y lt -x^2 + 8x + 3 For which of the following tables are all the values of x and...

1. TRANSLATE the problem requirements

• Given: The inequality \(\mathrm{y \lt -x^2 + 8x + 3}\)
• Find: Which table has ALL (x,y) pairs that satisfy this inequality
• Strategy: Check every pair in each table against the inequality

2. INFER the most efficient approach

• Instead of checking each individual pair separately, first calculate the "boundary value" \(\mathrm{(-x^2 + 8x + 3)}\) for each x-value that appears across all tables
• The x-values appearing are: 1, 4, and 7
• Then check if each table's y-values are less than these boundary values

3. SIMPLIFY to find boundary values

For x = 1:
\(\mathrm{-x^2 + 8x + 3 = -(1)^2 + 8(1) + 3}\)
\(\mathrm{= -1 + 8 + 3}\)
\(\mathrm{= 10}\)
So for \(\mathrm{x = 1}\), we need \(\mathrm{y \lt 10}\)

For x = 4:
\(\mathrm{-x^2 + 8x + 3 = -(4)^2 + 8(4) + 3}\)
\(\mathrm{= -16 + 32 + 3}\)
\(\mathrm{= 19}\)
So for \(\mathrm{x = 4}\), we need \(\mathrm{y \lt 19}\)

For x = 7:
\(\mathrm{-x^2 + 8x + 3 = -(7)^2 + 8(7) + 3}\)
\(\mathrm{= -49 + 56 + 3}\)
\(\mathrm{= 10}\)
So for \(\mathrm{x = 7}\), we need \(\mathrm{y \lt 10}\)

4. INFER by checking each table systematically

Table A: \(\mathrm{x=1, y=10}\). Is \(\mathrm{10 \lt 10}\)? No. → This table fails immediately

Table B: \(\mathrm{x=1, y=9}\). Is \(\mathrm{9 \lt 10}\)? ✓ Yes. \(\mathrm{x=4, y=20}\). Is \(\mathrm{20 \lt 19}\)? No. → This table fails

Table C: \(\mathrm{x=1, y=11}\). Is \(\mathrm{11 \lt 10}\)? No. → This table fails immediately

Table D: \(\mathrm{x=1, y=9}\). Is \(\mathrm{9 \lt 10}\)? ✓ Yes. \(\mathrm{x=4, y=18}\). Is \(\mathrm{18 \lt 19}\)? ✓ Yes. \(\mathrm{x=7, y=9}\). Is \(\mathrm{9 \lt 10}\)? ✓ Yes. → All pairs satisfy the inequality

Answer: D

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students check only one or two pairs from a table instead of verifying that ALL pairs satisfy the inequality. They might see that the first pair in Table B works \(\mathrm{(9 \lt 10)}\) and immediately select B without checking the remaining pairs.

This leads them to select Choice B because they stopped checking after finding one valid pair.

Second Most Common Error:

Poor SIMPLIFY execution: Students make arithmetic errors when evaluating the polynomial, especially with the negative signs. A common mistake is calculating \(\mathrm{-(4)^2}\) as \(\mathrm{(-4)^2 = 16}\) instead of -16, leading to incorrect boundary values.

This causes confusion about which y-values should work, leading to random guessing among the answer choices.

The Bottom Line:

This problem tests methodical checking skills - students must resist the urge to stop after finding one working pair and instead verify that every single pair in a table satisfies the inequality.