Question

Solve the following inequalities:

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

Answer 1

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

$\Rightarrow \sqrt{2x-1}<x-2$

$\Rightarrow 2x-1<{(x-2)}^2$

$\Rightarrow 2x-1-x^2+4x-4<0$

$\Rightarrow x^2-6x+5>0$

$\Rightarrow x^2-5x-x+5>0$

$\Rightarrow x(x-5)-1(x-5)>0$

$\Rightarrow (x-5)(x-1)>0$

$\therefore x\in (-\infty ,1)\cup (5,\infty )$