Solve (3-5x)(1-2x)leq0

$$(3-5x)(1-2x) \leq 0$$

Solve for x

$$x\in \begin{bmatrix}\frac{1}{2},\frac{3}{5}\end{bmatrix}$$

Solution Steps

For the product to be ≤0, one of the values 1-2x and 3-5x has to be ≥0 and the other has to be ≤0. Consider the case when 1-2x\geq 0 and 3-5x\leq 0.

$$1-2x\geq 0 3-5x\leq 0$$

This is false for any x.

$$x\in \emptyset$$

Consider the case when 1-2x\leq 0 and 3-5x\geq 0.

$$3-5x\geq 0 1-2x\leq 0$$

The solution satisfying both inequalities is x\in \left[\frac{1}{2},\frac{3}{5}\right].

$$x\in \begin{bmatrix}\frac{1}{2},\frac{3}{5}\end{bmatrix}$$

The final solution is the union of the obtained solutions.

$$x\in \begin{bmatrix}\frac{1}{2},\frac{3}{5}\end{bmatrix}$$