Tìm x biết: x-1/2011 x-2/2010-x-3/2009=x-4/2008

Lời giải:

\(\begin{array}{l}\frac{{x - 1}}{{2011}} + \frac{{x - 2}}{{2010}} - \frac{{x - 3}}{{2009}} = \frac{{x - 4}}{{2008}}\\\frac{{x - 1}}{{2011}} + \frac{{x - 2}}{{2010}} = \frac{{x - 4}}{{2008}} + \frac{{x - 3}}{{2009}}\\\left( {\frac{{x - 1}}{{2011}} - 1} \right) + \left( {\frac{{x - 2}}{{2010}} - 1} \right) = \left( {\frac{{x - 4}}{{2008}} - 1} \right) + \left( {\frac{{x - 3}}{{2009}} - 1} \right)\\\left( {\frac{{x - 1}}{{2011}} - \frac{{2011}}{{2011}}} \right) + \left( {\frac{{x - 2}}{{2010}} - \frac{{2010}}{{2010}}} \right) = \left( {\frac{{x - 4}}{{2008}} - \frac{{2008}}{{2008}}} \right) + \left( {\frac{{x - 3}}{{2009}} - \frac{{2009}}{{2009}}} \right)\\\frac{{x - 2012}}{{2011}} + \frac{{x - 2012}}{{2010}} = \frac{{x - 2012}}{{2009}} + \frac{{x - 2012}}{{2008}}\\\left( {x - 2012} \right).\left( {\frac{1}{{2011}} + \frac{1}{{2010}} - \frac{1}{{2008}} - \frac{1}{{2009}}} \right) = 0\end{array}\)

Mà \(\frac{1}{{2011}} + \frac{1}{{2010}} - \frac{1}{{2008}} - \frac{1}{{2009}} \ne 0\) nên \(x - 2012 = 0\,\,suy\,\,ra\,\,x = 2012\)