Solve x-6/x-2>0

$$\frac { x - 6 } { x - 2 } > 0$$

Solve for x

$$x\in \left(-\infty,2\right)\cup \left(6,\infty\right)$$

Solution Steps

For the quotient to be positive, x-6 and x-2 have to be both negative or both positive. Consider the case when x-6 and x-2 are both negative.

$$x-6<0 x-2<0$$

The solution satisfying both inequalities is x<2.

$$x<2$$

Consider the case when x-6 and x-2 are both positive.

$$x-2>0 x-6>0$$

The solution satisfying both inequalities is x>6.

$$x>6$$

The final solution is the union of the obtained solutions.

$$x<2\text{; }x>6$$