The graph of the function f (x) = (x - 3) (x + 1) is shown.

Which describes all of the values for which the graph is positive and decreasing?

A. all real values of x where x < - 1

B. all real values of x where x < 1

C. all real values of x where 1 < x < 3

D. all real values of x where x > 3

Option A. All the real values of x where x < - 1

Procedure

Solve the inequality:

(x - 3) (x+1) >0

That happens in two cases.

1) When both factors >0

x-3>0 and x+1>0

x>3 and x >-1

The intersection is x >3

2) When both factors <0

x-3<0 and x+1<0

x<3 and x<-1

the intersection is x<-1.

We have obtained that the function is positive for the intervals x 3. But in one of those intervals the function is decresing and in the other is increasing.

You can recognize that the function given is a parabola and, because the coefficient of the quadratic term is positive, the parabola opens upward. Then the function is decreasing in the first interval and increasing in the second interval.