Solve (x-1)(4-x)leq0

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

Solve for x

$$x\in (-\infty,1]\cup [4,\infty)$$

Solution Steps

For the product to be ≤0, one of the values x-1 and 4-x has to be ≥0 and the other has to be ≤0. Consider the case when x-1\geq 0 and 4-x\leq 0.

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

The solution satisfying both inequalities is x\geq 4.

$$x\geq 4$$

Consider the case when x-1\leq 0 and 4-x\geq 0.

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

The solution satisfying both inequalities is x\leq 1.

$$x\leq 1$$

The final solution is the union of the obtained solutions.

$$x\geq 4\text{; }x\leq 1$$