Solve x-2/x-4geq0

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

Solve for x

$$x\in (-\infty,2]\cup (4,\infty)$$

Solution Steps

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

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

The solution satisfying both inequalities is x\leq 2.

$$x\leq 2$$

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

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

The solution satisfying both inequalities is x>4.

$$x>4$$

The final solution is the union of the obtained solutions.

$$x\leq 2\text{; }x>4$$