Relations & Functions
Inequalities involving modulus
Grade None
Question:
<p>Given that \( |x - 1| \leq 5 \) and \( |x| \geq 2 \), the solution set is:</p>
<p>\( x \in [-4, 6] \)</p>
<p>\( x \in [-2, 2] \)</p>
<p>\( x \in [-4, -2] \cup [2, 6] \)</p>
<p>\( x \in (-\infty, -4] \cup [6, \infty) \)</p>
Step-by-Step Solution
Key Concept: Solve each absolute value inequality separately, then find the intersection of both solution sets. |x - 1| ≤ 5 gives -4 ≤ x ≤ 6, while |x| ≥ 2 gives x ≤ -2 or x ≥ 2.
<p><strong>Step 1:</strong> Solve |x - 1| ≤ 5</p><p>This means: -5 ≤ x - 1 ≤ 5</p><p>Adding 1: -4 ≤ x ≤ 6</p><p><strong>Step 2:</strong> Solve |x| ≥ 2</p><p>This means: x ≤ -2 or x ≥ 2</p><p><strong>Step 3:</strong> Find intersection of both conditions</p><p>We need x ∈ [-4, 6] AND (x ≤ -2 or x ≥ 2)</p><p>Intersection with x ≤ -2: [-4, -2]</p><p>Intersection with x ≥ 2: [2, 6]</p><p><strong>∴ Answer: C is [-4, -2] ∪ [2, 6]</strong></p>
Correct Answer: C