<p><strong>782.</strong> Find the number of integral values of <em>x</em> satisfying the inequality</p><p>\[\frac{\left(2^{\tan^{-1}x} - 4\right)(x-4)(x-10)}{x! - (x-1)!} < 0\]</p>
Step-by-Step Solution
Key Concept: The inequality requires analyzing the sign of numerator and denominator separately. The denominator x! - (x-1)! = (x-1)!(x-1) is defined only for x ≥ 1, and the numerator's sign depends on whether 2^(tan⁻¹x) < 4 (i.e., tan⁻¹x < 2).
<p><strong>Step 1: Simplify the denominator</strong></p><p>x! - (x-1)! = (x-1)!(x - 1)</p><p>This is defined for x ≥ 1 only, and equals 0 when x = 1.</p><p><strong>Step 2: Analyze the numerator factor 2^(tan⁻¹x) - 4</strong></p><p>Since tan⁻¹x ∈ (-π/2, π/2), we have 2^(tan⁻¹x) ∈ (0, 2^(π/2)) ≈ (0, 4.81)</p><p>2^(tan⁻¹x) = 4 when tan⁻¹x = 2, but 2 > π/2 ≈ 1.57, so this never occurs.</p><p>Therefore 2^(tan⁻¹x) - 4 < 0 for all real x.</p><p><strong>Step 3: Set up the inequality</strong></p><p>For (negative)(x-4)(x-10)/[(x-1)(x-1)!] ≥ 0, we need:</p><p>-(x-4)(x-10)/[(x-1)(x-1)!] ≥ 0</p><p>(x-4)(x-10)/[(x-1)(x-1)!] ≤ 0</p><p><strong>Step 4: Determine sign in each region (x > 1)</strong></p><p>• 1 < x < 4: (−)(−)/(+) = positive ✗</p><p>• x = 4: 0 ✓</p><p>• 4 < x < 10: (+)(−)/(+) = negative ✓</p><p>• x = 10: 0 ✓</p><p>• x > 10: (+)(+)/(+) = positive ✗</p><p><strong>Step 5: Count integral values</strong></p><p>Solution set: [4, 10]</p><p>Integral values: x ∈ {4, 5, 6, 7, 8, 9, 10}</p><p>∴ Answer: <strong>7</strong></p>
Correct Answer: 7