Solve (x-2)(x-3)geq0

$$( x - 2 ) ( x - 3 ) \geq 0$$

Solve for x

$$x\in (-\infty,2]\cup [3,\infty)$$

Solution Steps

For the product to be ≥0, x-2 and x-3 have to be both ≤0 or both ≥0. Consider the case when x-2 and x-3 are both ≤0.

$$x-2\leq 0 x-3\leq 0$$

The solution satisfying both inequalities is x\leq 2.

$$x\leq 2$$

Consider the case when x-2 and x-3 are both ≥0.

$$x-3\geq 0 x-2\geq 0$$

The solution satisfying both inequalities is x\geq 3.

$$x\geq 3$$

The final solution is the union of the obtained solutions.

$$x\leq 2\text{; }x\geq 3$$