Question

B3: Cho: 

\(M=\dfrac{x-7}{x-4\sqrt{x}+3}+\dfrac{1}{\sqrt{x}-1}-\dfrac{1}{\sqrt{x}-3}\)      với \(x\ge0,x\ne1,x\ne9\)

a,  Rút gọn M 

b,  Tìm x để M \(>\dfrac{3}{4}\)

Answer 1

a) Ta có: \(M=\dfrac{x-7}{x-4\sqrt{x}+3}+\dfrac{1}{\sqrt{x}-1}-\dfrac{1}{\sqrt{x}-3}\)

\(=\dfrac{x-7+\sqrt{x}-3-\sqrt{x}+1}{\left(\sqrt{x}-3\right)\left(\sqrt{x}-1\right)}\)

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

b) Để \(M>\dfrac{3}{4}\) thì \(M-\dfrac{3}{4}>0\)

\(\Leftrightarrow\dfrac{\sqrt{x}+3}{\sqrt{x}-1}-\dfrac{3}{4}>0\)

\(\Leftrightarrow\dfrac{4\sqrt{x}+12-3\sqrt{x}+3}{4\left(\sqrt{x}-1\right)}>0\)

\(\Leftrightarrow\sqrt{x}-1>0\)

\(\Leftrightarrow x>1\)

Kết hợp ĐKXĐ, ta được: \(\left\{{}\begin{matrix}x>1\\x\ne9\end{matrix}\right.\)