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Inequality with absolute values

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Hi folks,

I am looking for a good way to approach inequality problems that involve absolute value ratios, as given below. Any help is greatly appreciated.
Problem:

\frac{(x^2+6x-7)}{|x+4|} < 0

Options:
A) -7<x<-5 and -4<x<1
B) -7<x<-5 and -4<x<0
C) -7<x<-4 and -4<x<1
D) -7<x<-6 and -4<x<1

SOURCE: this is not from GMAT prep book; this problem is from a GRE practice book, which have many problems of this kind. I went through M-GMAT book inequalities chapter but haven't seen anything similar to this.

Thanks!

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