Question

A=(\(\dfrac{1}{\sqrt{x}-1}-\dfrac{1}{\sqrt{x}}\) )/( \(\dfrac{\sqrt{x}+1}{\sqrt{x}-2}-\dfrac{\sqrt{x}+2}{\sqrt{x}-1}\) )

a, rút gọn a 

b, tìm giá trị cuae x để A=0

c, tìm các giá trị của x để A<0

            Mọi Người giúp em vs ạ 

Answer 1

ĐKXĐ: \(x>0;x\ne\left\{1;4\right\}\)

\(A=\left(\dfrac{\sqrt{x}}{\sqrt{x}\left(\sqrt{x}-1\right)}-\dfrac{\sqrt{x}-1}{\sqrt{x}\left(\sqrt{x}-1\right)}\right):\left(\dfrac{\left(\sqrt{x}+1\right)\left(\sqrt{x}-1\right)}{\left(\sqrt{x}-2\right)\left(\sqrt{x}-1\right)}-\dfrac{\left(\sqrt{x}+2\right)\left(\sqrt{x}-2\right)}{\left(\sqrt{x}-2\right)\left(\sqrt{x}-1\right)}\right)\)

\(=\left(\dfrac{1}{\sqrt{x}\left(\sqrt{x}-1\right)}\right):\left(\dfrac{3}{\left(\sqrt{x}-2\right)\left(\sqrt{x}-1\right)}\right)\)

\(=\dfrac{1}{\sqrt{x}\left(\sqrt{x}-1\right)}.\dfrac{\left(\sqrt{x}-2\right)\left(\sqrt{x}-1\right)}{3}\)

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

b.

\(A=0\Rightarrow\dfrac{\sqrt{x}-2}{3\sqrt{x}}=0\Rightarrow\sqrt{x}-2=0\)

\(\Rightarrow\sqrt{x}=2\Rightarrow x=4\)

c.

\(A< 0\Rightarrow\dfrac{\sqrt{x}-2}{3\sqrt{x}}< 0\Rightarrow\sqrt{x}-2< 0\) (do \(3\sqrt{x}>0;\forall x\in TXĐ\))

\(\Rightarrow x< 4\)

Kết hợp ĐKXĐ \(\Rightarrow\left\{{}\begin{matrix}0< x< 4\\x\ne1\end{matrix}\right.\)