Solve 3-x/x-2<0

$$\frac { 3 - x } { x - 2 } < 0$$

Solve for x

$$x\in \left(-\infty,2\right)\cup \left(3,\infty\right)$$

Solution Steps

For the quotient to be negative, 3-x and x-2 have to be of the opposite signs. Consider the case when 3-x is positive and x-2 is negative.

$$3-x>0 x-2<0$$

The solution satisfying both inequalities is x<2.

$$x<2$$

Consider the case when x-2 is positive and 3-x is negative.

$$x-2>0 3-x<0$$

The solution satisfying both inequalities is x>3.

$$x>3$$

The final solution is the union of the obtained solutions.

$$x<2\text{; }x>3$$