Solve (x-3)(x-5)>0

$$( x - 3 ) ( x - 5 ) > 0$$

Solve for x

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

Solution Steps

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

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

The solution satisfying both inequalities is x<3.

$$x<3$$

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

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

The solution satisfying both inequalities is x>5.

$$x>5$$

The final solution is the union of the obtained solutions.

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