Dễ thấy: \(\left\{{}\begin{matrix}\left|x+1\right|\ge0\\\left|x+2\right|\ge0\\................\\\left|x+2010\right|\ge0\end{matrix}\right.\)\(\forall x\)
\(\Rightarrow |x+1| + |x+2| + |x+3| +....+ |x+2010|\ge0\)\(\forall x\)
\(\Rightarrow VT\ge0\forall x\Rightarrow VP\ge0\Rightarrow2011x\ge0\Rightarrow x\ge0\)
Vậy \(pt\Leftrightarrow\left(x+1\right)+\left(x+2\right)+...+\left(x+2010\right)=2011x\)
\(\Leftrightarrow\left(x+x+...+x\right)+\left(1+2+...+2010\right)=2011x\)
\(\Leftrightarrow2010x+\dfrac{2010\cdot\left(2010+1\right)}{2}=2011x\)
\(\Leftrightarrow2011x-2010x=\dfrac{2010\cdot\left(2010+1\right)}{2}\)
\(\Leftrightarrow x=2021055\)
