Solve x+6/x>0

$$\frac { x + 6 } { x } > 0$$

Solve for x

$$x\in \left(-\infty,-6\right)\cup \left(0,\infty\right)$$

Solution Steps

For the quotient to be positive, x+6 and x have to be both negative or both positive. Consider the case when x+6 and x are both negative.

$$x+6<0 x<0$$

The solution satisfying both inequalities is x<-6.

$$x<-6$$

Consider the case when x+6 and x are both positive.

$$x>0 x+6>0$$

The solution satisfying both inequalities is x>0.

$$x>0$$

The final solution is the union of the obtained solutions.

$$x<-6\text{; }x>0$$