Solve (2x+1)(2-5x)geq0

$$( 2 x + 1 ) ( 2 - 5 x ) \geq 0$$

Solve for x

$$x\in \begin{bmatrix}-\frac{1}{2},\frac{2}{5}\end{bmatrix}$$

Solution Steps

For the product to be ≥0, 2x+1 and 2-5x have to be both ≤0 or both ≥0. Consider the case when 2x+1 and 2-5x are both ≤0.

$$2x+1\leq 0 2-5x\leq 0$$

This is false for any x.

$$x\in \emptyset$$

Consider the case when 2x+1 and 2-5x are both ≥0.

$$2-5x\geq 0 2x+1\geq 0$$

The solution satisfying both inequalities is x\in \left[-\frac{1}{2},\frac{2}{5}\right].

$$x\in \begin{bmatrix}-\frac{1}{2},\frac{2}{5}\end{bmatrix}$$

The final solution is the union of the obtained solutions.

$$x\in \begin{bmatrix}-\frac{1}{2},\frac{2}{5}\end{bmatrix}$$