Question:y gt x^2 - 4For which of the following tables are all the values of x and their corresponding values...

1. TRANSLATE the problem requirements

• Given: The inequality \(\mathrm{y \gt x^2 - 4}\)
• Need to find: Which table has ALL coordinate pairs that satisfy this inequality
• Key insight: Every single \(\mathrm{(x, y)}\) pair in the correct table must make \(\mathrm{y \gt x^2 - 4}\) true

2. TRANSLATE the testing approach

• For each coordinate pair \(\mathrm{(x, y)}\), I need to:
  • Calculate \(\mathrm{x^2 - 4}\)
  • Check if \(\mathrm{y \gt x^2 - 4}\)
• The correct answer will be the table where ALL pairs pass this test

3. SIMPLIFY by testing each table systematically

Table A:

• \(\mathrm{(2, 0)}\): \(\mathrm{x^2 - 4 = 4 - 4 = 0}\). Is \(\mathrm{0 \gt 0}\)? No. ✗
• Since the first pair fails, Table A is incorrect.

Table B:

• \(\mathrm{(2, 2)}\): \(\mathrm{x^2 - 4 = 4 - 4 = 0}\). Is \(\mathrm{2 \gt 0}\)? Yes. ✓
• \(\mathrm{(3, 4)}\): \(\mathrm{x^2 - 4 = 9 - 4 = 5}\). Is \(\mathrm{4 \gt 5}\)? No. ✗
• Table B fails on the second pair.

Table C:

• \(\mathrm{(2, -1)}\): \(\mathrm{x^2 - 4 = 4 - 4 = 0}\). Is \(\mathrm{-1 \gt 0}\)? No. ✗
• Table C fails on the first pair.

Table D:

• \(\mathrm{(2, 1)}\): \(\mathrm{x^2 - 4 = 4 - 4 = 0}\). Is \(\mathrm{1 \gt 0}\)? Yes. ✓
• \(\mathrm{(3, 6)}\): \(\mathrm{x^2 - 4 = 9 - 4 = 5}\). Is \(\mathrm{6 \gt 5}\)? Yes. ✓
• \(\mathrm{(4, 13)}\): \(\mathrm{x^2 - 4 = 16 - 4 = 12}\). Is \(\mathrm{13 \gt 12}\)? Yes. ✓

4. CONSIDER ALL CASES to confirm the answer

• Table D is the only one where all three coordinate pairs satisfy the inequality

Answer: D

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak CONSIDER ALL CASES skill: Students often stop checking after finding the first coordinate pair that works or doesn't work in a table, rather than systematically testing all pairs.

For example, they might check Table B, see that \(\mathrm{(2, 2)}\) satisfies the inequality, and immediately conclude B is correct without checking the other pairs. This leads them to select Choice B.

Second Most Common Error:

Poor TRANSLATE reasoning: Students may confuse the strict inequality symbol \(\mathrm{(\gt)}\) with greater than or equal to \(\mathrm{(\geq)}\), leading them to accept coordinate pairs where y equals \(\mathrm{x^2 - 4}\).

This causes them to think Table A is correct since the pairs give equality rather than strict inequality, leading them to select Choice A.

The Bottom Line:

This problem requires methodical checking of every coordinate pair. The challenge isn't in the calculations—it's in the discipline to test all cases systematically rather than jumping to conclusions after partial checking.