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

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

Solve for x

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

Solution Steps

For the product to be positive, x-1 and x-4 have to be both negative or both positive. Consider the case when x-1 and x-4 are both negative.

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

The solution satisfying both inequalities is x<1.

$$x<1$$

Consider the case when x-1 and x-4 are both positive.

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

The solution satisfying both inequalities is x>4.

$$x>4$$

The final solution is the union of the obtained solutions.

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