\(\left|\dfrac{2x}{x-1}-1\right|>3\left(ĐK:x\ne1\right)\)
\(\Leftrightarrow\left|\dfrac{2x-\left(x-1\right)}{x-1}\right|>3\)
\(\Leftrightarrow\left|\dfrac{x+1}{x-1}\right|>3\)
\(TH1:\left|\dfrac{x+1}{x-1}\right|>3\)
\(\Leftrightarrow\dfrac{x+1}{x-1}-3>0\)
\(\Leftrightarrow\dfrac{4-2x}{x-1}>0\)
\(\Leftrightarrow1< x< 2\)
\(TH2:\dfrac{x+1}{x-1}< -3\)
\(\Leftrightarrow\dfrac{x+1}{x-1}+3< 0\)
\(\Leftrightarrow\dfrac{4x-2}{x-1}< 0\)
\(\Leftrightarrow\dfrac{1}{2}< x< 1\)
Kết hợp cả 2 TH ta có: \(\dfrac{1}{2}< x< 2\) ( với \(x\ne1\) )
\(\left|\dfrac{2x}{x-1}-1\right|>3\)
\(\Leftrightarrow\left|2x-x+1\right|>3\)
\(\Leftrightarrow\left|x+1\right|>3\)
\(\Leftrightarrow\left[{}\begin{matrix}x+1>3\\x+1< -3\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}x>2\\x< -4\end{matrix}\right.\)
Vậy tập nghiệm của bpt là x > 2 hoặc x < -4
