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Integration of Trigonometric Functions

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Question

The complete solution set of the inequality, $l o g_{\frac{\sqrt{3}}{2}} (x^{2} - 3 x + 2) \geq 2$ , is?

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Solution

$l o g_{\frac{\sqrt{3}}{2}} (x^{2} - 3 x + 2) > 2 x^{2} - 3 x + 2 > 0 x^{2} - 2 x - x + 2 > 0 x (x - 2) - 1 (x - 2) > 0 (x - 1) (x - 2) >\to (1) x^{2} - 3 x + 2 \leq (\frac{\sqrt{3}}{2})^{2} x^{2} - 3 x + 2 \leq 3 / 4 x^{2} - 3 x + \frac{5}{4} \leq 0 4 x^{2} - 12 x + 5 \leq 0 4 x^{2} - 10 x - 2 x + 5 \leq 0 2 x (2 x - 5) - 1 (2 x - 5) \leq 0 (2 x - 1) (2 x - 5) \leq 0 (x - 1 / 2) (x - 5 / 2) \leq 0 \to (2) ∴ x ϵ [\frac{1}{2}, 1) U (2, \frac{5}{2}]$

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