Question

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

Answer 1

Either `x>0` or `x<-4`

Explanation:

We can multiply both sides of the inequality by `x^2` to get

`x(x+4) > 0`

Note that we multiply by `x^2` since it is always positive and multiplying by a positive quantity does not change the sign of the inequality (on the other hand, multiplying by `x` would have lead apparently to `x+4<0`, which is wrong!)

Now, for `x(x+4)>0`, either both factors have to be positive, or both have to be negative. In the first case, `x` has to be greater than 0, in the second, it has to be smaller than -4.