y lt x^2 + 5 For which of the following tables are all the values of x and their corresponding...

1. TRANSLATE the inequality requirement

• Given: \(\mathrm{y \lt x² + 5}\)
• This means: For each (x,y) pair, the y-value must be less than \(\mathrm{x² + 5}\)
• We need ALL pairs in a table to satisfy this condition

2. SIMPLIFY to find the boundary values

• Calculate \(\mathrm{x² + 5}\) for each x-value:

• For \(\mathrm{x = 3}\): \(\mathrm{3² + 5 = 9 + 5 = 14}\)
• For \(\mathrm{x = 5}\): \(\mathrm{5² + 5 = 25 + 5 = 30}\)
• For \(\mathrm{x = 7}\): \(\mathrm{7² + 5 = 49 + 5 = 54}\)

3. CONSIDER ALL CASES by checking each table systematically

Table A:

• (3, 14): Is \(\mathrm{14 \lt 14}\)? No ✗
• Since the first pair fails, Table A is eliminated

Table B:

• (3, 10): Is \(\mathrm{10 \lt 14}\)? Yes ✓
• (5, 34): Is \(\mathrm{34 \lt 30}\)? No ✗
• Since one pair fails, Table B is eliminated

Table C:

• (3, 10): Is \(\mathrm{10 \lt 14}\)? Yes ✓
• (5, 26): Is \(\mathrm{26 \lt 30}\)? Yes ✓
• (7, 50): Is \(\mathrm{50 \lt 54}\)? Yes ✓
• All pairs satisfy the inequality!

Table D:

• (3, 18): Is \(\mathrm{18 \lt 14}\)? No ✗
• Since the first pair fails, Table D is eliminated

Answer: C

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak CONSIDER ALL CASES skill: Students check only the first pair in each table and select based on partial information.

For example, they might check Table A, see that \(\mathrm{14 \lt 14}\) is false, then check Table B, see that \(\mathrm{10 \lt 14}\) is true, and immediately select B without checking the remaining pairs. Since \(\mathrm{34 \lt 30}\) is false, this leads them to select Choice B (Table B).

Second Most Common Error:

Poor TRANSLATE reasoning: Students misinterpret the inequality direction, thinking \(\mathrm{y \gt x² + 5}\) instead of \(\mathrm{y \lt x² + 5}\).

This reversal makes them look for y-values that are greater than the calculated boundary values. Under this misconception, Table A looks perfect since \(\mathrm{14 \geq 14}\), \(\mathrm{30 \geq 30}\), and \(\mathrm{54 \geq 54}\), leading them to select Choice A (Table A).

The Bottom Line:

This problem tests systematic checking skills more than mathematical calculation. Students must resist the urge to jump to conclusions after checking just one or two pairs from each table.