y leq 3x + 1x - y gt 1Which of the following ordered pairs (x, y) satisfies the system of...

1. TRANSLATE the problem information

• Given system of inequalities:
  • \(\mathrm{y \leq 3x + 1}\)
  • \(\mathrm{x - y \gt 1}\)
• Four ordered pairs to test: (-2, -1), (-1, 3), (1, 5), (2, -1)
• What this tells us: We need to find which point makes BOTH inequalities true

2. INFER the approach

• Since we have a system of inequalities, a solution must satisfy every inequality in the system
• Strategy: Test each answer choice by substituting the x and y values into both inequalities
• If ANY inequality is false, that point is not a solution

3. TRANSLATE and SIMPLIFY each test

Choice A: (-2, -1)
Substitute \(\mathrm{x = -2}\), \(\mathrm{y = -1}\) into first inequality:

\(\mathrm{-1 \leq 3(-2) + 1}\)
\(\mathrm{-1 \leq -5}\)

This is FALSE, so Choice A fails immediately.

Choice B: (-1, 3)
Substitute \(\mathrm{x = -1}\), \(\mathrm{y = 3}\) into first inequality:

\(\mathrm{3 \leq 3(-1) + 1}\)
\(\mathrm{3 \leq -2}\)

This is FALSE, so Choice B fails immediately.

Choice C: (1, 5)
Substitute \(\mathrm{x = 1}\), \(\mathrm{y = 5}\) into first inequality:

\(\mathrm{5 \leq 3(1) + 1}\)
\(\mathrm{5 \leq 4}\)

This is FALSE, so Choice C fails immediately.

Choice D: (2, -1)
First inequality:

\(\mathrm{-1 \leq 3(2) + 1}\)
\(\mathrm{-1 \leq 7}\) ✓ TRUE

Second inequality:

\(\mathrm{2 - (-1) \gt 1}\)
\(\mathrm{3 \gt 1}\) ✓ TRUE

Both inequalities are satisfied!

4. APPLY CONSTRAINTS to select final answer

Since (2, -1) is the only point that satisfies both inequalities, it's our solution.

Answer: D. (2, -1)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Making sign errors when substituting negative numbers or when performing operations with negatives.

For example, when testing (-2, -1), a student might incorrectly evaluate:

\(\mathrm{3(-2) + 1 = -6 + 1 = -5}\), but then mistakenly think "\(\mathrm{-1 \leq -5}\)" is true because they confuse the direction of inequalities with negative numbers.

This may lead them to select Choice A ((-2, -1)) incorrectly.

Second Most Common Error:

Insufficient INFER reasoning: Only checking one inequality instead of verifying that both must be satisfied.

A student might substitute a point into just the first inequality, see that it works, and immediately select that answer without checking the second inequality. This rushed approach misses the fundamental requirement that ALL inequalities in a system must be satisfied.

This leads to confusion and potentially selecting any choice that satisfies only one inequality.

The Bottom Line:

This problem tests careful substitution arithmetic combined with understanding that systems require ALL conditions to be met. The key insight is being methodical—check every inequality for every choice until you find the one that works completely.