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