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nCr Definitions and Properties

Solve the fol...

Question

Solve the following inequality:
$1 + {log}_{2} (x - 2) > {log}_{2} (x^{2} - 3 x + 2)$

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Solution

$1 + {log}_{2} (x - 2) > {log}_{2} (x^{2} - 3 x + 2)$
${log}_{2} 2 + {log}_{2} (x - 2) - {log}_{2} (x - 1) (x - 2) > 0$
${log}_{2} 2 + {log}_{2} (x - 2) - {log}_{2} (x - 1) - {log}_{2} (x - 2) > 0$
${log}_{2} (\frac{2}{x - 1}) > 0 ⟹ \frac{2}{x - 1} > 1 ⟹ \frac{2 - x + 1}{x - 1} > 0 ⟹ \frac{3 - x}{x - 1} > 0 ⟹ (x - 3) (x - 1) < 0 ⟹ x \in (1, 3)$

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