y leq |x| + 7 y geq -2x - 1 Which point (x, y) is a solution to the given...

1. INFER the solution strategy

• Given information:
  • System with two inequalities: \(\mathrm{y \leq |x| + 7}\) and \(\mathrm{y \geq -2x - 1}\)
  • Four answer choices with specific coordinate points
• What this tells us: We need to find which point satisfies BOTH inequalities simultaneously by substituting each point's coordinates

2. TRANSLATE each answer choice systematically

For each point (x, y), substitute into both inequalities:

• Point A: \(\mathrm{(-14, 0)}\) means \(\mathrm{x = -14, y = 0}\)
• Point B: \(\mathrm{(0, -14)}\) means \(\mathrm{x = 0, y = -14}\)
• Point C: \(\mathrm{(0, 14)}\) means \(\mathrm{x = 0, y = 14}\)
• Point D: \(\mathrm{(14, 0)}\) means \(\mathrm{x = 14, y = 0}\)

3. SIMPLIFY calculations for each point

Testing Point A: (-14, 0)

• First inequality: \(\mathrm{0 \leq |-14| + 7}\)
  \(\mathrm{= 14 + 7}\)
  \(\mathrm{= 21}\) ✓
• Second inequality: \(\mathrm{0 \geq -2(-14) - 1}\)
  \(\mathrm{= 28 - 1}\)
  \(\mathrm{= 27}\) ✗

Point A fails the second inequality.

Testing Point B: (0, -14)

• First inequality: \(\mathrm{-14 \leq |0| + 7}\)
  \(\mathrm{= 0 + 7}\)
  \(\mathrm{= 7}\) ✓
• Second inequality: \(\mathrm{-14 \geq -2(0) - 1}\)
  \(\mathrm{= -1}\) ✗

Point B fails the second inequality.

Testing Point C: (0, 14)

• First inequality: \(\mathrm{14 \leq |0| + 7}\)
  \(\mathrm{= 7}\) ✗

Point C fails the first inequality.

Testing Point D: (14, 0)

• First inequality: \(\mathrm{0 \leq |14| + 7}\)
  \(\mathrm{= 14 + 7}\)
  \(\mathrm{= 21}\) ✓
• Second inequality: \(\mathrm{0 \geq -2(14) - 1}\)
  \(\mathrm{= -28 - 1}\)
  \(\mathrm{= -29}\) ✓

Point D satisfies both inequalities!

Answer: D

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Students make calculation errors when evaluating absolute values or handling negative numbers in the linear inequality.

For example, when checking point A, they might calculate \(\mathrm{-2(-14) - 1}\) incorrectly as \(\mathrm{-28 - 1 = -29}\) instead of \(\mathrm{28 - 1 = 27}\), leading them to think point A works when it doesn't. This confusion causes them to select the first point that seems to work rather than systematically checking all options.

Second Most Common Error:

Incomplete INFER reasoning: Students may check only the first inequality for each point, assuming that's sufficient to determine the answer.

This partial approach might lead them to think that points A, B, and D all work (since they all satisfy the first inequality), causing confusion and potentially leading them to select Choice A (-14, 0) as it appears first in their partial checking.

The Bottom Line:

Systems of inequalities require patience and systematic checking. The key insight is that BOTH conditions must be satisfied simultaneously - one "yes" and one "no" means the point doesn't work, period.