a: \(P=\dfrac{\sqrt{x}+2}{\sqrt{x}+1}+\dfrac{2-\sqrt{x}}{\sqrt{x}-1}\)
\(=\dfrac{\sqrt{x}+2}{\sqrt{x}+1}-\dfrac{\sqrt{x}-2}{\sqrt{x}-1}\)
\(=\dfrac{\left(\sqrt{x}+2\right)\left(\sqrt{x}-1\right)-\left(\sqrt{x}-2\right)\left(\sqrt{x}+1\right)}{x-1}\)
\(=\dfrac{x+\sqrt{x}-2-\left(x-\sqrt{x}-2\right)}{x-1}\)
\(=\dfrac{x+\sqrt{x}-2-x+\sqrt{x}+2}{x-1}=\dfrac{2\sqrt{x}}{x-1}\)
b: P<=1
=>P-1<=0
=>\(\dfrac{2\sqrt{x}-x+1}{x-1}< =0\)
=>\(\dfrac{-\left(x-2\sqrt{x}-1\right)}{x-1}< =0\)
=>\(\dfrac{\left(\sqrt{x}-1\right)^2-2}{x-1}>=0\)
TH1: \(\left\{{}\begin{matrix}\left(\sqrt{x}-1\right)^2-2>=0\\x-1>0\end{matrix}\right.\)
=>\(\left\{{}\begin{matrix}\left(\sqrt{x}-1\right)^2>=2\\x>1\end{matrix}\right.\)
=>\(\left\{{}\begin{matrix}\left[{}\begin{matrix}\sqrt{x}-1>=\sqrt{2}\\\sqrt{x}-1< =-\sqrt{2}\end{matrix}\right.\\x>1\end{matrix}\right.\)
=>\(\left[{}\begin{matrix}\left\{{}\begin{matrix}x>1\\\sqrt{x}>=\sqrt{2}+1\end{matrix}\right.\\\left\{{}\begin{matrix}x>1\\\sqrt{x}< =-\sqrt{2}+1< 0\end{matrix}\right.\Leftrightarrow Loại\end{matrix}\right.\)
=>\(\left\{{}\begin{matrix}x>1\\x>=3+2\sqrt{2}\end{matrix}\right.\Leftrightarrow x>=3+2\sqrt{2}\)
TH2:
\(\left\{{}\begin{matrix}\left(\sqrt{x}-1\right)^2-2< =0\\x-1< 0\end{matrix}\right.\)
=>\(\left\{{}\begin{matrix}\left(\sqrt{x}-1\right)^2< =2\\x< 1\end{matrix}\right.\)
=>\(\left\{{}\begin{matrix}-\sqrt{2}< =\sqrt{x}-1< =\sqrt{2}\\x< 1\end{matrix}\right.\)
=>\(\left\{{}\begin{matrix}-\sqrt{2}+1< =\sqrt{x}< =\sqrt{2}+1\\x< 1\end{matrix}\right.\)
=>\(\left\{{}\begin{matrix}0< =\sqrt{x}< =\sqrt{2}+1\\x< 1\end{matrix}\right.\)
=>\(\left\{{}\begin{matrix}0< =x< =3+2\sqrt{2}\\x< 1\end{matrix}\right.\Leftrightarrow0< =x< 1\)
