The point \((8, 2)\) in the xy-plane is a solution to which of the following systems of inequalities?

1. TRANSLATE the problem information

• Given information:
  • Point: \((8, 2)\)
  • This means \(x = 8\) and \(y = 2\)
  • Need to find which system of inequalities this point satisfies

2. INFER the approach

• For a point to be a solution to a system of inequalities, it must make BOTH inequalities true
• Strategy: Test each system by substituting \(x = 8\) and \(y = 2\) into both inequalities

3. APPLY CONSTRAINTS to test each system systematically

Choice A: \(x \gt 0\) and \(y \gt 0\)

• Substitute: Is \(8 \gt 0\)? Yes ✓
• Substitute: Is \(2 \gt 0\)? Yes ✓
• Both satisfied → This could be our answer

Choice B: \(x \gt 0\) and \(y \lt 0\)

• Substitute: Is \(8 \gt 0\)? Yes ✓
• Substitute: Is \(2 \lt 0\)? No ✗
• Second inequality fails → Not the answer

Choice C: \(x \lt 0\) and \(y \gt 0\)

• Substitute: Is \(8 \lt 0\)? No ✗
• Substitute: Is \(2 \gt 0\)? Yes ✓
• First inequality fails → Not the answer

Choice D: \(x \lt 0\) and \(y \lt 0\)

• Substitute: Is \(8 \lt 0\)? No ✗
• Substitute: Is \(2 \lt 0\)? No ✗
• Both fail → Not the answer

Answer: A

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students might think they only need to satisfy ONE inequality instead of BOTH in a system.

For example, looking at Choice B (\(x \gt 0\) and \(y \lt 0\)), they correctly see that \(8 \gt 0\) is true, but don't carefully check that \(2 \lt 0\) is false. They might rush and select Choice B because "one part works."

This may lead them to select Choice B (\(x \gt 0\), \(y \lt 0\)) or cause confusion when multiple choices seem to "partially work."

Second Most Common Error:

Missing conceptual knowledge: Students might confuse which coordinate is x and which is y in the ordered pair \((8, 2)\).

If they think \(x = 2\) and \(y = 8\), they would get different results when testing the inequalities. For instance, testing Choice A with wrong coordinates: "Is \(2 \gt 0\)? Yes. Is \(8 \gt 0\)? Yes." They'd still get Choice A, but this error could cause problems in other similar questions.

The Bottom Line:

This problem tests whether students understand that systems require ALL conditions to be met simultaneously. The key insight is being systematic about checking both inequalities completely before moving to the next choice.