Which point is a solution to the given system of inequalities in the xy-plane?y leq x + 7y geq -2x...

1. TRANSLATE the problem requirement

• Given information:
  • System of inequalities: \(\mathrm{y \leq x + 7}\) and \(\mathrm{y \geq -2x - 1}\)
  • Four answer choices: \(\mathrm{A(-14,0)}\), \(\mathrm{B(0,-14)}\), \(\mathrm{C(0,14)}\), \(\mathrm{D(14,0)}\)
• What this tells us: We need to find which point makes BOTH inequalities true

2. INFER the solution approach

• Key insight: A point is a solution to a system of inequalities only if it satisfies ALL inequalities in the system
• Strategy: Test each answer choice by substituting its coordinates into both inequalities

3. SIMPLIFY by testing each choice systematically

Testing Choice A: (-14, 0)

• First inequality: \(\mathrm{0 \leq (-14) + 7}\) → \(\mathrm{0 \leq -7}\) ✗
• Since the first inequality fails, A is not the answer

Testing Choice B: (0, -14)

• First inequality: \(\mathrm{-14 \leq 0 + 7}\) → \(\mathrm{-14 \leq 7}\) ✓
• Second inequality: \(\mathrm{-14 \geq -2(0) - 1}\) → \(\mathrm{-14 \geq -1}\) ✗
• Since the second inequality fails, B is not the answer

Testing Choice C: (0, 14)

• First inequality: \(\mathrm{14 \leq 0 + 7}\) → \(\mathrm{14 \leq 7}\) ✗
• Since the first inequality fails, C is not the answer

Testing Choice D: (14, 0)

• First inequality: \(\mathrm{0 \leq 14 + 7}\) → \(\mathrm{0 \leq 21}\) ✓
• Second inequality: \(\mathrm{0 \geq -2(14) - 1}\) → \(\mathrm{0 \geq -29}\) ✓
• Both inequalities are satisfied!

Answer: D

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER reasoning: Students test only one inequality instead of both, forgetting that "system" means ALL conditions must be satisfied simultaneously.

For example, they might test just \(\mathrm{y \leq x + 7}\) and find that choices B and D both work, then guess between them. This incomplete approach can lead to selecting Choice B (0, -14) if they happen to test that one first.

Second Most Common Error:

Poor SIMPLIFY execution: Students make sign errors when working with negative numbers, especially when substituting negative coordinates or evaluating expressions like \(\mathrm{-2(-14)}\).

A common mistake is calculating \(\mathrm{-2(-14) - 1}\) as \(\mathrm{-28 - 1 = -29}\) instead of \(\mathrm{+28 - 1 = 27}\). This computational error can cause them to incorrectly reject the right answer or accept a wrong answer, leading to confusion and guessing.

The Bottom Line:

This problem requires systematic testing and careful arithmetic. Students must remember that "system" means checking ALL inequalities, and they must be extra careful with negative number operations to avoid computational errors that derail their solution.