Basic Mathematics & Logarithm
Modulus Inequalities
Grade 11
Question:
<p>Complete set of values of \(x\) satisfying inequality \(\||x-1|-5| < 2x - 5\)</p>
<p>\((5/2, \infty)\)</p>
<p>\((11/3, \infty)\)</p>
<p>\((-1, \infty)\)</p>
<p>\((-\infty, 1/3)\)</p>
Step-by-Step Solution
Key Concept: Recognize that nested absolute values must be unwrapped from outside to inside by considering both positive and negative cases at each layer. The critical insight is that |A| < B (where B > 0) is equivalent to -B < A < B.
<p><strong>Step 1: Remove outer absolute value</strong></p><p>||x-1|-5| < 3 means: -3 < |x-1|-5 < 3</p><p><strong>Step 2: Solve the compound inequality</strong></p><p>Adding 5 to all parts: 2 < |x-1| < 8</p><p><strong>Step 3: Split based on |x-1| < 8</strong></p><p>From |x-1| < 8: -8 < x-1 < 8, so -7 < x < 9</p><p><strong>Step 4: Split based on |x-1| > 2</strong></p><p>From |x-1| > 2: either x-1 > 2 or x-1 < -2</p><p>This gives: x > 3 or x < -1</p><p><strong>Step 5: Find intersection</strong></p><p>Combine both conditions:</p><p>• For x > 3: need 3 < x < 9</p><p>• For x < -1: need -7 < x < -1</p><p>∴ Answer: x ∈ (-7, -1) ∪ (3, 9), which is <strong>B</strong></p>
Correct Answer: B