Question

How do you solve `(x-4)(x+1)>=0`?

Answer 1

`x <= -1` and `4<=x`

Explanation:

Zeros of the function `(x-4)(x+1)` are `{-1,4}` and they divide the real number line in three parts.

`x<=-1` - In this region the two binomials `(x+1)` and `(x-4)` both are negative (or zero) and hence their product is positive. So this region forms a solution.

`-1 < x < 4` - In this region while `(x+1)` is posiitive and `(x-4)` is negative and hence their product is negative. So this region does not form a solution.

`4 <= x` - In this region both `(x+1)` and `(x-4)` are positive (or zero) and hence their product is positive. So this region forms a solution.

This can also be checked from the following graph.

graph{(x-4)(x+1) [-5, 5, -10, 10]}