Which of the following ordered pairs \((\mathrm{x}, \mathrm{y})\) satisfies the inequality |2x + 3y| gt 6?\((1, 2)\)\((2, -1)\)\((-1, 3)\)I onlyII...

1. TRANSLATE the problem requirements

• Given information:
  • Inequality: \(|2x + 3y| \gt 6\)
  • Three ordered pairs to test: \((1, 2)\), \((2, -1)\), \((-1, 3)\)
• What this tells us: We need to substitute each pair and check which ones make the inequality true

2. SIMPLIFY each substitution systematically

Testing pair I: (1, 2)

• Substitute: \(|2(1) + 3(2)| \gt 6\)
• Calculate inside absolute value: \(|2 + 6| = |8| = 8\)
• Check inequality: \(8 \gt 6\) ✓ (True)

Testing pair II: (2, -1)

• Substitute: \(|2(2) + 3(-1)| \gt 6\)
• Calculate inside absolute value: \(|4 - 3| = |1| = 1\)
• Check inequality: \(1 \gt 6\) ✗ (False)

Testing pair III: (-1, 3)

• Substitute: \(|2(-1) + 3(3)| \gt 6\)
• Calculate inside absolute value: \(|-2 + 9| = |7| = 7\)
• Check inequality: \(7 \gt 6\) ✓ (True)

3. APPLY CONSTRAINTS to determine the final answer

• Pairs I and III satisfy the inequality
• Pair II does not satisfy the inequality
• Looking at answer choices: "I and III only" corresponds to choice D

Answer: D

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make arithmetic errors when evaluating expressions inside the absolute value bars, especially with negative numbers.

For example, when testing pair II (2, -1), they might calculate \(2(2) + 3(-1)\) as \(4 + 3 = 7\) instead of \(4 - 3 = 1\). This would give \(|7| = 7\), making them think \(7 \gt 6\) is true, leading them to incorrectly include pair II in their answer. This may lead them to select Choice C (I and II only).

Second Most Common Error:

Poor TRANSLATE reasoning: Students might misinterpret what the absolute value inequality means or forget to check all three pairs systematically.

Some students might only test one or two pairs and make assumptions about the others, or they might confuse the inequality direction (thinking they need values less than 6). This leads to confusion and guessing among the available choices.

The Bottom Line:

Success requires careful arithmetic with positive and negative numbers inside absolute value expressions, plus systematic testing of each given option. The absolute value eliminates sign concerns, but students must still handle the underlying arithmetic correctly.