3x - 4 gt 2y + 1For which of the following tables are all the values of x and their...

1. SIMPLIFY the inequality to isolate one variable

• Given: \(\mathrm{3x - 4 \gt 2y + 1}\)
• SIMPLIFY by collecting like terms and isolating y:
  • \(\mathrm{3x - 4 \gt 2y + 1}\)
  • \(\mathrm{3x - 5 \gt 2y}\)
  • \(\mathrm{\frac{3x - 5}{2} \gt y}\)
  • Therefore: \(\mathrm{y \lt \frac{3x - 5}{2}}\)

2. APPLY CONSTRAINTS by finding boundary values

• For each x-value in the tables, calculate what y must be less than:
  • When \(\mathrm{x = 3}\): \(\mathrm{y \lt \frac{9 - 5}{2} = 2}\)
  • When \(\mathrm{x = 5}\): \(\mathrm{y \lt \frac{15 - 5}{2} = 5}\)
  • When \(\mathrm{x = 7}\): \(\mathrm{y \lt \frac{21 - 5}{2} = 8}\)

3. APPLY CONSTRAINTS systematically to each table

• Check every coordinate pair in each table:

Choice A: All points fail because y equals the boundary (not less than)
Choice B: All points satisfy: \(\mathrm{1 \lt 2}\) ✓, \(\mathrm{4 \lt 5}\) ✓, \(\mathrm{7 \lt 8}\) ✓
Choice C: Fails at \(\mathrm{(5,5)}\) because 5 is not less than 5
Choice D: Fails at \(\mathrm{(3,2)}\) because 2 is not less than 2

Answer: B

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak APPLY CONSTRAINTS reasoning: Students check only some coordinate pairs rather than ALL pairs in each table, or they misunderstand that every single point must satisfy the inequality.

They might see that most points in Choice C or D work and assume that's sufficient, leading them to select Choice C or Choice D without verifying all coordinate pairs.

Second Most Common Error:

Poor SIMPLIFY execution: Students make errors when rearranging the inequality, particularly with signs or fraction operations.

Common mistakes include forgetting to reverse inequality signs (when dividing by negative numbers, though not applicable here) or arithmetic errors in the rearrangement process. This leads to wrong boundary conditions and incorrect evaluation of the coordinate pairs, causing confusion and potentially random answer selection.

The Bottom Line:

This problem requires both careful algebraic manipulation and systematic checking. Success depends on properly isolating the variable and then methodically verifying that ALL coordinate pairs in a table satisfy the constraint—partial satisfaction isn't sufficient.