Solve (x-4)(1-x)<0

$$( x - 4 ) ( 1 - x ) < 0$$

Solve for x

$$x\in \left(-\infty,1\right)\cup \left(4,\infty\right)$$

Solution Steps

For the product to be negative, 1-x and x-4 have to be of the opposite signs. Consider the case when 1-x is positive and x-4 is negative.

$$1-x>0 x-4<0$$

The solution satisfying both inequalities is x<1.

$$x<1$$

Consider the case when x-4 is positive and 1-x is negative.

$$x-4>0 1-x<0$$

The solution satisfying both inequalities is x>4.

$$x>4$$

The final solution is the union of the obtained solutions.

$$x>4\text{; }x<1$$