Which of the following ordered pairs \((\mathrm{x}, \mathrm{y})\) satisfies the inequality 5x - 3y lt 4? \((1, 1)\) \((2, 5)\)...

1. TRANSLATE the problem setup

• Given information:
  • Inequality: \(5\mathrm{x} - 3\mathrm{y} \lt 4\)
  • Three ordered pairs to test: \((1,1)\), \((2,5)\), \((3,2)\)
  • Need to determine which satisfy the inequality

• What this tells us: We need to substitute each pair's x and y values into the left side of the inequality and check if the result is less than 4.

2. SIMPLIFY each substitution systematically

Testing pair I: (1, 1)

• Substitute \(\mathrm{x} = 1, \mathrm{y} = 1\):

\(5(1) - 3(1) = 5 - 3 = 2\)

• Check inequality: Is \(2 \lt 4\)? Yes ✓

Testing pair II: (2, 5)

• Substitute \(\mathrm{x} = 2, \mathrm{y} = 5\):

\(5(2) - 3(5) = 10 - 15 = -5\)

• Check inequality: Is \(-5 \lt 4\)? Yes ✓

Testing pair III: (3, 2)

• Substitute \(\mathrm{x} = 3, \mathrm{y} = 2\):

\(5(3) - 3(2) = 15 - 6 = 9\)

• Check inequality: Is \(9 \lt 4\)? No ✗

3. APPLY CONSTRAINTS to select the final answer

• Pairs I and II satisfy the inequality
• Looking at answer choices, "I and II only" corresponds to choice C

Answer: C. I and II only

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make arithmetic errors during substitution, particularly with negative results like in pair II where \(5(2) - 3(5) = 10 - 15 = -5\). They might calculate this as 5 instead of -5, leading them to think \(-5 \lt 4\) is false when it's actually true.

This may lead them to select Choice A (I only) or Choice D (I and III only) depending on which other errors they make.

Second Most Common Error:

Poor APPLY CONSTRAINTS reasoning: Students correctly evaluate all three pairs but get confused when matching their results to the Roman numeral combinations in the answer choices. They might know that pairs (1,1) and (2,5) work but select the wrong letter choice.

This leads to confusion and guessing among the available options.

The Bottom Line:

This problem tests careful arithmetic execution and systematic checking. Success depends on methodically substituting each pair, performing accurate calculations (especially with negative numbers), and correctly interpreting the answer format.