\(\left|x+\dfrac{1}{101}\right|+\left|x+\dfrac{2}{101}\right|+\left|x+\dfrac{3}{101}\right|+...+\left|x+\dfrac{100}{101}\right|=101x\)
Ta có : \(\left\{{}\begin{matrix}\left|x+\dfrac{1}{101}\right|\ge0\\\left|x+\dfrac{1}{102}\right|\ge0\\....\\\left|x+\dfrac{100}{101}\right|\ge0\end{matrix}\right.\)
\(\Rightarrow\left|x+\dfrac{1}{101}\right|+\left|x+\dfrac{2}{101}\right|+\left|x+\dfrac{3}{101}\right|+...+\left|x+\dfrac{100}{101}\right|\ge0\)
\(\Rightarrow101x\ge0\)
\(\Rightarrow x\ge0\)
\(\Rightarrow\left\{{}\begin{matrix}\left|x+\dfrac{1}{101}\right|=x+\dfrac{1}{101}\\\left|x+\dfrac{2}{101}\right|=x+\dfrac{2}{101}\\....\\\left|x+\dfrac{100}{101}\right|=x+\dfrac{100}{101}\end{matrix}\right.\)
\(\Rightarrow x+\dfrac{1}{101}+x+\dfrac{2}{101}+x+\dfrac{3}{101}+...+x+\dfrac{100}{101}=101x\)
\(\Rightarrow100x+\dfrac{1+2+3+...+100}{101}=101x\)
\(\Rightarrow100x+\dfrac{5050}{101}=101x\)
\(\Rightarrow100x+50=101x\)
\(\Rightarrow101x-100x=50\)
\(\Rightarrow x=50\)
Vậy \(x=50\)
