Solve x-2/x>0

$$\frac{ x-2 }{ x } > 0$$

Solve for x

$$x\in \left(-\infty,0\right)\cup \left(2,\infty\right)$$

Solution Steps

For the quotient to be positive, x-2 and x have to be both negative or both positive. Consider the case when x-2 and x are both negative.

$$x-2<0 x<0$$

The solution satisfying both inequalities is x<0.

$$x<0$$

Consider the case when x-2 and x are both positive.

$$x>0 x-2>0$$

The solution satisfying both inequalities is x>2.

$$x>2$$

The final solution is the union of the obtained solutions.

$$x<0\text{; }x>2$$