Solve x-1/6-x>0

$$\frac { x - 1 } { 6 - x } > 0$$

Solve for x

$$x\in \left(1,6\right)$$

Solution Steps

For the quotient to be positive, x-1 and 6-x have to be both negative or both positive. Consider the case when x-1 and 6-x are both negative.

$$x-1<0 6-x<0$$

This is false for any x.

$$x\in \emptyset$$

Consider the case when x-1 and 6-x are both positive.

$$6-x>0 x-1>0$$

The solution satisfying both inequalities is x\in \left(1,6\right).

$$x\in \left(1,6\right)$$

The final solution is the union of the obtained solutions.

$$x\in \left(1,6\right)$$