Solve x-2/x-1>=0

$$\frac{ x-2 }{ x-1 } > =0$$

Solve for x

$$x\in \left(-\infty,1\right)\cup \left(2,\infty\right)$$

Solution Steps

For the quotient to be positive, x-2 and x-1 have to be both negative or both positive. Consider the case when x-2 and x-1 are both negative.

$$x-2<0 x-1<0$$

The solution satisfying both inequalities is x<1.

$$x<1$$

Consider the case when x-2 and x-1 are both positive.

$$x-1>0 x-2>0$$

The solution satisfying both inequalities is x>2.

$$x>2$$

The final solution is the union of the obtained solutions.

$$x<1\text{; }x>2$$