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x+5<0 x-3<0
For the product to be positive, x+5 and x-3 have to be both negative or both positive. Consider the case when x+5 and x-3 are both negative.
x<-5
The solution satisfying both inequalities is x<-5.
x-3>0 x+5>0
Consider the case when x+5 and x-3 are both positive.
x>3
The solution satisfying both inequalities is x>3.
x<-5\text{; }x>3
The final solution is the union of the obtained solutions.