<p>The solution set of the inequality \(\dfrac{|x+2|-x}{x} < 2\) is</p>
Step-by-Step Solution
Key Concept: Solve by analyzing the absolute value expression by cases (x ≥ -2 and x < -2), then consider the sign of denominator x. The critical insight is that the denominator x ≠ 0 creates a boundary, and |x+2| behaves differently on either side of x = -2.
<p><strong>Step 1:</strong> Identify domain: x ≠ 0</p><p><strong>Step 2:</strong> Split into cases based on |x+2|</p><p><strong>Case 1: x ≥ -2</strong></p><p>Then |x+2| = x+2, so: (x+2-x)/x < 1 ⟹ 2/x < 1</p><p>• If x > 0: 2 < x, so x > 2</p><p>• If -2 ≤ x < 0: 2 > x (always true in this range), so -2 ≤ x < 0</p><p>Combined from Case 1: x ∈ [-2, 0) ∪ (2, ∞)</p><p><strong>Case 2: x < -2</strong></p><p>Then |x+2| = -(x+2), so: (-x-2-x)/x < 1 ⟹ (-2x-2)/x < 1</p><p>Rearranging: (-2x-2)/x - 1 < 0 ⟹ (-3x-2)/x < 0</p><p>This is negative when numerator and denominator have opposite signs:</p><p>• x > 0 and -3x-2 < 0: contradicts x < -2</p><p>• x < 0 and -3x-2 > 0: gives x < -2/3, so x < -2 satisfies this</p><p>Combined from Case 2: x ∈ (-∞, -2)</p><p><strong>Step 3:</strong> Combine both cases: (-∞, -2) ∪ [-2, 0) ∪ (2, ∞) = (-∞, 0) ∪ (2, ∞)</p><p>∴ Answer: C</p>
Correct Answer: C