\(\left(\dfrac{-x+1}{x-9}-\dfrac{\sqrt{x}}{3-\sqrt{x}}\right):\dfrac{1+3\sqrt{x}}{5\sqrt{x}+15}\left(x\ge0\right).\\ =\left(\dfrac{-x+1}{\left(\sqrt{x}-3\right)\left(\sqrt{x}+3\right)}+\dfrac{\sqrt{x}}{\sqrt{x}-3}\right):\dfrac{1+3\sqrt{x}}{5\left(\sqrt{x}+3\right)}.\\ =\dfrac{-x+1+x+3\sqrt{x}}{\left(\sqrt{x}-3\right)\left(\sqrt{x}+3\right)}.\dfrac{5\left(\sqrt{x}+3\right)}{1+3\sqrt{x}}.\\ =\dfrac{1+3\sqrt{x}}{\sqrt{x}-3}.\dfrac{5}{1+3\sqrt{x}}.\\ =\dfrac{5}{\sqrt{x}-3}.\)
\(\Leftrightarrow\left(\dfrac{-\left(x-1\right)}{x-9}-\dfrac{\sqrt{x}}{3-\sqrt{x}}\right):\left(\dfrac{1+3\sqrt{x}}{5\sqrt{x}+15}\right)\\ \Leftrightarrow\left(\dfrac{x-1}{9-x}-\dfrac{\sqrt{x}}{3-\sqrt{x}}\right):\left(\dfrac{1+3\sqrt{x}}{5\sqrt{x}+15}\right)\\ \Leftrightarrow\left(\dfrac{x-1}{\left(3-\sqrt{x}\right)\left(3+\sqrt{x}\right)}-\dfrac{\sqrt{x}}{3-\sqrt{x}}\right):\left(\dfrac{1+3\sqrt{x}}{5\sqrt{x}+15}\right)\\ \Leftrightarrow\left(\dfrac{x-1}{\left(3-\sqrt{x}\right)\left(3+\sqrt{x}\right)}-\dfrac{\sqrt{x}\left(3+\sqrt{x}\right)}{\left(3-\sqrt{x}\right)\left(3+\sqrt{x}\right)}\right):\left(\dfrac{1+3\sqrt{x}}{5\sqrt{x}+15}\right)\\ \Leftrightarrow\left(\dfrac{x-1}{9-x}-\dfrac{3\sqrt{x}+x}{9-x}\right):\left(\dfrac{1+3\sqrt{x}}{5\sqrt{x}+15}\right)\\ \Leftrightarrow\left(\dfrac{x-1-3\sqrt{x}-x}{9-x}\right):\left(\dfrac{1+3\sqrt{x}}{5\sqrt{x}+15}\right)\\ \)
\(\Leftrightarrow\left(\dfrac{-\left(3\sqrt{x}+1\right)}{\left(3-\sqrt{x}\right)\left(3+\sqrt{x}\right)}\right).\left(\dfrac{5\left(\sqrt{x}+3\right)}{1+3\sqrt{x}}\right)\)
\(\Leftrightarrow\dfrac{-\left(3\sqrt{x}+1\right)5\left(\sqrt{x}+3\right)}{\left(3-\sqrt{x}\right)\left(3+\sqrt{x}\right).\left(1+3\sqrt{x}\right)}=\dfrac{-5}{3-\sqrt{x}}\)
