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