jmc

If $x+4\ge 0$ (or $x\ge -4$), then the given inequality is the same as $x+4<9$ which means $x<5$. If $x+4<0$ (or $x<-4$), we have $-(x+4)<9$ which means $x+4>-9$ which yields $x>-13$. Thus, the solution is $-13<x<5$. Thus, the integers in this solution are -1 through -12 (12 integers), 1 through 4 (4 integers), and 0 (1 integer). Thus, the total is $12+4+1=\boxed{17}$ integers.