$\dfrac{x+1}{2007} + \dfrac{x+2}{2006} = \dfrac{x+2001}{7} + \dfrac{x+2002}{6}$
$⇔ \dfrac{x+1}{2007} + \dfrac{x+2}{2006} + 2 = \dfrac{x+2001}{7} + \dfrac{x+2002}{6} + 2$
$⇔ \dfrac{x+1}{2007} + \dfrac{x+2}{2006} + 1 + 1 = \dfrac{x+2001}{7} + \dfrac{x+2002}{6} + 1 + 1$
$⇔ (\dfrac{x+1}{[2007}+1) + (\dfrac{x+2}{2006}+1) = (\dfrac{x+2001}{7}+1) + (\dfrac{x+2002}{6}+1)$
$⇔ \dfrac{x+2008}{2007} + \dfrac{x+2008}{2006} = \dfrac{x+2008}{7} + \dfrac{x+2008}{6}$
$⇔ \dfrac{x+2008}{2007} + \dfrac{x+2008}{2006} - \dfrac{x+2008}{7} - \dfrac{x+2008}{6} = 0$
$⇔ ( x + 2008 ) + ( \dfrac{1}{2007} + \dfrac{1}{2006} - \dfrac{1}{7} - \dfrac{1}{6} ) = 0 $
$⇔ x + 2008 = 0$
$⇔ x = -2008$
