Ta lập bảng xét dấu:
+) Nếu x < 1 thì \(\left\{{}\begin{matrix}\left|x-1\right|=-x+1\\\left|2x-4\right|=-2x+4\end{matrix}\right.\)
Do đó: \(\left|x-1\right|+\left|2x-4\right|=3x\)
\(\Leftrightarrow\left(-x+1\right)+\left(-2x+4\right)=3x\)
\(\Leftrightarrow-x+1-2x+4=3x\)
\(\Leftrightarrow\left(-x-2x\right)+\left(1+4\right)=3x\)
\(\Leftrightarrow-3x+5=3x\)
\(\Leftrightarrow3x+3x=5\)
\(\Leftrightarrow6x=5\)
\(\Leftrightarrow x=\dfrac{5}{6}\)(Thỏa mãn)
+) Nếu \(1< x< 2\) thì \(\left\{{}\begin{matrix}\left|x-1\right|=x-1\\\left|2x-4\right|=-2x+4\end{matrix}\right.\)
Do đó: \(\left|x-1\right|+\left|2x-4\right|=3x\)
\(\Leftrightarrow\left(x-1\right)+\left(-2x+4\right)=3x\)
\(\Leftrightarrow x-1-2x+4=3x\)
\(\Leftrightarrow\left(x-2x\right)+\left(-1+4\right)=3x\)
\(\Leftrightarrow-x+3=3x\)
\(\Leftrightarrow3x+x=3\)
\(\Leftrightarrow4x=3\)
\(\Leftrightarrow x=\dfrac{3}{4}\) (k thỏa mãn nên loại)
+) Nếu \(x>2\) thì: \(\left\{{}\begin{matrix}\left|x-1\right|=x-1\\\left|2x-4\right|=2x-4\end{matrix}\right.\)
Do đó: \(\left|x-1\right|+\left|2x-4\right|=3x\)
\(\Leftrightarrow\left(x-1\right)+\left(2x-4\right)=3x\)
\(\Leftrightarrow x-1+2x-4=3x\)
\(\Leftrightarrow\left(x-2x\right)+\left(-1-4\right)=3x\)
\(\Leftrightarrow-x-5=3x\)
\(\Leftrightarrow2x=5\)
\(\Rightarrow x=\dfrac{5}{2}\) (thỏa mãn). Vậy \(\left[{}\begin{matrix}x=\dfrac{2}{3}\\x=\dfrac{5}{2}\end{matrix}\right.\)
