\(M=\left(\frac{1}{\sqrt{x}-1}+\frac{\sqrt{x}}{x-1}\right)\div\left(\frac{\sqrt{x}}{\sqrt{x}-1}-1\right)\)

\(=\frac{\sqrt{x}+1+\sqrt{x}}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}\div\frac{\sqrt{x}-\sqrt{x}+1}{\sqrt{x}-1}\)

\(=\frac{2\sqrt{x}+1}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}.\frac{\sqrt{x-1}}{1}\)

\(=\frac{2\sqrt{x}+1}{\sqrt{x}+1}\)

\(M< \frac{3}{2}\Leftrightarrow\frac{2\sqrt{x}+1}{\sqrt{x}+1}< \frac{3}{2}\Leftrightarrow2\left(2\sqrt{x}+1\right)< 3\left(\sqrt{x}+1\right)\)

\(\Leftrightarrow4\sqrt{x}+2< 3\sqrt{x}+3\Leftrightarrow\sqrt{x}< 1\Leftrightarrow0\le x< 1\).

\(\frac{\sqrt{x}-1}{\sqrt{x}+1}< \frac{1}{2}\Leftrightarrow\frac{\sqrt{x}-1}{\sqrt{x}+1}-\frac{1}{2}< 0\)

\(\Leftrightarrow\frac{2\sqrt{x}-2-\sqrt{x}-1}{2\left(\sqrt{x}+1\right)}< 0\Leftrightarrow\frac{\sqrt{x}-3}{2\left(\sqrt{x}+1\right)}< 0\)

\(\Leftrightarrow\sqrt{x}-3< 0\Leftrightarrow x< 9\Rightarrow0\le x< 9\)

anh đi anh nhớ quê nha 

nhớ canh rau muống nhớ cà dầm tương 

nhớ thằng đẩy bố xuống mương 

bố mà bắt được bố tương vỡ mồm

\(A=\sqrt{6+3\sqrt{3}}-\sqrt{14-3\sqrt{3}}-2\sqrt{2}\)

\(A\sqrt{2}=\sqrt{12+6\sqrt{3}}-\sqrt{28-6\sqrt{3}}-4\)

\(=\sqrt{9+2.3.\sqrt{3}+3}-\sqrt{27-2.3\sqrt{3}.1+1}-4\)

\(=\sqrt{\left(3+\sqrt{3}\right)^2}-\sqrt{\left(3\sqrt{3}-1\right)^2}-4\)

\(=3+\sqrt{3}-3\sqrt{3}+1-4\)

\(=-2\sqrt{3}\)

Suy ra \(A=-\sqrt{6}\).

ta có 

\(A=\frac{\sqrt{x}-\sqrt{x-1}-\left(\sqrt{x}+\sqrt{x-1}\right)}{\left(\sqrt{x}+\sqrt{x-1}\right)\left(\sqrt{x}-\sqrt{x-1}\right)}-\frac{0}{1-\sqrt{x}}\)

\(=-\frac{2\sqrt{x-1}}{x-\left(x-1\right)}=-2\sqrt{x-1}\) dễ thấy \(A\le0\) với mọi x

a) \(A=\frac{a+2\sqrt{ab}+b}{\sqrt{a}+\sqrt{b}}-\frac{\sqrt{ab}\left(\sqrt{a}-\sqrt{b}\right)}{\sqrt{ab}}\)

\(=\sqrt{a}+\sqrt{b}-\sqrt{a}+\sqrt{b}=2\sqrt{b}\)

b) \(B=\left(\frac{\sqrt{x}-\sqrt{y}\left(x+\sqrt{xy}+y\right)}{\sqrt{x}-\sqrt{y}}+\sqrt{xy}\right):\left(x-y\right)-\frac{2\sqrt{y}}{\sqrt{x}-\sqrt{y}}\)

\(=\frac{x+y+2\sqrt{xy}}{x-y}-\frac{2\sqrt{y}}{\sqrt{x}-\sqrt{y}}\)\(=\frac{\sqrt{x}+\sqrt{y}}{\sqrt{x}-\sqrt{y}}-\frac{2\sqrt{y}}{\sqrt{x}-\sqrt{y}}=\frac{\sqrt{x}-\sqrt{y}}{\sqrt{x}-\sqrt{y}}=1\)

\(ĐKXĐ:x\ge0;x\ne1\)

\(\frac{1}{\sqrt{x}-1}-\frac{2\sqrt{x}}{x\sqrt{x}-x+\sqrt{x}-1}\)

\(\frac{1}{\sqrt{x}-1}-\frac{2\sqrt{x}}{\left(\sqrt{x}-1\right)\left(x+1\right)}\)

\(\frac{x+1-2\sqrt{x}}{\left(\sqrt{x}-1\right)\left(x+1\right)}=\frac{\left(\sqrt{x}-1\right)^2}{\left(\sqrt{x}-1\right)\left(x+1\right)}=\frac{\sqrt{x}-1}{x+1}\)

\(=\frac{1}{\sqrt{x}-1}-\frac{2\sqrt{x}}{\left(\sqrt{x}-1\right)\left(x+1\right)}\)

\(=\frac{x+1-2\sqrt{x}}{\left(\sqrt{x}-1\right)\left(x+1\right)}=\frac{\left(\sqrt{x}-1\right)^2}{\left(\sqrt{x}-1\right)\left(x+1\right)}=\frac{\sqrt{x}-1}{x+1}\)

\(=\frac{x+\sqrt{x}}{x\left(\sqrt{x}+1\right)+\left(\sqrt{x}+1\right)}+\frac{1}{x+1}\)

\(=\frac{\sqrt{x}\left(\sqrt{x}+1\right)}{\left(\sqrt{x}+1\right)\left(x+1\right)}+\frac{1}{x+1}\)

\(=\frac{\sqrt{x}}{x+1}+\frac{1}{x+1}=\frac{\sqrt{x}+1}{x+1}\)

=)))))