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

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

Solve for x

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

Solution Steps

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

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

The solution satisfying both inequalities is x\leq -1.

$$x\leq -1$$

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

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

The solution satisfying both inequalities is x\geq 4.

$$x\geq 4$$

The final solution is the union of the obtained solutions.

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