What is the equation of the line, that passes through (3,-1) and (-1,5)

y = -3/2x +7/2

Step-by-step explanation:

When we have two points, we need to find the slope.

Slope is given by

m = (y2-y1)/ (x2-x1)

   = (5--1)/ (-1-3)

   = (5+1)/ (-4)

   = 6/-4

   = -3/2

We have the slope and a point

Using the point slope form of a line

y-y1 = m(x-x1)

y--1 = -3/2( x-3)

y+1 = -3/2(x-3)

They usually want the line written in slope intercept form (y=mx+b)

Distribute the -3/2

y+1 = -3/2x +9/2

Subtract 1

y+1-1 = -3/2x +9/2 -2/2

y = -3/2x +7/2

Step-by-step explanation:

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 < -1 and 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.