Solve (2x+5)(4-3x)geq0

$$( 2 x + 5 ) ( 4 - 3 x ) \geq 0$$

Solve for x

$$x\in \begin{bmatrix}-\frac{5}{2},\frac{4}{3}\end{bmatrix}$$

Solution Steps

For the product to be ≥0, 2x+5 and 4-3x have to be both ≤0 or both ≥0. Consider the case when 2x+5 and 4-3x are both ≤0.

$$2x+5\leq 0 4-3x\leq 0$$

This is false for any x.

$$x\in \emptyset$$

Consider the case when 2x+5 and 4-3x are both ≥0.

$$4-3x\geq 0 2x+5\geq 0$$

The solution satisfying both inequalities is x\in \left[-\frac{5}{2},\frac{4}{3}\right].

$$x\in \begin{bmatrix}-\frac{5}{2},\frac{4}{3}\end{bmatrix}$$

The final solution is the union of the obtained solutions.

$$x\in \begin{bmatrix}-\frac{5}{2},\frac{4}{3}\end{bmatrix}$$