Question: A system of inequalities is given by 2x - y leq 5 and y lt x - 2. Which...

1. TRANSLATE the problem requirements

• Given information:
  • System of inequalities: \(\mathrm{2x - y \leq 5}\) and \(\mathrm{y \lt x - 2}\)
  • Four coordinate pairs to test: \(\mathrm{A(-2, 5)}\), \(\mathrm{B(0, -3)}\), \(\mathrm{C(2, 1)}\), \(\mathrm{D(4, -1)}\)
  • Need to find which point satisfies BOTH inequalities

2. INFER the testing strategy

• Since this is a system, a solution must satisfy both inequalities simultaneously
• Test each coordinate pair by substituting x and y values into both inequalities
• If any inequality is false, that point is not a solution

3. TRANSLATE and test each choice systematically

Choice A: (-2, 5)

• First inequality: \(\mathrm{2(-2) - 5 = -9 \leq 5}\) ✓
• Second inequality: \(\mathrm{5 \lt (-2) - 2}\) → \(\mathrm{5 \lt -4}\) ✗
• Not a solution

Choice B: (0, -3)

• First inequality: \(\mathrm{2(0) - (-3) = 3 \leq 5}\) ✓
• Second inequality: \(\mathrm{-3 \lt 0 - 2}\) → \(\mathrm{-3 \lt -2}\) ✓
• This works!

4. SIMPLIFY the remaining calculations to verify

Choice C: (2, 1)

• First inequality: \(\mathrm{2(2) - 1 = 3 \leq 5}\) ✓
• Second inequality: \(\mathrm{1 \lt 2 - 2}\) → \(\mathrm{1 \lt 0}\) ✗
• Not a solution

Choice D: (4, -1)

• First inequality: \(\mathrm{2(4) - (-1) = 9 \leq 5}\) ✗
• Since the first inequality fails, this cannot be a solution

5. APPLY CONSTRAINTS to identify the final answer

• Only choice B satisfies both conditions of the system
• A solution to a system must work for ALL inequalities

Answer: \(\mathrm{B (0, -3)}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak APPLY CONSTRAINTS reasoning: Students test only the first inequality and stop when they find it's satisfied, forgetting that systems require ALL conditions to be met.

For example, they might test choice A, see that \(\mathrm{-9 \leq 5}\) is true, and immediately select it without checking the second inequality. This leads them to select Choice \(\mathrm{A (-2, 5)}\) even though it fails the second condition.

Second Most Common Error:

Poor SIMPLIFY execution: Students make sign errors when substituting negative coordinates, especially with expressions like \(\mathrm{2x - (-3)}\) or \(\mathrm{y \lt x - 2}\) when x is negative.

This commonly happens with choice B where \(\mathrm{2(0) - (-3)}\) becomes \(\mathrm{2(0) - 3 = -3}\) instead of \(\mathrm{2(0) + 3 = 3}\), making them incorrectly reject the right answer and resort to guessing.

The Bottom Line:

Systems of inequalities problems require systematic testing and careful attention to ALL conditions. Students who rush through or don't double-check their arithmetic often miss the correct solution.