<p>Number of integral values of <i>x</i>, such that <span>\(8 < g(x) \leq 18\)</span> is:</p>
Step-by-Step Solution
Key Concept: Transform the inequality into a quadratic form by substituting t = 2^x, then solve the resulting rational inequality while respecting the constraint that t > 0 (since 2^x is always positive).
<p><strong>Step 1:</strong> Let t = 2^x where t > 0. The inequality becomes:</p><p>8 < t + 16/t ≤ 10</p><p><strong>Step 2:</strong> Split into two inequalities:</p><p><strong>Part A:</strong> t + 16/t > 8</p><p>⟹ t² - 8t + 16 > 0 (multiply by t > 0)</p><p>⟹ (t - 4)² > 0</p><p>⟹ t ≠ 4, and t > 0</p><p><strong>Part B:</strong> t + 16/t ≤ 10</p><p>⟹ t² - 10t + 16 ≤ 0 (multiply by t > 0)</p><p>⟹ (t - 2)(t - 8) ≤ 0</p><p>⟹ 2 ≤ t ≤ 8</p><p><strong>Step 3:</strong> Combine both conditions:</p><p>t ∈ [2, 8] AND t ≠ 4</p><p>⟹ t ∈ [2, 4) ∪ (4, 8]</p><p><strong>Step 4:</strong> Convert back to x using t = 2^x:</p><p>2^x ∈ [2, 4) ∪ (4, 8]</p><p>⟹ 2¹ ≤ 2^x < 2² or 2² < 2^x ≤ 2³</p><p>⟹ 1 ≤ x < 2 or 2 < x ≤ 3</p><p>⟹ x ∈ [1, 2) ∪ (2, 3]</p><p><strong>Step 5:</strong> Integral values of x are: x = 1 and x = 3</p><p>∴ Number of integral values = <strong>2</strong></p>
Correct Answer: B