Question:y gt 2x^2 - 5For which of the following tables are all the values of x and their corresponding values...

1. INFER the problem requirements

• Given: The inequality \(\mathrm{y \gt 2x² - 5}\)
• Need: A table where ALL coordinate pairs (x, y) satisfy this inequality
• Strategy: Test every single pair in each table - if even one fails, that table is wrong

2. SIMPLIFY each coordinate pair systematically

Starting with Table A, test each pair:

For (-2, 4):

• Substitute: \(\mathrm{4 \gt 2(-2)² - 5}\)
• SIMPLIFY: \(\mathrm{4 \gt 2(4) - 5}\) → \(\mathrm{4 \gt 8 - 5}\) → \(\mathrm{4 \gt 3}\) ✓

For (0, -4):

• Substitute: \(\mathrm{-4 \gt 2(0)² - 5}\)
• SIMPLIFY: \(\mathrm{-4 \gt 0 - 5}\) → \(\mathrm{-4 \gt -5}\) ✓

For (3, 14):

• Substitute: \(\mathrm{14 \gt 2(3)² - 5}\)
• SIMPLIFY: \(\mathrm{14 \gt 2(9) - 5}\) → \(\mathrm{14 \gt 18 - 5}\) → \(\mathrm{14 \gt 13}\) ✓

3. CONSIDER ALL CASES in Table A

All three pairs work! But let me verify the other tables fail by testing their first points:

• Table B: \(\mathrm{2 \gt 2(-2)² - 5}\) → \(\mathrm{2 \gt 3}\) ✗
• Table C: \(\mathrm{3 \gt 2(-2)² - 5}\) → \(\mathrm{3 \gt 3}\) ✗ (equal, not greater)
• Table D: \(\mathrm{0 \gt 2(-2)² - 5}\) → \(\mathrm{0 \gt 3}\) ✗

Answer: A

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER reasoning: Students test only one or two coordinate pairs from a table instead of checking ALL pairs systematically. They might test the first pair in Table A, see it works, and immediately choose A without verifying the other pairs.

This incomplete checking could lead them to accidentally select a wrong table if they happened to test a "working" pair from an otherwise incorrect table, or miss that Table A actually works completely.

Second Most Common Error:

Poor SIMPLIFY execution: Students make calculation errors when evaluating \(\mathrm{2x² - 5}\), especially with negative values or order of operations. For example, calculating \(\mathrm{2(-2)²}\) as -8 instead of +8, or forgetting to square before multiplying by 2.

This may lead them to incorrectly reject the right answer Choice A or incorrectly accept a wrong choice.

The Bottom Line:

This problem requires methodical verification - you must check EVERY coordinate pair in a table, and you must be precise with your algebraic calculations. Rushing through either step will likely lead to the wrong answer.