\(\dfrac{x+1}{2012}+\dfrac{x+2}{2011}=\dfrac{x+3}{2010}+\dfrac{x+4}{2009}\)
\(\Leftrightarrow1+\dfrac{x+1}{2012}+1+\dfrac{x+2}{2011}=1+\dfrac{x+3}{2010}+1+\dfrac{x+4}{2009}\) \(\Leftrightarrow\dfrac{x+1+2012}{2012}+\dfrac{x+2+2011}{2011}=\dfrac{x+3+2010}{2010}+\dfrac{x+4+2009}{2009}\) \(\Leftrightarrow\dfrac{x+2013}{2012}+\dfrac{x+2013}{2011}-\dfrac{x+2013}{2010}-\dfrac{x+2013}{2009}=0\) \(\Leftrightarrow\left(x+2013\right)\left(\dfrac{1}{2012}+\dfrac{1}{2011}-\dfrac{1}{2010}-\dfrac{1}{2009}\right)=0\)
Vì \(\dfrac{1}{2012}+\dfrac{1}{2011}-\dfrac{1}{2010}-\dfrac{1}{2009}\ne0\)
\(\Rightarrow x+2013=0\)
\(\Rightarrow x=-2013\)
Vậy........
\(\dfrac{x+1}{2012}+\dfrac{x+2}{2011}=\dfrac{x+3}{2010}+\dfrac{x+4}{2009}\)
\(\Leftrightarrow\dfrac{x+1}{2012}+1+\dfrac{x+2}{2011}+1=\dfrac{x+3}{2010}+1+\dfrac{x+4}{2009}+1\)
\(\Leftrightarrow\dfrac{x+2013}{2012}+\dfrac{x+2013}{2011}-\dfrac{x+2013}{2010}-\dfrac{x+2013}{2009}=0\)
\(\Leftrightarrow\left(x+2013\right)\left(\dfrac{1}{2012}+\dfrac{1}{2011}-\dfrac{1}{2010}-\dfrac{1}{2009}\right)=0\)
\(\Leftrightarrow x=-2013\)(vì \(\dfrac{1}{2012}+\dfrac{1}{2011}-\dfrac{1}{2010}-\dfrac{1}{2009}\ne0\))
