Solve (-x+1)(x-5)geq0

$$(-x+1)(x-5) \geq 0$$

Solve for x

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

Solution Steps

For the product to be ≥0, 1-x and x-5 have to be both ≤0 or both ≥0. Consider the case when 1-x and x-5 are both ≤0.

$$1-x\leq 0 x-5\leq 0$$

The solution satisfying both inequalities is x\in \left[1,5\right].

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

Consider the case when 1-x and x-5 are both ≥0.

$$x-5\geq 0 1-x\geq 0$$

This is false for any x.

$$x\in \emptyset$$

The final solution is the union of the obtained solutions.

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