Which real numbers satisfies the inequality (x+3)(x+4)>0?

To satisfy the inequality, both factors of the product
have to have the same sign, that means that, if (x+3) is positive, (x+4)>, too,
and reverse.

Let's solve both
cases.

First, let's consider both factors as being
positive:

x+3>0

x+3-3>-3

x+0>-3

x>-3,
which is the interval (-3,
inf.)

x+4>0

x+4-4>-4

x>-4,
which is the interval (-4, inf.)

The common interval which
satisfies the inequality is (-3, inf.).

The other case is
when both factors are
negative.

x+3<0

x<-3,
which is the interval (-inf,
-3)

and

x+4<0

x<-4,
which is the interval (-inf, -4).

The common interval which
satisfies the inequality is (-inf, -4).