$\cfrac{2x+m-1}{x+1}>0$
$\to\cfrac{2x+m-1}{x+1}.\cfrac12>0.\cfrac12$
$\to\cfrac{x+\cfrac{m-1}2}{x+1}>0$
$\to\begin{cases}-\cfrac{m-1}{2}>-1\to\left[ \begin{array}{l}x<-1\\x>-\cfrac{m-1}{2}\end{array} \right.\\-\cfrac{m-1}2<-1\to\left[ \begin{array}{l}x>-1\\x<-\cfrac{m-1}{2}\end{array} \right.\end{cases}$\(\)