Solve (1-2x)(x+3)geq0

$$( 1 - 2 x ) ( x + 3 ) \geq 0$$

Solve for x

$$x\in \begin{bmatrix}-3,\frac{1}{2}\end{bmatrix}$$

Solution Steps

For the product to be ≥0, x+3 and 1-2x have to be both ≤0 or both ≥0. Consider the case when x+3 and 1-2x are both ≤0.

$$x+3\leq 0 1-2x\leq 0$$

This is false for any x.

$$x\in \emptyset$$

Consider the case when x+3 and 1-2x are both ≥0.

$$1-2x\geq 0 x+3\geq 0$$

The solution satisfying both inequalities is x\in \left[-3,\frac{1}{2}\right].

$$x\in \begin{bmatrix}-3,\frac{1}{2}\end{bmatrix}$$

The final solution is the union of the obtained solutions.

$$x\in \begin{bmatrix}-3,\frac{1}{2}\end{bmatrix}$$