\(P=\frac{\sqrt{x}}{\sqrt{x}+2}+\frac{-x+x\sqrt{x}+6}{x+\sqrt{x}-2}-\frac{\sqrt{x}+1}{\sqrt{x}-1}\)
\(P=\frac{\sqrt{x}\left(\sqrt{x}-1\right)}{x+\sqrt{x}-2}+\frac{-x+x\sqrt{x}+6}{x+\sqrt{x}-2}-\frac{\left(\sqrt{x}+1\right)\left(\sqrt{x}+2\right)}{x+\sqrt{x}-2}\)
\(P=\frac{x-\sqrt{x}-x+x\sqrt{x}+6-x-3\sqrt{x}-2}{x+\sqrt{x}-2}\)
\(P=\frac{-x+x\sqrt{x}+4-4\sqrt{x}}{x+\sqrt{x}-2}\)
\(P=\frac{x\left(\sqrt{x}-1\right)-4\left(\sqrt{x}-1\right)}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+2\right)}\)
\(P=\frac{\left(\sqrt{x}-1\right)\left(x-4\right)}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+2\right)}\)
\(P=\frac{\left(\sqrt{x}-2\right)\left(\sqrt{x}+2\right)}{\sqrt{x}+2}\)
\(P=\sqrt{x}-2\)
@Trần Ngọc Thảo
\(Q=\frac{\left(x+27\right)\cdot P}{\left(\sqrt{x}+3\right)\left(\sqrt{x}-2\right)}\ge6\)
\(\Leftrightarrow Q=\frac{\left(x+27\right)\left(\sqrt{x}-2\right)}{\left(\sqrt{x}+3\right)\left(\sqrt{x}-2\right)}\ge6\)
\(\Leftrightarrow\frac{x+27}{\sqrt{x}+3}\ge6\)
\(\Leftrightarrow x+27\ge6\left(\sqrt{x}+3\right)\)
\(\Leftrightarrow x+27-6\sqrt{x}-18\ge0\)
\(\Leftrightarrow x-6\sqrt{x}+9\ge0\)
\(\Leftrightarrow\left(\sqrt{x}-3\right)^2\ge0\)( luôn đúng )
Vậy \(x\ge0\)thì bất phương trình luôn đúng
\(\)