y leq x y leq -x Which of the following ordered pairs is a solution to the system of inequalities...

1. TRANSLATE the problem requirements

• Given system:
  • \(\mathrm{y \leq x}\)
  • \(\mathrm{y \leq -x}\)
• Need to find: Which ordered pair satisfies BOTH inequalities

2. APPLY CONSTRAINTS to understand the solution criteria

• A valid solution must make both inequalities true
• If an ordered pair fails either inequality, it's not a solution to the system

3. CONSIDER ALL CASES by testing each choice systematically

Choice A: (1,0)

• First inequality: \(\mathrm{y \leq x}\) becomes \(0 \leq 1\) ✓
• Second inequality: \(\mathrm{y \leq -x}\) becomes \(0 \leq -1\) ✗
• Conclusion: Not a solution

Choice B: (−1,0)

• First inequality: \(\mathrm{y \leq x}\) becomes \(0 \leq -1\) ✗
• Second inequality: \(\mathrm{y \leq -x}\) becomes \(0 \leq -(-1) = 0 \leq 1\) ✓
• Conclusion: Not a solution

Choice C: (0,1)

• First inequality: \(\mathrm{y \leq x}\) becomes \(1 \leq 0\) ✗
• Second inequality: \(\mathrm{y \leq -x}\) becomes \(1 \leq 0\) ✗
• Conclusion: Not a solution

Choice D: (0,−1)

• First inequality: \(\mathrm{y \leq x}\) becomes \(-1 \leq 0\) ✓
• Second inequality: \(\mathrm{y \leq -x}\) becomes \(-1 \leq 0\) ✓
• Conclusion: This is our solution!

Answer: D. (0,−1)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak APPLY CONSTRAINTS reasoning: Students find an ordered pair that satisfies one inequality and assume it's the answer without checking the second inequality.

For example, they see that (1,0) makes \(\mathrm{y \leq x}\) true (since \(0 \leq 1\)) and immediately select Choice A without verifying that \(0 \leq -1\) is false. This incomplete checking leads them to select Choice A (1,0).

Second Most Common Error:

Poor CONSIDER ALL CASES execution: Students get confused about the meaning of \(\mathrm{y \leq -x}\) and incorrectly evaluate the second inequality.

They might think \(\mathrm{y \leq -x}\) means the same as \(\mathrm{y \geq x}\), or they make sign errors when substituting negative values. This conceptual confusion about inequality manipulation causes them to get stuck and guess randomly.

The Bottom Line:

Systems of inequalities require methodical verification - every inequality must be satisfied. The key insight is that "solution to the system" means satisfying ALL conditions, not just one.