The point \((6, 8)\) in the xy-plane is a solution to which of the following systems of inequalities? y gt...

1. TRANSLATE the problem information

• Given information:
  • Point: \((6, 8)\), so \(\mathrm{x = 6}\) and \(\mathrm{y = 8}\)
  • Need to find which system this point satisfies
• What this tells us: We need to substitute these values into each system

2. INFER the solution approach

• For a point to be a solution to a system of inequalities, it must satisfy ALL inequalities in that system
• Strategy: Test each option by substituting \(\mathrm{x = 6}\) and \(\mathrm{y = 8}\)

3. SIMPLIFY by testing each system systematically

Option A: \(\mathrm{y \gt x}\) and \(\mathrm{y \gt 0}\)

• First inequality: \(\mathrm{8 \gt 6}\)? ✓ True
• Second inequality: \(\mathrm{8 \gt 0}\)? ✓ True
• Result: Both satisfied ✓

Option B: \(\mathrm{y \lt x}\) and \(\mathrm{y \gt 0}\)

• First inequality: \(\mathrm{8 \lt 6}\)? ✗ False
• Result: System fails (don't need to check second)

Option C: \(\mathrm{y \gt x}\) and \(\mathrm{y \lt 0}\)

• First inequality: \(\mathrm{8 \gt 6}\)? ✓ True
• Second inequality: \(\mathrm{8 \lt 0}\)? ✗ False
• Result: System fails

Option D: \(\mathrm{y \lt x}\) and \(\mathrm{y \lt 0}\)

• First inequality: \(\mathrm{8 \lt 6}\)? ✗ False
• Result: System fails (don't need to check second)

Answer: A

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students may think that satisfying just ONE inequality in a system is enough, rather than understanding that ALL inequalities must be satisfied.

For example, looking at Option C, they see that \(\mathrm{8 \gt 6}\) is true and might select C without checking that \(\mathrm{8 \lt 0}\) is false. This leads them to select Choice C instead of the correct answer.

Second Most Common Error:

Poor TRANSLATE reasoning: Students might confuse the inequality symbols, especially mixing up which direction \(\mathrm{\gt}\) and \(\mathrm{\lt}\) point.

They might read "\(\mathrm{y \gt x}\)" as "y is less than x" or vice versa, leading to incorrect evaluations. This confusion can cause them to eliminate the correct answer and select any of the wrong choices, or leads to confusion and guessing.

The Bottom Line:

The key insight is that "solution to a system" means the point must work for EVERY inequality, not just some of them. Students must check each inequality carefully and ensure all are satisfied before selecting an answer.