y lt -4x + 4 Which point \(\mathrm{(x, y)}\) is a solution to the given inequality in the xy-plane?...

1. TRANSLATE the problem requirements

• Given: The inequality \(\mathrm{y \lt -4x + 4}\)
• Need to find: Which point (x, y) makes this inequality true
• Strategy: Test each answer choice by substituting coordinates

2. TRANSLATE and test each point systematically

Testing Choice A: (−4,0)

• Substitute \(\mathrm{x = -4}\) and \(\mathrm{y = 0}\) into \(\mathrm{y \lt -4x + 4}\):
• \(\mathrm{0 \lt -4(-4) + 4}\)

3. SIMPLIFY the arithmetic for Choice A

• \(\mathrm{0 \lt -4(-4) + 4}\)
• \(\mathrm{0 \lt 16 + 4}\) [since \(\mathrm{-4 \times -4 = +16}\)]
• \(\mathrm{0 \lt 20}\) ✓ This is TRUE

4. TRANSLATE and test Choice B: (0,5)

• Substitute \(\mathrm{x = 0}\) and \(\mathrm{y = 5}\):
• \(\mathrm{5 \lt -4(0) + 4}\)

5. SIMPLIFY for Choice B

• \(\mathrm{5 \lt 0 + 4}\)
• \(\mathrm{5 \lt 4}\) ✗ This is FALSE

6. TRANSLATE and test Choice C: (2,1)

• Substitute \(\mathrm{x = 2}\) and \(\mathrm{y = 1}\):
• \(\mathrm{1 \lt -4(2) + 4}\)

7. SIMPLIFY for Choice C

• \(\mathrm{1 \lt -8 + 4}\)
• \(\mathrm{1 \lt -4}\) ✗ This is FALSE

8. TRANSLATE and test Choice D: (2,−1)

• Substitute \(\mathrm{x = 2}\) and \(\mathrm{y = -1}\):
• \(\mathrm{-1 \lt -4(2) + 4}\)

9. SIMPLIFY for Choice D

• \(\mathrm{-1 \lt -8 + 4}\)
• \(\mathrm{-1 \lt -4}\) ✗ This is FALSE

Answer: A. (−4,0)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Sign errors when multiplying negative numbers

Students often calculate \(\mathrm{-4(-4)}\) as \(\mathrm{-16}\) instead of \(\mathrm{+16}\), making them think \(\mathrm{0 \lt -16 + 4}\), or \(\mathrm{0 \lt -12}\), which is false. This leads them to incorrectly eliminate Choice A and then guess among the remaining options.

Second Most Common Error:

Poor TRANSLATE reasoning: Misunderstanding which coordinate represents which variable

Some students mix up x and y coordinates, substituting the values in the wrong order. For example, they might test point (−4,0) by substituting \(\mathrm{x = 0}\) and \(\mathrm{y = -4}\), leading to incorrect calculations and wrong answer selection.

The Bottom Line:

This problem requires careful attention to arithmetic with negative numbers and systematic substitution of coordinates. Success depends on methodically checking each option rather than trying shortcuts.