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

1. TRANSLATE the problem information

• Given information:
  • System of inequalities: \(2\mathrm{x} - \mathrm{y} \leq 5\) and \(\mathrm{y} \lt \mathrm{x} - 2\)
  • Four coordinate points to test: \((-2, 5)\), \((0, -3)\), \((2, 1)\), \((4, -1)\)
• What this tells us: We need to find which point makes both inequalities true

2. INFER the testing approach

• For a point to be a solution to the system, it must satisfy BOTH inequalities
• We'll substitute each point's x and y values into both expressions
• Only check the second inequality if the first one passes

3. SIMPLIFY by testing each point systematically

Testing (A) (-2, 5):

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

Testing (B) (0, -3):

• First: \(2(0) - (-3) = 3 \leq 5\) ✓
• Second: \(-3 \lt 0 - 2\) → \(-3 \lt -2\) ✓
• Result: This is our solution!

Testing (C) (2, 1):

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

Testing (D) (4, -1):

• First: \(2(4) - (-1) = 9 \leq 5\) ✗
• Result: Not a solution (no need to check second)

Answer: B

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students test only one inequality instead of both, or they think satisfying one inequality is sufficient.

For example, they might check that point (A) satisfies the first inequality (-9 ≤ 5) and conclude it's correct without checking the second inequality. This leads them to select Choice A ((-2, 5)).

Second Most Common Error:

Poor SIMPLIFY execution: Students make sign errors when working with negative numbers, especially when subtracting negative values or comparing negative numbers.

Common mistakes include thinking 2(0) - (-3) = -3 instead of +3, or believing that -3 < -4 is true. These calculation errors can lead to confusion and guessing among the remaining choices.

The Bottom Line:

Systems of inequalities require systematic checking of ALL conditions. The key insight is that "solution to the system" means the point works for every single inequality—there's no partial credit in systems!