ĐK: $\left\{\begin{array}{l}x\ne-1\\x\ne-2\end{array}\right.$
BPT tương đương với:
$x+1+\dfrac{1}{x+1}>x+2+\dfrac{1}{x+2}-1$
$\Leftrightarrow \dfrac{1}{x+1}-\dfrac{1}{x+2}>0$
$\Leftrightarrow \dfrac{1}{(x+1)(x+2)}>0$
$\Leftrightarrow (x+1)(x+2)>0$
$\Leftrightarrow \left[\begin{array}{l}x>-1\\x<-2\end{array}\right.$