Solve x-6/2-xgeq0

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

Solve for x

$$x\in (2,6]$$

Solution Steps

For the quotient to be ≥0, x-6 and 2-x have to be both ≤0 or both ≥0, and 2-x cannot be zero. Consider the case when x-6\leq 0 and 2-x is negative.

$$x-6\leq 0 2-x<0$$

The solution satisfying both inequalities is x\in \left(2,6\right].

$$x\in (2,6]$$

Consider the case when x-6\geq 0 and 2-x is positive.

$$x-6\geq 0 2-x>0$$

This is false for any x.

$$x\in \emptyset$$

The final solution is the union of the obtained solutions.

$$x\in (2,6]$$