A= (2/x-√x - 1/√x-1) : x-4/x√x+√x - 2x với x>0, x khác 1, x khác 4 a) rút gọn A b) tìm x để A > -1/2

Đáp án:

$\begin{array}{l}
Dkxd:x > 0;x \ne 1;x \ne 4\\
a)A = \left( {\dfrac{2}{{x - \sqrt x }} - \dfrac{1}{{\sqrt x  - 1}}} \right):\dfrac{{x - 4}}{{x\sqrt x  + \sqrt x  - 2x}}\\
 = \dfrac{{2 - \sqrt x }}{{\sqrt x \left( {\sqrt x  - 1} \right)}}.\dfrac{{\sqrt x \left( {x - 2\sqrt x  + 1} \right)}}{{\left( {\sqrt x  - 2} \right)\left( {\sqrt x  + 2} \right)}}\\
 = \dfrac{{ - 1}}{{\sqrt x  - 1}}.\dfrac{{{{\left( {\sqrt x  - 1} \right)}^2}}}{{\sqrt x  + 2}}\\
 = \dfrac{{1 - \sqrt x }}{{\sqrt x  + 2}}\\
b)A >  - \dfrac{1}{2}\\
 \Leftrightarrow \dfrac{{1 - \sqrt x }}{{\sqrt x  + 2}} + \dfrac{1}{2} > 0\\
 \Leftrightarrow \dfrac{{2 - 2\sqrt x  + \sqrt x  + 2}}{{2\left( {\sqrt x  + 2} \right)}} > 0\\
 \Leftrightarrow 4 - \sqrt x  > 0\\
 \Leftrightarrow \sqrt x  < 4\\
 \Leftrightarrow x < 16\\
Vậy\,0 < x < 16;x \ne 1;x \ne 4
\end{array}$