y gt 7x - 4 For which of the following tables are all the values of x and their corresponding...

1. TRANSLATE the problem information

• Given: The inequality \(\mathrm{y \gt 7x - 4}\)
• Find: Which table has ALL coordinate pairs that satisfy this inequality
• What this tells us: For each x-value, y must be strictly greater than \(\mathrm{7x - 4}\)

2. INFER the solution strategy

• Since all tables use the same x-values (3, 5, and 8), I can substitute each x-value into the inequality to find what y must be greater than
• Then check each table systematically - ALL pairs must work for the table to be correct

3. TRANSLATE each x-value into a constraint

• For \(\mathrm{x = 3}\): \(\mathrm{y \gt 7(3) - 4 = 21 - 4 = 17}\), so \(\mathrm{y \gt 17}\)
• For \(\mathrm{x = 5}\): \(\mathrm{y \gt 7(5) - 4 = 35 - 4 = 31}\), so \(\mathrm{y \gt 31}\)
• For \(\mathrm{x = 8}\): \(\mathrm{y \gt 7(8) - 4 = 56 - 4 = 52}\), so \(\mathrm{y \gt 52}\)

4. APPLY CONSTRAINTS to check each table

Choice A: \(\mathrm{(3,18)}\), \(\mathrm{(5,27)}\), \(\mathrm{(8,48)}\)

• \(\mathrm{x = 3}\): Is \(\mathrm{18 \gt 17}\)? Yes ✓
• \(\mathrm{x = 5}\): Is \(\mathrm{27 \gt 31}\)? No ✗
• \(\mathrm{x = 8}\): Is \(\mathrm{48 \gt 52}\)? No ✗

This table fails.

Choice B: \(\mathrm{(3,17)}\), \(\mathrm{(5,31)}\), \(\mathrm{(8,52)}\)

• \(\mathrm{x = 3}\): Is \(\mathrm{17 \gt 17}\)? No ✗ (17 equals 17, not greater than)
• \(\mathrm{x = 5}\): Is \(\mathrm{31 \gt 31}\)? No ✗
• \(\mathrm{x = 8}\): Is \(\mathrm{52 \gt 52}\)? No ✗

This table fails.

Choice C: \(\mathrm{(3,21)}\), \(\mathrm{(5,27)}\), \(\mathrm{(8,52)}\)

• \(\mathrm{x = 3}\): Is \(\mathrm{21 \gt 17}\)? Yes ✓
• \(\mathrm{x = 5}\): Is \(\mathrm{27 \gt 31}\)? No ✗
• \(\mathrm{x = 8}\): Is \(\mathrm{52 \gt 52}\)? No ✗

This table fails.

Choice D: \(\mathrm{(3,21)}\), \(\mathrm{(5,35)}\), \(\mathrm{(8,56)}\)

• \(\mathrm{x = 3}\): Is \(\mathrm{21 \gt 17}\)? Yes ✓
• \(\mathrm{x = 5}\): Is \(\mathrm{35 \gt 31}\)? Yes ✓
• \(\mathrm{x = 8}\): Is \(\mathrm{56 \gt 52}\)? Yes ✓

This table works for all pairs!

Answer: D

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak APPLY CONSTRAINTS: Students confuse \(\mathrm{\gt}\) (strictly greater than) with \(\mathrm{\geq}\) (greater than or equal to)

When checking Choice B, they see \(\mathrm{y = 17}\) when the requirement is \(\mathrm{y \gt 17}\), and incorrectly think "17 equals 17, so that works." They don't recognize that strict inequality means the value must be bigger, not equal. The same error occurs with the other boundary values (31 and 52).

This may lead them to select Choice B or cause confusion when multiple tables seem to "work."

Second Most Common Error:

Poor INFER reasoning: Students check only the first coordinate pair or stop after finding one table that partially works

They might check Choice A, see that \(\mathrm{18 \gt 17}\) works, and immediately select it without verifying the other coordinate pairs. They don't realize that ALL pairs in the table must satisfy the inequality.

This may lead them to select Choice A (18, 27, 48).

The Bottom Line:

This problem tests whether students truly understand strict inequalities and can systematically verify that every solution in a set meets the given constraint. The key insight is recognizing that even one failing coordinate pair eliminates an entire table.