Question

Solve the given inequality for $x$:

$\sqrt{\frac{x-2}{1-2x}}>1$

Answer 1

First of all, term inside the square root must be positive

$\Rightarrow \frac{x-2}{1-2x}\ge 0$

$\Rightarrow \frac{x-2}{x-\frac{1}{2}}\le 0$

$\Rightarrow x\in [\frac{1}{2},2]$ ....... $\left(1\right)$

Given $\sqrt{\frac{x-2}{1-2x}}>1$

Taking square on both sides ;

$\Rightarrow \frac{x-2}{1-2x}>1$

$\Rightarrow x-2<1-2x$ ...... $[\because 1-2x\le 0]$

$\Rightarrow 3x<3$

$\Rightarrow x<1$....... $\left(2\right)$

Combining both equations, we get ;

$x\in [\frac{1}{2},1)$