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

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

Solve for x

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

Solution Steps

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

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

This is false for any x.

$$x\in \emptyset$$

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

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

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

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

The final solution is the union of the obtained solutions.

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