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

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

Solve for x

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

Solution Steps

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

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

This is false for any x.

$$x\in \emptyset$$

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

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

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

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

The final solution is the union of the obtained solutions.

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