Question

The solution set of the inequality
$ \log_{\sin(\pi/3)} (x^2-3x+2)\ge 2 $ is

• A. $ \left(\frac{1}{2},2\right) $
• B. $ \left[\frac{1}{2},2\right] $
• C. $ [\frac{1}{2},1) \cup (2,\frac{5}{2}] $
• D. $ (\frac{1}{2},\frac {5}{2}) $

Answer 1

Answer: (C)

We have,
$\log _{\left(\sin \frac{\pi}{3}\right)}\left(x^{2}-3 x+2\right) \geq 2$
$\Rightarrow \log _{\frac{\sqrt{3}}{2}}\left(x^{2}-3 x+2\right) \geq 2 $
$\Rightarrow \left(x^{2}-3 x+2\right) \leq\left(\frac{\sqrt{3}}{2}\right)^{2} $
and $ x^{2}-3 x+2>0 $
$\Rightarrow x^{2}-3 x+2 \leq \frac{3}{4} $ and $x^{2}-2 x-x+2>0$
$\Rightarrow 4 x^{2}-12 x+5 \leq 0 $ and $(x-2)(x-1)>0 $
$\Rightarrow(2 x-5)(2 x-1) \leq 0 $ and $ x \in(-\infty, 1) \cup(2, \infty)$
$ x \in\left[\frac{1}{2}, \frac{5}{2}\right] $ and $ x \in(-\infty, 1) \cup(2, \infty) $
$\therefore x \in\left[\frac{1}{2}, 1\right) \cup\left(2, \frac{5}{2}\right]$