Solve -x+2/x-4geq0

$$\frac { - x + 2 } { x - 4 } \geq 0$$

Solve for x

$$x\in [2,4)$$

Solution Steps

For the quotient to be ≥0, 2-x and x-4 have to be both ≤0 or both ≥0, and x-4 cannot be zero. Consider the case when 2-x\leq 0 and x-4 is negative.

$$2-x\leq 0 x-4<0$$

The solution satisfying both inequalities is x\in \left[2,4\right).

$$x\in [2,4)$$

Consider the case when 2-x\geq 0 and x-4 is positive.

$$2-x\geq 0 x-4>0$$

This is false for any x.

$$x\in \emptyset$$

The final solution is the union of the obtained solutions.

$$x\in [2,4)$$