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Integration of Irrational Algebraic Fractions - 1

Solve the fol...

Question

Solve the following inequality:
${log}_{{log}_{2} (0.5 x)} (x^{2} - 10 x + 22) > 0$

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Solution

$\frac{log (x^{2} - 10 x + 22)}{log log 0.5 x - log log 2} > 0 H e n c e, F o r log (x^{2} - 10 x + 22) x^{2} - 10 x + 22 > 0 x \in (- \infty, 5 - \sqrt{3}) \cup (5 + \sqrt{3}, \infty) A l s o, F o r {log}_{2} (0.5 x) {log}_{2} (0.5 x) > 0 log (0.5 x) > 0 0.5 x > 1 x > 2$
Taking intersection, we get $x \in (5 + \sqrt{3}, \infty)$
For $(5 + \sqrt{3}, \infty) log {log}_{2} (0.5 x) > 0$
Hence, For $x \in (5 + \sqrt{3}, \infty)$
$log (x^{2} - 10 x + 22) > 0 x^{2} - 10 x + 22 > 1 x^{2} - 10 x + 21 > 0 (x - 3) (x - 7) > 0 x \in (- \infty, 3) \cup (7, \infty)$
Hence, $x \in (7, \infty)$

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