ĐKXĐ: \(x>1\)
a) \(P=\frac{15\sqrt{x}-11}{x+2\sqrt{x}-3}+\frac{3\sqrt{x}-2}{1-\sqrt{x}}-\frac{2\sqrt{x}+3}{\sqrt{x}+3}\)
\(P=\frac{15\sqrt{x}-11}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+3\right)}+\frac{\left(2-3\sqrt{x}\right)\left(\sqrt{x}+3\right)}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+3\right)}-\frac{\left(2\sqrt{x}+3\right)\left(\sqrt{x}-1\right)}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+3\right)}\)
\(P=\frac{15\sqrt{x}-11-3x+6-7\sqrt{x}-2x-\sqrt{x}+3}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+3\right)}\)
\(P=\frac{-5x-2+7\sqrt{x}}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+3\right)}\)
\(P=\frac{-\left(\sqrt{x}-1\right)\left(5\sqrt{x}-2\right)}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+3\right)}\)
\(P=\frac{-5\sqrt{x}+2}{\sqrt{x}+3}\)
b) \(P=\frac{1}{2}\Leftrightarrow\frac{-5\sqrt{x}+2}{\sqrt{x}+3}=\frac{1}{2}\)
\(\Leftrightarrow2\left(-5\sqrt{x}+2\right)=\sqrt{x}+3\)
\(\Leftrightarrow-10\sqrt{x}+2-\sqrt{x}-3=0\)
\(\Leftrightarrow-11\sqrt{x}-1=0\)
\(\Leftrightarrow\sqrt{x}=\frac{-1}{11}\)( vô lý )
Vậy không có giá trị của x thỏa mãn.
c) \(P\le\frac{2}{3}\)
\(\Leftrightarrow\frac{-5\sqrt{x}+2}{\sqrt{x}+3}\le\frac{2}{3}\)
\(\Leftrightarrow3\left(-5\sqrt{x}+2\right)\le2\left(\sqrt{x}+3\right)\)
\(\Leftrightarrow-15\sqrt{x}+6\le2\sqrt{x}+6\)
\(\Leftrightarrow-17\sqrt{x}\le0\) ( luôn đúng )
Ta có đpcm.