Solve (2x-5)(x+3)geq0

$$( 2 x - 5 ) ( x + 3 ) \geq 0$$

Solve for x

$$x\in (-\infty,-3]\cup [\frac{5}{2},\infty)$$

Solution Steps

For the product to be ≥0, x+3 and 2x-5 have to be both ≤0 or both ≥0. Consider the case when x+3 and 2x-5 are both ≤0.

$$x+3\leq 0 2x-5\leq 0$$

The solution satisfying both inequalities is x\leq -3.

$$x\leq -3$$

Consider the case when x+3 and 2x-5 are both ≥0.

$$2x-5\geq 0 x+3\geq 0$$

The solution satisfying both inequalities is x\geq \frac{5}{2}.

$$x\geq \frac{5}{2}$$

The final solution is the union of the obtained solutions.

$$x\leq -3\text{; }x\geq \frac{5}{2}$$