Cho biểu thức A=(1/x+căn x-1/căn x+1):(căn x)/(x+2 căn x+1) (x>0)

* Hướng dẫn giải

a) $A=\left(\frac{1}{x+\sqrt{x}}-\frac{1}{\sqrt{x}+1}\right):\frac{\sqrt{x}}{x+2\sqrt{x}+1}$

$=\frac{1-\sqrt{x}}{\sqrt{x}.\left(\sqrt{x}+1\right)}\cdot \frac{{\left(\sqrt{x}+1\right)}^2}{\sqrt{x}}=\frac{\left(1-\sqrt{x}\right)\left(1+\sqrt{x}\right)}{\sqrt{x}.\sqrt{x}}=\frac{1-x}{x}$

b) $A>\frac{1}{2}\iff \frac{1-x}{x}>\frac{1}{2}\iff \frac{1-x}{x}-\frac{1}{2}>0\iff \frac{2-2x-x}{2x}>0$

$\iff \frac{2-3x}{2x}>0$ mà $x>0\iff 2-3x>0\iff x<\frac{2}{3}$

Vậy $0<x<\frac{2}{3}$ thì $A>\frac{1}{2}\cdot$