How do you solve #(x+4)/x>0#?

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Answer 1 · 2016-07-21T06:21:22

#x in (-oo, -4) uu (0, oo)#

If #x = 0# then the denominator is #0# so the quotient is undefined. So #x = 0# is not part of the solution set.

That leaves two cases:

#color(white)()#
Case #bb(x > 0)#

Multiply both sides of the inequality by #x# to get:

#x+4 > 0#

Subtract #4# from both sides to get:

#x > -4#

Since #x > 0#, this holds already, so the whole of #(0, oo)# is part of the solution set.

#color(white)()#
Case #bb(x < 0)#

Multiply both sides of the inequality by #0# and reverse the inequality (since the multiplier is negative) to get:

#x+4 < 0#

Subtract #4# from both sides to get:

#x < -4#

So #(-oo, -4)# is part of the solution set.

#color(white)()#
Conclusion

The solution set is the union of these two cases, namely:

#(-oo, -4) uu (0, oo)#