x^2 + y^2 lt 30For which of the following tables are all the values of x and their corresponding values...

1. TRANSLATE the problem information

• Given information:
  • Inequality: \(\mathrm{x^2 + y^2 \lt 30}\)
  • Four tables of \(\mathrm{(x,y)}\) coordinate pairs
  • Need to find which table has ALL pairs satisfying the inequality

This means: For each coordinate pair, I need to square both x and y, add them together, and check if the sum is less than 30.

2. SIMPLIFY by checking each table systematically

I'll work through each table, computing \(\mathrm{x^2 + y^2}\) for every pair:

Table A:

• \(\mathrm{(3, -1)}\): \(\mathrm{3^2 + (-1)^2 = 9 + 1 = 10 \lt 30}\) ✓
• \(\mathrm{(4, 3)}\): \(\mathrm{4^2 + 3^2 = 16 + 9 = 25 \lt 30}\) ✓
• \(\mathrm{(5, -4)}\): \(\mathrm{5^2 + (-4)^2 = 25 + 16 = 41 \gt 30}\) ✗

Since \(\mathrm{41 \gt 30}\), not all pairs work.

Table B:

• \(\mathrm{(3, -1)}\): \(\mathrm{3^2 + (-1)^2 = 9 + 1 = 10 \lt 30}\) ✓
• \(\mathrm{(5, 3)}\): \(\mathrm{5^2 + 3^2 = 25 + 9 = 34 \gt 30}\) ✗
• \(\mathrm{(4, -4)}\): \(\mathrm{4^2 + (-4)^2 = 16 + 16 = 32 \gt 30}\) ✗

Two pairs fail, so this table doesn't work.

Table C:

• \(\mathrm{(5, -1)}\): \(\mathrm{5^2 + (-1)^2 = 25 + 1 = 26 \lt 30}\) ✓
• \(\mathrm{(3, 3)}\): \(\mathrm{3^2 + 3^2 = 9 + 9 = 18 \lt 30}\) ✓
• \(\mathrm{(4, -4)}\): \(\mathrm{4^2 + (-4)^2 = 16 + 16 = 32 \gt 30}\) ✗

Since \(\mathrm{32 \gt 30}\), not all pairs work.

Table D:

• \(\mathrm{(5, -1)}\): \(\mathrm{5^2 + (-1)^2 = 25 + 1 = 26 \lt 30}\) ✓
• \(\mathrm{(4, 3)}\): \(\mathrm{4^2 + 3^2 = 16 + 9 = 25 \lt 30}\) ✓
• \(\mathrm{(3, -4)}\): \(\mathrm{3^2 + (-4)^2 = 9 + 16 = 25 \lt 30}\) ✓

3. APPLY CONSTRAINTS to select the final answer

Since the question asks for the table where ALL values satisfy the inequality, only Table D qualifies—every single pair produces a sum less than 30.

Answer: D

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak APPLY CONSTRAINTS reasoning: Students find one or two pairs in a table that work and immediately select that table, forgetting that ALL pairs must satisfy the inequality.

For example, they might check Table A, see that \(\mathrm{(3, -1)}\) gives \(\mathrm{10 \lt 30}\) and \(\mathrm{(4, 3)}\) gives \(\mathrm{25 \lt 30}\), then select A without checking the third pair \(\mathrm{(5, -4)}\) which gives \(\mathrm{41 \gt 30}\). This leads them to select Choice A instead of continuing their systematic check.

Second Most Common Error:

Poor SIMPLIFY execution: Students make calculation errors, especially when squaring negative numbers, thinking \(\mathrm{(-4)^2 = -16}\) instead of \(\mathrm{+16}\).

This type of arithmetic mistake can cause them to incorrectly evaluate whether pairs satisfy the inequality, leading to confusion and potentially selecting any of the wrong answer choices.

The Bottom Line:

This problem requires careful, complete checking—students must resist the urge to stop once they find some satisfying pairs and must systematically verify every single coordinate pair in each table.