|x - 5| gt 82x + y lt 20The point \((\mathrm{x}, 2)\) is a solution to the system of inequalities...

1. TRANSLATE the point condition into usable information

• Given: Point \((x, 2)\) is a solution to the system
• This means when \(y = 2\), both inequalities must be satisfied
• Substitute \(y = 2\) into both inequalities

2. SIMPLIFY the linear inequality

• Start with: \(2x + y \lt 20\)
• Substitute \(y = 2\): \(2x + 2 \lt 20\)
• Subtract 2: \(2x \lt 18\)
• Divide by 2: \(x \lt 9\)

3. SIMPLIFY the absolute value inequality

• Start with: \(|x - 5| \gt 8\)
• This means the expression inside can be positive or negative:
  • Case 1: \(x - 5 \gt 8\), so \(x \gt 13\)
  • Case 2: \(x - 5 \lt -8\), so \(x \lt -3\)
• Combined: \(x \gt 13\) OR \(x \lt -3\)

4. INFER the intersection of solution sets

• For the system to be satisfied: \((x \gt 13 \text{ OR } x \lt -3) \text{ AND } x \lt 9\)
• Since \(x \gt 13\) AND \(x \lt 9\) is impossible (no number can be both greater than 13 and less than 9)
• We need: \(x \lt -3\) AND \(x \lt 9\)
• This simplifies to: \(x \lt -3\)

5. APPLY CONSTRAINTS to select from answer choices

• Test each choice against \(x \lt -3\):
  • (A) -5: Yes, \(-5 \lt -3\) ✓
  • (B) -1: No, \(-1 \gt -3\) ✗
  • (C) 3: No, \(3 \gt -3\) ✗
  • (D) 15: No, \(15 \gt -3\) ✗

Answer: A

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students correctly solve each inequality individually but fail to properly find their intersection. They might think the solution is '\(x \gt 13\) OR \(x \lt -3\) OR \(x \lt 9\)' instead of recognizing that ALL conditions must be satisfied simultaneously. This leads to confusion about which values actually work, causing them to guess or select Choice D (15) because it satisfies one of the conditions.

Second Most Common Error:

Inadequate SIMPLIFY execution: Students make errors when solving the absolute value inequality, perhaps forgetting that \(|x - 5| \gt 8\) creates two separate cases, or incorrectly setting up the cases as \(x - 5 \gt 8\) AND \(x - 5 \lt -8\). This leads them to get an incorrect solution set and potentially select Choice B (-1) or Choice C (3).

The Bottom Line:

This problem tests whether students understand that systems of inequalities require intersection (AND logic), not union (OR logic), and whether they can correctly handle the two-case nature of absolute value inequalities. The key insight is recognizing that when solution sets conflict, only the overlapping region is valid.