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

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

Solve for x

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

Solution Steps

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

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

The solution satisfying both inequalities is x\leq -\frac{3}{2}.

$$x\leq -\frac{3}{2}$$

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

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

The solution satisfying both inequalities is x\geq 5.

$$x\geq 5$$

The final solution is the union of the obtained solutions.

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