a) Ta có: \(M=\left(\frac{\sqrt{x}}{\sqrt{x}-1}-\frac{1}{x-\sqrt{x}}\right):\left(\frac{1}{\sqrt{x}+1}+\frac{2}{\sqrt{x}-1}\right)\)
\(=\left(\frac{x-1}{\sqrt{x}\left(\sqrt{x}-1\right)}\right):\left(\frac{\sqrt{x}-1+2\left(\sqrt{x}+1\right)}{\left(\sqrt{x}+1\right)\left(\sqrt{x}-1\right)}\right)\)
\(=\frac{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}{\sqrt{x}\left(\sqrt{x}-1\right)}:\frac{\sqrt{x}-1+2\sqrt{x}+2}{\left(\sqrt{x}+1\right)\left(\sqrt{x}-1\right)}\)
\(=\frac{\sqrt{x}+1}{\sqrt{x}}:\frac{3\sqrt{x}+1}{\left(\sqrt{x}+1\right)\left(\sqrt{x}-1\right)}\)
\(=\frac{\left(\sqrt{x}+1\right)^2\cdot\left(\sqrt{x}-1\right)}{\sqrt{x}\left(3\sqrt{x}+1\right)}\)
b) Để M>0 thì \(\frac{\left(\sqrt{x}+1\right)^2\cdot\left(\sqrt{x}-1\right)}{\sqrt{x}\left(3\sqrt{x}+1\right)}>0\)
mà \(\forall\left\{{}\begin{matrix}x>0\\x\ne1\end{matrix}\right.\), ta luôn có: \(\sqrt{x}\left(3\sqrt{x}+1\right)>0\)
nên \(\left(\sqrt{x}+1\right)^2\cdot\left(\sqrt{x}-1\right)>0\)
mà \(\left(\sqrt{x}+1\right)^2>0\forall0< x\ne1\)
nên \(\sqrt{x}-1>0\)
\(\Leftrightarrow\sqrt{x}>1\)
hay x>1(nhận)
Vậy: để M>0 thì x>1
