Question

1. Cho P = \(\left(\dfrac{2\sqrt{x}}{\sqrt{x}+3}+\dfrac{\sqrt{x}}{\sqrt{x}-3}-\dfrac{3x+3}{x-9}\right):\left(\dfrac{2\sqrt{x}-2}{\sqrt{x}-3}-1\right)\)

a) tìm x để P < \(\dfrac{-1}{2}\)

b) Tìm GTNN của P

Answer 1

1.ĐK:\(x\ge0,x\ne9\)

\(P=\left(\dfrac{2\sqrt{x}\left(\sqrt{x}-3\right)+\sqrt{x}\left(\sqrt{x}+3\right)-3x-3}{\left(\sqrt{x}+3\right)\left(\sqrt{x}-3\right)}\right):\dfrac{2\sqrt{x}-2-\sqrt{x}-3}{\sqrt{x}-3}\)

\(=\left[\dfrac{-3\sqrt{x}-3}{\left(\sqrt{x}+3\right)\left(\sqrt{x}-3\right)}\right].\dfrac{\sqrt{x}-3}{\sqrt{x}-5}\)

\(=\dfrac{-3\left(\sqrt{x}+1\right)}{\left(\sqrt{x}+3\right)\left(\sqrt{x}-5\right)}.\)

Để \(P< \dfrac{-1}{2}\Leftrightarrow\dfrac{-3\left(\sqrt{x}+1\right)}{\left(\sqrt{x}+3\right)\left(\sqrt{x}-5\right)}< \dfrac{-1}{2}\)