Solve 0.25x/0.25x-1>0

$$\frac{ 0.25x }{ 0.25x-1 } > 0$$

Solve for x

$$x\in \left(-\infty,0\right)\cup \left(4,\infty\right)$$

Solution Steps

For the quotient to be positive, \frac{x}{4} and \frac{x}{4}-1 have to be both negative or both positive. Consider the case when \frac{x}{4} and \frac{x}{4}-1 are both negative.

$$\frac{x}{4}<0 \frac{x}{4}-1<0$$

The solution satisfying both inequalities is x<0.

$$x<0$$

Consider the case when \frac{x}{4} and \frac{x}{4}-1 are both positive.

$$\frac{x}{4}-1>0 \frac{x}{4}>0$$

The solution satisfying both inequalities is x>4.

$$x>4$$

The final solution is the union of the obtained solutions.

$$x<0\text{; }x>4$$