Solve 5-x/3-x>0

$$\frac { 5 - x } { 3 - x } > 0$$

Solve for x

$$x\in \left(-\infty,3\right)\cup \left(5,\infty\right)$$

Solution Steps

For the quotient to be positive, 5-x and 3-x have to be both negative or both positive. Consider the case when 5-x and 3-x are both negative.

$$5-x<0 3-x<0$$

The solution satisfying both inequalities is x>5.

$$x>5$$

Consider the case when 5-x and 3-x are both positive.

$$3-x>0 5-x>0$$

The solution satisfying both inequalities is x<3.

$$x<3$$

The final solution is the union of the obtained solutions.

$$x>5\text{; }x<3$$