Question

Solution set of
${\log }_3(x^2-2)<{\log }_3(\frac{3}{2}|x|-1)\text{is}$

• A. $(-2,-\sqrt{2})$
• B. $(-\sqrt{2},-1)$
• C. $(-\sqrt{2},2)$
• D. None of these

Answer 1

The correct option is D None of these

We must have
$x^2-2>0,\frac{3}{2}|x|-1>0\text{and}{x}^2-2<\frac{3}{2}|x|-1$

$\Rightarrow x^2>2,|x|>\frac{2}{3}\text{and}|x{|}^2-\frac{3}{2}|x|-1<0$

$\Rightarrow |x{|}^2>2,|x|>\frac{2}{3}\text{and}2|x{|}^2-3|x|-2<0$

$\Rightarrow |x|>\sqrt{2},\text{and}(2|x|+1)(|x|-2)<0$

$\Rightarrow |x|>\sqrt{2}\text{and}|x|<2$
$[\therefore 2|x|+1>0]$

$\Rightarrow \sqrt{2}<|x|<2$
$\Rightarrow x\in (-2,-\sqrt{2})\cup (\sqrt{2},2)$

Hence, The correct answer is option D.