1: \(P=\dfrac{x\sqrt{x}-1}{x-\sqrt{x}}+\dfrac{x\sqrt{x}+1}{x+\sqrt{x}}-\dfrac{4}{\sqrt{x}}\)
\(=\dfrac{\left(\sqrt{x}-1\right)\left(x+\sqrt{x}+1\right)}{\sqrt{x}\left(\sqrt{x}-1\right)}+\dfrac{\left(\sqrt{x}+1\right)\left(x-\sqrt{x}+1\right)}{\sqrt{x}\left(\sqrt{x}+1\right)}-\dfrac{4}{\sqrt{x}}\)
\(=\dfrac{x+\sqrt{x}+1+x-\sqrt{x}+1-4}{\sqrt{x}}\)
\(=\dfrac{2x-2}{\sqrt{x}}\)
2: \(P\cdot Q\sqrt{x}< 8\)
=>\(\left(2x-2\right)\cdot\dfrac{\sqrt{x}-1}{\sqrt{x}+1}< 8\)
=>\(\left(2x-2\right)\left(\sqrt{x}-1\right)-8\left(\sqrt{x}+1\right)< 0\)
=>\(\left(x-1\right)\left(\sqrt{x}-1\right)-4\left(\sqrt{x}+1\right)< 0\)
=>\(\left(\sqrt{x}+1\right)\left[\left(\sqrt{x}-1\right)^2-4\right]< 0\)
=>\(\left(\sqrt{x}-1\right)^2-4< 0\)
=>\(\left(\sqrt{x}-1-2\right)\left(\sqrt{x}-1+2\right)< 0\)
=>\(\left(\sqrt{x}-3\right)\left(\sqrt{x}+1\right)< 0\)
=>\(\sqrt{x}-3< 0\)
=>\(\sqrt{x}< 3\)
=>0<=x<9
Kết hợp ĐKXĐ, ta được: \(\left\{{}\begin{matrix}0< x< 9\\x\ne1\end{matrix}\right.\)
