Solve (x-4)(1-x)geq0

$$( x - 4 ) ( 1 - x ) \geq 0$$

Solve for x

$$x\in \begin{bmatrix}1,4\end{bmatrix}$$

Solution Steps

For the product to be ≥0, 1-x and x-4 have to be both ≤0 or both ≥0. Consider the case when 1-x and x-4 are both ≤0.

$$1-x\leq 0 x-4\leq 0$$

The solution satisfying both inequalities is x\in \left[1,4\right].

$$x\in \begin{bmatrix}1,4\end{bmatrix}$$

Consider the case when 1-x and x-4 are both ≥0.

$$x-4\geq 0 1-x\geq 0$$

This is false for any x.

$$x\in \emptyset$$

The final solution is the union of the obtained solutions.

$$x\in \begin{bmatrix}1,4\end{bmatrix}$$