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: Linear inequality \(\mathrm{y \gt 4x + 8}\)
• Need to find: Which table has ALL (x,y) pairs that satisfy this inequality
• Method: Substitute each x and y value to check if \(\mathrm{y \gt 4x + 8}\) is true

2. INFER an efficient strategy

• Since ALL pairs must work, if even one pair fails, that entire table is eliminated
• Test the first pair (x=2) across all options first - this can eliminate multiple tables quickly
• Only verify remaining pairs in tables that pass the first test

3. TRANSLATE and test x = 2 across all options

Option A: (2, 19)

• Does \(\mathrm{19 \gt 4(2) + 8}\)?
• Does \(\mathrm{19 \gt 16}\)? ✓ YES

Option B: (2, 8)

• Does \(\mathrm{8 \gt 4(2) + 8}\)?
• Does \(\mathrm{8 \gt 16}\)? ✗ NO - Eliminate Option B

Option C: (2, 13)

• Does \(\mathrm{13 \gt 4(2) + 8}\)?
• Does \(\mathrm{13 \gt 16}\)? ✗ NO - Eliminate Option C

Option D: (2, 13)

• Does \(\mathrm{13 \gt 4(2) + 8}\)?
• Does \(\mathrm{13 \gt 16}\)? ✗ NO - Eliminate Option D

4. APPLY CONSTRAINTS and verify remaining pairs in Option A

Only Option A survived the first test, but we must verify all its pairs:

For x = 4, y = 30:

• Does \(\mathrm{30 \gt 4(4) + 8}\)?
• Does \(\mathrm{30 \gt 24}\)? ✓ YES

For x = 6, y = 41:

• Does \(\mathrm{41 \gt 4(6) + 8}\)?
• Does \(\mathrm{41 \gt 32}\)? ✓ YES

Answer: A

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE reasoning: Students misinterpret "greater than" as "greater than or equal to" and accept y-values that exactly equal \(\mathrm{4x + 8}\).

For example, if a student sees \(\mathrm{y = 16}\) when \(\mathrm{x = 2}\), they might think "\(\mathrm{16 = 4(2) + 8}\), so this works" instead of recognizing that the inequality requires \(\mathrm{y \gt 16}\), not \(\mathrm{y = 16}\). This conceptual confusion about strict vs. non-strict inequalities can lead to accepting incorrect answer choices that contain boundary values.

Second Most Common Error:

Incomplete INFER process: Students test only one or two pairs per table instead of verifying that ALL pairs satisfy the inequality.

They might find that the first pair in Option A works, immediately select it, and move on without checking the remaining pairs. While this happens to work for this problem, the incomplete verification process reflects poor systematic thinking that would fail on problems where the first pair works but later pairs don't.

The Bottom Line:

This problem tests whether students understand that inequality solutions require systematic verification of ALL given pairs, and that "greater than" means strictly greater than, not greater than or equal to.