Ta có: \(\frac{x+1}{2009}+\frac{x+3}{2007}=\frac{x+5}{2005}+\frac{x+7}{2003}\)
\(\Leftrightarrow\frac{x+1}{2009}+1+\frac{x+3}{2007}+1=\frac{x+5}{2005}+1+\frac{x+7}{2003}+1\)
\(\Leftrightarrow\frac{x+1+2009}{2009}+\frac{x+3+2007}{2007}=\frac{x+5+2005}{2005}+\frac{x+7+2003}{2003}\)
\(\Leftrightarrow\frac{x+2010}{2009}+\frac{x+2010}{2007}=\frac{x+2010}{2005}+\frac{x+2010}{2003}\)
\(\Leftrightarrow\frac{x+2010}{2009}+\frac{x+2010}{2007}-\frac{x+2010}{2005}-\frac{x+2010}{2003}=0\)
\(\Leftrightarrow\left(x+2010\right)\left(\frac{1}{2009}+\frac{1}{2007}-\frac{1}{2005}-\frac{1}{2003}\right)=0\)
Vì \(\frac{1}{2009}+\frac{1}{2007}-\frac{1}{2005}-\frac{1}{2003}\ne0\)
=> x + 2010 = 0
=> x = -2010
Vậy x = -2010
\(\frac{x+1}{2009}+\frac{x+3}{2007}=\frac{x+5}{2005}+\frac{x+7}{2003}\)
\(\Leftrightarrow\left(\frac{x+1}{2009}+1\right)+\left(\frac{x+3}{2007}+1\right)=\left(\frac{x+5}{2005}+1\right)+\left(\frac{x+7}{2003}+1\right)\)
\(\Leftrightarrow\left(\frac{x+2010}{2009}\right)+\left(\frac{x+2010}{2007}\right)=\left(\frac{x+2010}{2005}\right)+\left(\frac{x+2010}{2003}\right)\)
\(\Leftrightarrow\left(x+2010\right)\left(\frac{1}{2009}+\frac{1}{2007}-\frac{1}{2005}-\frac{1}{2003}\right)=0\)
\(\Leftrightarrow x+2010=0\) ( Vì \(\frac{1}{2009}+\frac{1}{2007}-\frac{1}{2005}-\frac{1}{2003}\ne0\))
\(\Leftrightarrow x=-2010\)
Vậy tập nghiệm của phương trình là S = { -2010 } .
