Solve x-2/x-1geq0

$$\frac { x - 2 } { x - 1 } \geq 0$$

Solve for x

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

Solution Steps

For the quotient to be ≥0, x-2 and x-1 have to be both ≤0 or both ≥0, and x-1 cannot be zero. Consider the case when x-2\leq 0 and x-1 is negative.

$$x-2\leq 0 x-1<0$$

The solution satisfying both inequalities is x<1.

$$x<1$$

Consider the case when x-2\geq 0 and x-1 is positive.

$$x-2\geq 0 x-1>0$$

The solution satisfying both inequalities is x\geq 2.

$$x\geq 2$$

The final solution is the union of the obtained solutions.

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