Question

1. Giải bpt: \(\sqrt{x-2}-2\ge\sqrt{2x-5}-\sqrt{x+1}\)

2. Với \(x\in\left(0;1\right)\) tìm Min \(P=\dfrac{\sqrt{1-x}\left(1+\sqrt{1-x}\right)}{x}+\dfrac{5}{\sqrt{1-x}}\)

Answer 1

`sqrt{x-2}-2>=sqrt{2x-5}-sqrt{x+1}`

`đk:x>=5/2`

`bpt<=>\sqrt{x-2}+\sqrt{x+1}>=\sqrt{2x-5}+2`

`<=>x-2+x+1+2\sqrt{(x-2)(x+1)}>=2x-5+4+4\sqrt{2x-5}`

`<=>2x-1+2\sqrt{(x-2)(x+1)}>=2x-1+4\sqrt{2x-5}`

`<=>2\sqrt{(x-2)(x+1)}>=4\sqrt{2x-5}`

`<=>sqrt{x^2-x-2}>=2sqrt{2x-5}`

`<=>x^2-x-2>=4(2x-5)`

`<=>x^2-x-2>=8x-20`

`<=>x^2-9x+18>=0`

`<=>(x-3)(x-6)>=0`

`<=>` \(\left[ \begin{array}{l}x \ge 6\\x \le 3\end{array} \right.\) 

Kết hợp đkxđ:

`=>` \(\left[ \begin{array}{l}x \ge 6\\\dfrac52 \le x \le 3\end{array} \right.\)