Question:y lt x^2x + y gt 2Which of the following ordered pairs (x, y) satisfies the system of inequalities above?\(\mathrm{(-1,...

1. TRANSLATE the problem information

• Given system:
  • \(\mathrm{y \lt x^2}\)
  • \(\mathrm{x + y \gt 2}\)
• Given choices: Four ordered pairs (x, y) to test
• Need to find: Which ordered pair satisfies BOTH inequalities

2. INFER the approach

• For a system of inequalities, both inequalities must be true simultaneously
• Strategy: Test each ordered pair by substituting the x and y values into both inequalities
• If any inequality is false, that ordered pair is not a solution

3. TRANSLATE each ordered pair and test systematically

Testing Choice A: (-1, 2)

• First inequality: \(\mathrm{y \lt x^2}\) becomes \(\mathrm{2 \lt (-1)^2 = 2 \lt 1}\)
• Since \(\mathrm{2 \lt 1}\) is false, Choice A fails immediately

Testing Choice B: (1, 0)

• First inequality: \(\mathrm{y \lt x^2}\) becomes \(\mathrm{0 \lt 1^2 = 0 \lt 1}\) ✓ (true)
• Second inequality: \(\mathrm{x + y \gt 2}\) becomes \(\mathrm{1 + 0 \gt 2 = 1 \gt 2}\)
• Since \(\mathrm{1 \gt 2}\) is false, Choice B fails

Testing Choice C: (1, 3)

• First inequality: \(\mathrm{y \lt x^2}\) becomes \(\mathrm{3 \lt 1^2 = 3 \lt 1}\)
• Since \(\mathrm{3 \lt 1}\) is false, Choice C fails immediately

Testing Choice D: (2, 1)

• First inequality: \(\mathrm{y \lt x^2}\) becomes \(\mathrm{1 \lt 2^2 = 1 \lt 4}\) ✓ (true)
• Second inequality: \(\mathrm{x + y \gt 2}\) becomes \(\mathrm{2 + 1 \gt 2 = 3 \gt 2}\) ✓ (true)

4. APPLY CONSTRAINTS to identify the solution

• Only Choice D satisfies both inequalities
• Both \(\mathrm{1 \lt 4}\) and \(\mathrm{3 \gt 2}\) are true statements

Answer: D

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students substitute values incorrectly or make sign errors, especially with negative numbers like \(\mathrm{(-1)^2}\). They might calculate \(\mathrm{(-1)^2}\) as -1 instead of +1, leading them to think \(\mathrm{2 \lt -1}\) is true when evaluating Choice A.

This may lead them to incorrectly select Choice A ((-1, 2)) because they think it satisfies the first inequality.

Second Most Common Error:

Poor APPLY CONSTRAINTS reasoning: Students test only one inequality instead of both, or they don't understand that BOTH must be satisfied simultaneously. They might find that Choice B satisfies \(\mathrm{y \lt x^2}\) and stop there, not checking the second inequality.

This may lead them to select Choice B ((1, 0)) because they only verified one constraint.

The Bottom Line:

This problem requires careful systematic testing and understanding that a system means ALL conditions must be met. Students often rush through calculations or don't fully grasp the "system" concept, leading to incomplete analysis.