Basic Mathematics & Logarithm
Inequalities involving absolute values
Grade 11
Question:
<p>For \(|x-1|\leq 5\), \(|x|\geq 2\), the correct domain of the solution is:</p>
<p>\(x<[-4,-2]\cup[4,6]\)</p>
<p>\(x<[-8,-4]\cup[2,6]\)</p>
<p>\(x<[-4,-2]\cup[2,6]\)</p>
<p>\(x<[-4,-2]\cup[1,5]\)</p>
Step-by-Step Solution
Key Concept: Solve each absolute value inequality separately, then find their intersection by analyzing the number line regions: |x-1|≤5 gives -4≤x≤6, and |x|≥2 gives x≤-2 or x≥2.
<p><strong>Step 1: Solve |x-1|≤5</strong></p><p>-5 ≤ x-1 ≤ 5</p><p>-4 ≤ x ≤ 6</p><p><strong>Step 2: Solve |x|≥2</strong></p><p>x ≤ -2 or x ≥ 2</p><p><strong>Step 3: Find intersection of both conditions</strong></p><p>We need: [-4, 6] AND ((-∞, -2] ∪ [2, ∞))</p><p>• For x ∈ [-4, -2]: satisfies both conditions</p><p>• For x ∈ (-2, 2): violates |x|≥2, excluded</p><p>• For x ∈ [2, 6]: satisfies both conditions</p><p><strong>Step 4: Combine regions</strong></p><p>Domain = [-4, -2] ∪ [2, 6]</p><p>∴ Answer: <strong>x ∈ [-4, -2] ∪ [2, 6]</strong> (Option C)</p>
Correct Answer: C