Xét `P-9`, ta có:

`6/(\sqrt{x}+1)-9`

`=(6-9(\sqrt{x}+1))/(\sqrt{x}+1)`

`=(6-9\sqrt{x}-9)/(\sqrt{x}+1)`

`=(-9\sqrt{x}-3)/(\sqrt{x}+1)`

`=(-3(3\sqrt{x}+1))/(\sqrt{x}+1)`

Ta có: `\sqrt{x}>=0 AA x`

`=>3\sqrt{x}>=0 AA x`

`=>3\sqrt{x}+1>=1 AA x`

`=>-3(3\sqrt{x}+1)<=-3 AA x`

`=>-3(3\sqrt{x}+1)<0 AA x (1)`

Ta có: `\sqrt{x}>=0 AA x`

`=>\sqrt{x}+1>=1 AA x`

`=>\sqrt{x}+1>0 AA x (2)`

 Từ `(1);(2)=>(-3(3\sqrt{x}+1))/(\sqrt{x}+1)<0 AA x`

`=>P-9<0`

`=>P<9`

(Rút gọn P sai nên không thể chứng minh)

__________

`P=(6/(\sqrt{x}+1)+6/(\sqrt{x}-1)-(2x)/(x-1)):2/(x-1)`

`P=[6/(\sqrt{x}+1)+6/(\sqrt{x}-1)-(2x)/((\sqrt{x}-1)(\sqrt{x}+1))] . (x-1)/2`

`P=(6(\sqrt{x}-1)+6(\sqrt{x}+1)-2x)/((\sqrt{x}-1)(\sqrt{x}+1)) . (x-1)/2`

`P=(6\sqrt{x}-6+6\sqrt{x}+6-2x)/(x-1) . (x-1)/2`

`P=(12\sqrt{x}-2x)/2`

`P=(2(6\sqrt{x}-x))/2`

`P=6\sqrt{x}-x`

Xét `P-9`, ta có:

`6\sqrt{x}-x-9`

`=-x+6\sqrt{x}-9`

`=-(x-6\sqrt{x}+9)`

`=-(\sqrt{x}-3)^2<=0 AA x>=0;x\ne1`

`=>P-9<=0`

`=>P<=9 (đpcm)`