Explanation

(c)
$\log_{\sin\left( \frac{\pi}{3} \right)}{(x^{2} - 3x + 2) \geq 2}$
$\Rightarrow x^{2} - 3x + 2 \leq \frac{3}{4}\ \ ($If${\ log}_{a}{b = c \Rightarrow b = c^{a})}$
$\Rightarrow x^{2} - 3x + \frac{5}{4} \leq 0$
$\Rightarrow 4x^{2} - 12x + 5 \leq 0$
$\Rightarrow (2x - 5)(2x - 1) \leq 0$
$\Rightarrow \frac{1}{2} \leq x \leq \frac{5}{2}$ …(i)
Also, $x^{2} - 3x + 2 > 0$
$\Rightarrow (x - 1)(x - 2) > 0$
$\Rightarrow x < 1\$or$\ x > 2$ …(ii)
From relation (i) and (ii), we get
$x \in \left\lbrack \left. \ \frac{1}{2},1 \right) \cup \left( 2,\frac{5}{2} \right.\ \right\rbrack$