Solve (x-5)(x-12)<0

$$( x - 5 ) ( x - 12 ) < 0$$

Solve for x

$$x\in \left(5,12\right)$$

Solution Steps

For the product to be negative, x-5 and x-12 have to be of the opposite signs. Consider the case when x-5 is positive and x-12 is negative.

$$x-5>0 x-12<0$$

The solution satisfying both inequalities is x\in \left(5,12\right).

$$x\in \left(5,12\right)$$

Consider the case when x-12 is positive and x-5 is negative.

$$x-12>0 x-5<0$$

This is false for any x.

$$x\in \emptyset$$

The final solution is the union of the obtained solutions.

$$x\in \left(5,12\right)$$