For which of the following tables are all the values of x and their corresponding values of y solutions to...

1. TRANSLATE the problem requirements

• Given: Inequality \(\mathrm{2x - y \gt 883}\) and four tables with \(\mathrm{(x,y)}\) pairs
• Need to find: Which table has ALL pairs satisfying the inequality

2. INFER the systematic approach

• Since all tables use the same three x-values (440, 441, 442), I can find what y-constraint each x-value creates
• Then check which table's y-values satisfy ALL three constraints

3. SIMPLIFY to find y-constraints for each x-value

For x = 440:

• \(\mathrm{2(440) - y \gt 883}\)
• \(\mathrm{880 - y \gt 883}\)
• \(\mathrm{-y \gt 3}\)
• \(\mathrm{y \lt -3}\) (flip sign when dividing by -1)

For x = 441:

• \(\mathrm{2(441) - y \gt 883}\)
• \(\mathrm{882 - y \gt 883}\)
• \(\mathrm{-y \gt 1}\)
• \(\mathrm{y \lt -1}\)

For x = 442:

• \(\mathrm{2(442) - y \gt 883}\)
• \(\mathrm{884 - y \gt 883}\)
• \(\mathrm{-y \gt -1}\)
• \(\mathrm{y \lt 1}\)

4. APPLY CONSTRAINTS to evaluate each table

Choice A: \(\mathrm{x=440, y=0}\); \(\mathrm{x=441, y=-2}\); \(\mathrm{x=442, y=-4}\)

• For \(\mathrm{x=440}\): Is \(\mathrm{0 \lt -3}\)? No ✗ (fails immediately)

Choice B: \(\mathrm{x=440, y=0}\); \(\mathrm{x=442, y=-2}\); \(\mathrm{x=441, y=-4}\)

• For \(\mathrm{x=440}\): Is \(\mathrm{0 \lt -3}\)? No ✗ (fails immediately)

Choice C: \(\mathrm{x=442, y=0}\); \(\mathrm{x=440, y=-2}\); \(\mathrm{x=441, y=-4}\)

• For \(\mathrm{x=440}\): Is \(\mathrm{-2 \lt -3}\)? No ✗ (fails)

Choice D: \(\mathrm{x=442, y=0}\); \(\mathrm{x=441, y=-2}\); \(\mathrm{x=440, y=-4}\)

• For \(\mathrm{x=442}\): Is \(\mathrm{0 \lt 1}\)? Yes ✓
• For \(\mathrm{x=441}\): Is \(\mathrm{-2 \lt -1}\)? Yes ✓
• For \(\mathrm{x=440}\): Is \(\mathrm{-4 \lt -3}\)? Yes ✓ (all pass!)

Answer: D

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Forgetting to flip the inequality sign when dividing by -1

When solving \(\mathrm{-y \gt 3}\), students write \(\mathrm{y \gt -3}\) instead of \(\mathrm{y \lt -3}\). This creates the wrong constraint (\(\mathrm{y \gt -3}\) instead of \(\mathrm{y \lt -3}\)), making them think that larger y-values satisfy the constraint when actually smaller ones do. With incorrect constraints, they might select Choice A or B since these have \(\mathrm{y = 0}\) for \(\mathrm{x = 440}\), which would satisfy \(\mathrm{y \gt -3}\) but not the correct constraint \(\mathrm{y \lt -3}\).

Second Most Common Error:

Poor INFER reasoning: Not checking all three pairs systematically

Students might check just one or two pairs instead of verifying that ALL pairs satisfy the inequality. They might see that Choice A has \(\mathrm{(441,-2)}\) and \(\mathrm{(442,-4)}\) working and select it without noticing that \(\mathrm{(440,0)}\) fails the constraint. This leads to selecting Choice A prematurely.

The Bottom Line:

This problem tests whether students can correctly manipulate inequalities with negative coefficients and then systematically apply multiple constraints - both skills that require careful attention to detail.