1200 - 3x lt 4yFor which of the following tables are all the values of x and their corresponding values...

1. TRANSLATE the problem requirements

• Given information:
  • Inequality: \(\mathrm{1200 - 3x \lt 4y}\)
  • Four tables with different \(\mathrm{(x,y)}\) coordinate pairs
  • Need to find which table has ALL pairs satisfying the inequality

• What this tells us: We need to substitute each \(\mathrm{(x,y)}\) pair into the inequality and check if the left side is strictly less than the right side.

2. INFER the testing strategy

• Since we need ALL pairs in a table to work, we can eliminate a table as soon as we find one pair that doesn't satisfy the inequality
• We should test systematically, checking each pair completely before moving to the next choice
• The inequality \(\mathrm{1200 - 3x \lt 4y}\) can be evaluated by calculating the left side, calculating the right side, then comparing

3. SIMPLIFY the calculations for each table

Choice A:

• (108, 219): \(\mathrm{1200 - 3(108) = 1200 - 324 = 876}\), and \(\mathrm{4(219) = 876}\)
• Check: \(\mathrm{876 \lt 876}\)? No, this is false.
• Since one pair fails, eliminate Choice A.

Choice B:

• (104, 222): \(\mathrm{1200 - 3(104) = 1200 - 312 = 888}\), and \(\mathrm{4(222) = 888}\)
• Check: \(\mathrm{888 \lt 888}\)? No, this is false.
• Since one pair fails, eliminate Choice B.

Choice D:

• (100, 225): \(\mathrm{1200 - 3(100) = 1200 - 300 = 900}\), and \(\mathrm{4(225) = 900}\)
• Check: \(\mathrm{900 \lt 900}\)? No, this is false.
• Since one pair fails, eliminate Choice D.

Choice C:

• (104, 223): \(\mathrm{1200 - 312 = 888}\), and \(\mathrm{4(223) = 892}\) → \(\mathrm{888 \lt 892}\) ✓
• (100, 226): \(\mathrm{1200 - 300 = 900}\), and \(\mathrm{4(226) = 904}\) → \(\mathrm{900 \lt 904}\) ✓
• (108, 220): \(\mathrm{1200 - 324 = 876}\), and \(\mathrm{4(220) = 880}\) → \(\mathrm{876 \lt 880}\) ✓

4. APPLY CONSTRAINTS to confirm the answer

• All three pairs in Choice C satisfy the strict inequality
• Choice C is the only table where every pair works

Answer: C

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak APPLY CONSTRAINTS skill: Students confuse the strict inequality symbol (\(\mathrm{\lt}\)) with "less than or equal to" (\(\mathrm{\leq}\))

When they encounter cases like \(\mathrm{876 \lt 876}\) or \(\mathrm{888 \lt 888}\), they incorrectly think "876 equals 876, so that's close enough" and mark these as true. This leads them to incorrectly accept choices that have equality cases instead of strict inequality.

This may lead them to select Choice A, B, or D depending on which one they test first, since all three contain pairs where the left side equals the right side.

Second Most Common Error:

Poor SIMPLIFY execution: Students make arithmetic errors during the multi-step calculations

They might calculate \(\mathrm{1200 - 3(108)}\) incorrectly as \(\mathrm{876 - 324 = 552}\) instead of \(\mathrm{1200 - 324 = 876}\), or miscalculate products like \(\mathrm{4(219)}\). These calculation errors lead to wrong inequality comparisons and incorrect elimination of the right choice or acceptance of wrong choices.

This leads to confusion and guessing among the remaining choices.

The Bottom Line:

This problem tests whether students can systematically verify multiple conditions while maintaining precision with strict inequalities and multi-step arithmetic - it's not just about knowing the inequality symbol, but applying it correctly across several test cases.