a) \(P=\dfrac{\sqrt{x}+5}{\sqrt{x}-2}=\dfrac{\sqrt{9}+5}{\sqrt{9}-2}=\dfrac{3+5}{3-2}=8\)
b) \(Q=\dfrac{\sqrt{x}-1}{\sqrt{x}+2}-\dfrac{5\sqrt{x}-2}{4-x}=\dfrac{\left(\sqrt{x}-1\right)\left(\sqrt{x}-2\right)+5\sqrt{x}-2}{\left(\sqrt{x}-2\right)\left(\sqrt{x}+2\right)}=\dfrac{x-3\sqrt{x}+2+5\sqrt{x}-2}{\left(\sqrt{x}-2\right)\left(\sqrt{x}+2\right)}=\dfrac{\sqrt{x}\left(\sqrt{x}+2\right)}{\left(\sqrt{x}-2\right)\left(\sqrt{x}+2\right)}=\dfrac{\sqrt{x}}{\sqrt{x}-2}\)
c) \(M=\dfrac{Q}{P}=\dfrac{\sqrt{x}}{\sqrt{x}-2}:\dfrac{\sqrt{x}+5}{\sqrt{x}-2}=\dfrac{\sqrt{x}}{\sqrt{x}-2}.\dfrac{\sqrt{x}-2}{\sqrt{x}+5}=\dfrac{\sqrt{x}}{\sqrt{x}+5}< \dfrac{1}{2}\)
\(\Leftrightarrow2\sqrt{x}< 3\sqrt{x}+15\Leftrightarrow\sqrt{x}>-15\left(đúng\forall x\ge0,x\ne4\right)\)
d) \(M=\dfrac{\sqrt{x}}{\sqrt{x}+5}=1-\dfrac{5}{\sqrt{x}+5}\in Z\)
\(\Rightarrow\sqrt{x}+5\inƯ\left(5\right)=\left\{-5;-1;1;5\right\}\)
Do \(x\ge0,x\ne4\)
\(\Rightarrow x\in\left\{0\right\}\)