`1)P=(x+2+x-1-x-sqrtx-1)/(xsqrtx-1).(x+sqrtx+1)/(sqrtx-1)`
`=(x-sqrtx)/(xsqrtx-1).(x+sqrtx+1)/(sqrtx-1)`
`=sqrtx/(x+sqrtx+1).(x+sqrtx+1)/(sqrtx-1)`
`=sqrtx/(sqrtx-1)`
`2)P<1/2`
`=>sqrtx/(sqrtx-1)-1/2<0`
`=>(2sqrtx-sqrtx+1)/(2sqrtx-2)<0`
`=>(sqrtx+1)/(2sqrtx-2)<0`
Vì `sqrtx+1>=1>0`
`=>2sqrtx-2<0`
`=>sqrtx-1<0`
`=>x<1`
KHĐKXĐ
`=>0<x<1`
\(P=\dfrac{x+2+\left(\sqrt{x}+1\right)\left(\sqrt{x}-1\right)-\left(x+\sqrt{x}+1\right)}{\left(\sqrt{x}-1\right)\left(x+\sqrt{x}+1\right)}.\dfrac{x+\sqrt{x}+1}{\sqrt{x}-1}\)
\(P=\dfrac{x+2+x-1-x-\sqrt{x}-1}{\left(\sqrt{x}-1\right)^2}\)
\(P=\dfrac{x-\sqrt{x}}{\left(\sqrt{x}-1\right)^2}\)
\(P=\dfrac{\sqrt{x}\left(\sqrt{x}-1\right)}{\left(\sqrt{x}-1\right)^2}\)
\(P=\dfrac{\sqrt{x}}{\sqrt{x}-1}\)
\(P< \dfrac{1}{2}\rightarrow\dfrac{\sqrt{x}}{\sqrt{x}-1}< \dfrac{1}{2}\)
\(\dfrac{\sqrt{x}}{\sqrt{x}-1}-\dfrac{\sqrt{x}-1}{2\left(\sqrt{x}-1\right)}< 0\)
\(\dfrac{2\sqrt{x}-\sqrt{x}+1}{2\left(\sqrt{x}-1\right)}< 0\)
\(\dfrac{\sqrt{x}+1}{\sqrt{x}-1}< 0\)
\(\sqrt{x}+1>0\rightarrow\sqrt{x}-1< 0\)
\(\sqrt{x}< 1\)
x<1
Kết hợp lại với đk, ta có : \(0\le x< 1\)
