\(\dfrac{x-1}{2013}+\dfrac{x-2}{2012}+\dfrac{x-3}{2011}=\dfrac{x-4}{2010}+\dfrac{x-5}{2009}+\dfrac{x-6}{2008}\)

\(\Leftrightarrow\dfrac{x-1}{2013}-1+\dfrac{x-2}{2012}-1+\dfrac{x-3}{2011}-1=\dfrac{x-4}{2010}-1+\dfrac{x-5}{2009}-1+\dfrac{x-6}{2008}-1\)

=>x-2014=0

hay x=2014

x+5/2009 + x+4/2010 = x+3/2011 + x+2/2012

=> 1 + x+5/2009 + 1 + x+4/2000 = 1 + x+3/2011 + 1 + x+2/2012

=> x+2014/2009 + x+2014/2000 = x+2004/2011 + x+2014/2012

=> x+2014/2009 + x+2014/2000 - x+2014/2011 - x+2014/2012 = 0

=> (x+2014).(1/2009 + 1/2010 - 1/2011 - 1/2012) = 0

Do 1/2009 > 1/2011; 1/2010 > 1/2012

=> 1/2009 + 1/2010 - 1/2011 - 1/2012 khác 0

=> x + 2014 = 0

=> x = -2014

`Answer:`

\(\left(\frac{x+1}{2013}\right)+\left(\frac{x+2}{2012}\right)+\left(\frac{x+3}{2011}\right)=\left(\frac{x+4}{2010}\right)+\left(\frac{x+5}{2009}\right)+\left(\frac{x+6}{2008}\right)\)

\(\Leftrightarrow\frac{x+1}{2013}+1+\frac{x+2}{2012}+1+\frac{x+3}{2011}+1=\frac{x+4}{2010}+1+\frac{x+5}{2009}+1+\frac{x+6}{2008}+1\)

\(\Leftrightarrow\frac{x+2014}{2013}+\frac{x+2014}{2012}+\frac{x+2014}{2011}=\frac{x+2014}{2010}+\frac{x+2014}{2009}+\frac{x+2014}{2008}\)

\(\Leftrightarrow\frac{x+2014}{2013}+\frac{x+2014}{2012}+\frac{x+2014}{2011}-\frac{x+2014}{2010}-\frac{x+2014}{2009}-\frac{x+2014}{2008}=0\)

\(\Leftrightarrow\left(x+2014\right)\left(\frac{1}{2013}+\frac{1}{2012}+\frac{1}{2011}-\frac{1}{2010}-\frac{1}{2009}-\frac{1}{2008}\right)=0\)

\(\Rightarrow x+2014=0\)

\(\Leftrightarrow x=-2014\)

\(\dfrac{x+1}{2012}+\dfrac{x+2}{2011}=\dfrac{x+3}{2010}+\dfrac{x+4}{2009}\)

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

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 \(\dfrac{1}{2012}+\dfrac{1}{2011}-\dfrac{1}{2010}-\dfrac{1}{2009}\ne0\))