Solve (2-3x)(4x-7)geq0

$$(2-3x)(4x-7) \geq 0$$

Solve for x

$$x\in \begin{bmatrix}\frac{2}{3},\frac{7}{4}\end{bmatrix}$$

Solution Steps

For the product to be ≥0, 2-3x and 4x-7 have to be both ≤0 or both ≥0. Consider the case when 2-3x and 4x-7 are both ≤0.

$$2-3x\leq 0 4x-7\leq 0$$

The solution satisfying both inequalities is x\in \left[\frac{2}{3},\frac{7}{4}\right].

$$x\in \begin{bmatrix}\frac{2}{3},\frac{7}{4}\end{bmatrix}$$

Consider the case when 2-3x and 4x-7 are both ≥0.

$$4x-7\geq 0 2-3x\geq 0$$

This is false for any x.

$$x\in \emptyset$$

The final solution is the union of the obtained solutions.

$$x\in \begin{bmatrix}\frac{2}{3},\frac{7}{4}\end{bmatrix}$$