Solve x-2/5-x>0

$$\frac { x - 2 } { 5 - x } > 0$$

Solve for x

$$x\in \left(2,5\right)$$

Solution Steps

For the quotient to be positive, x-2 and 5-x have to be both negative or both positive. Consider the case when x-2 and 5-x are both negative.

$$x-2<0 5-x<0$$

This is false for any x.

$$x\in \emptyset$$

Consider the case when x-2 and 5-x are both positive.

$$5-x>0 x-2>0$$

The solution satisfying both inequalities is x\in \left(2,5\right).

$$x\in \left(2,5\right)$$

The final solution is the union of the obtained solutions.

$$x\in \left(2,5\right)$$