Solve -x+6/x-9geq0

$$\frac { - x + 6 } { x - 9 } \geq 0$$

Solve for x

$$x\in [6,9)$$

Solution Steps

For the quotient to be ≥0, 6-x and x-9 have to be both ≤0 or both ≥0, and x-9 cannot be zero. Consider the case when 6-x\leq 0 and x-9 is negative.

$$6-x\leq 0 x-9<0$$

The solution satisfying both inequalities is x\in \left[6,9\right).

$$x\in [6,9)$$

Consider the case when 6-x\geq 0 and x-9 is positive.

$$6-x\geq 0 x-9>0$$

This is false for any x.

$$x\in \emptyset$$

The final solution is the union of the obtained solutions.

$$x\in [6,9)$$