Solve x-5/x-1<0

$$\frac{ x-5 }{ x-1 } < 0$$

Solve for x

$$x\in \left(1,5\right)$$

Solution Steps

For the quotient to be negative, x-5 and x-1 have to be of the opposite signs. Consider the case when x-5 is positive and x-1 is negative.

$$x-5>0 x-1<0$$

This is false for any x.

$$x\in \emptyset$$

Consider the case when x-1 is positive and x-5 is negative.

$$x-1>0 x-5<0$$

The solution satisfying both inequalities is x\in \left(1,5\right).

$$x\in \left(1,5\right)$$

The final solution is the union of the obtained solutions.

$$x\in \left(1,5\right)$$