When x+4>0, i.e., x>-4 we can write x^2-9<0 so (x-3)(x+3)<0. That implies x-3<0 so x<3 or x+3<0 and x<-3. But we need to take care here. Let's make the problem a bit easier. Draw a straight line and mark in order from left (negative) to right (positive) the points -4, -3 and 3. We have divided the line into 4 segments:
- -infinity to -4
- -4 to -3
- -3 to +3
- 3 to +infinity.
Put a cross at -4, because it's a no-no point.
Now we place x in these four segments one at a time and see what happens to the expression. We're only interested in the main about whether a quantity is positive or negative (+) or (-).
Segment 1
(+)/(-)<0 satisfies the inequality so x<-4 is a solution.
Segment 2
(-)/(+)<0 OK, so -4<x<-3 because x=-3 gives zero which also satisfies the inequality.
Segment 3
(-)/(+)<0, so -3<x<3 because x=3 also valid.
Segment 4
(+)/(+)>0 not valid.
x=-4 must be excluded because we would be dividing by zero.
Conclusion
x<-4
-4<x<3 (segments 2 and 3)