE each ordered pair into the inequality

For each pair \(\mathrm{(x,y)}\), substitute and calculate \(\mathrm{2x + 5y}\), then compare to 29.

Table A: \(\mathrm{(3,5), (6,3), (10,4)}\)

• \(\mathrm{(3,5)}\): \(\mathrm{2(3) + 5(5) = 6 + 25 = 31 \geq 29}\) ✓
• \(\mathrm{(6,3)}\): \(\mathrm{2(6) + 5(3) = 12 + 15 = 27 \lt 29}\) ✗ → Table A fails

Table B: \(\mathrm{(2,6), (4,5), (8,2)}\)

• \(\mathrm{(2,6)}\): \(\mathrm{2(2) + 5(6) = 4 + 30 = 34 \geq 29}\) ✓
• \(\mathrm{(4,5)}\): \(\mathrm{2(4) + 5(5) = 8 + 25 = 33 \geq 29}\) ✓
• \(\mathrm{(8,2)}\): \(\mathrm{2(8) + 5(2) = 16 + 10 = 26 \lt 29}\) ✗ → Table B fails

Table C: \(\mathrm{(5,4), (1,6), (0,5)}\)

• \(\mathrm{(5,4)}\): \(\mathrm{2(5) + 5(4) = 10 + 20 = 30 \geq 29}\) ✓
• \(\mathrm{(1,6)}\): \(\mathrm{2(1) + 5(6) = 2 + 30 = 32 \geq 29}\) ✓
• \(\mathrm{(0,5)}\): \(\mathrm{2(0) + 5(5) = 0 + 25 = 25 \lt 29}\) ✗ → Table C fails

Table D: \(\mathrm{(3,6), (7,3), (9,4)}\)

• \(\mathrm{(3,6)}\): \(\mathrm{2(3) + 5(6) = 6 + 30 = 36 \geq 29}\) ✓
• \(\mathrm{(7,3)}\): \(\mathrm{2(7) + 5(3) = 14 + 15 = 29 \geq 29}\) ✓ (equality counts!)
• \(\mathrm{(9,4)}\): \(\mathrm{2(9) + 5(4) = 18 + 20 = 38 \geq 29}\) ✓

4. APPLY CONSTRAINTS to select final answer

• Only Table D has all three pairs satisfying the inequality
• Tables A, B, and C each have exactly one failing pair

Answer: D

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak CONSIDER ALL CASES skill: Students check only the first one or two ordered pairs in each table, finding pairs that work, and incorrectly select an answer without verifying all pairs.

For example, they might test \(\mathrm{(3,5)}\) from Table A, get \(\mathrm{31 \geq 29}\) ✓, and think "this table works!" without checking \(\mathrm{(6,3)}\) which gives only 27. This leads them to select Choice A instead of continuing their systematic check.

Second Most Common Error:

Poor TRANSLATE execution: Students make arithmetic errors when substituting values, particularly with the coefficient 5 in larger y-values, or forget to multiply both terms correctly.

This causes confusion about which pairs actually satisfy the inequality, leading to guessing among the choices.

The Bottom Line:

This problem tests thoroughness more than complex math skills. Success requires disciplined checking of every single ordered pair, not just finding some that work.