HEL On a coordinate plane, 2 solid straight lines are shown. The first line has a negative slope and goes through (negative 4, negative 2) and (0, negative 3). Everything above the line is shaded. The second line has a positive slope and goes through (0, negative 2) and (2, 2). Everything above the line is shaded. Which number completes the system of linear inequalities represented by the graph? y > 2x – 2 and x + 4y >

y > 2x - 2 and x + 4y > -12

Explanation:

Given

Let the first line be A. So, the coordinates are:

(-4,-2) and (0,-3)

Let the second line be B. So the coordinates are:.

(0,-2) and (2,2)

Required

Complete the inequality

[First, we need to determine the slope (m) of A]

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

[Where]

(x1,y1) = (-4,-2) and (x2,y2) = (0,-3)

[The equation becomes]

m = (-3 -(-2))/(0 - (-4))

m = (-3 + 2)/(0 + 4)

m = (-1)/4)

m = -¼

[Next, we calculate the equation of A using]

y - y1 > m(x - x1)

[We used > because from the question, we understand that everything above the line is shaded.]

So, the inequality becomes.

y - (-3) > -¼(x - 0)

y + 3 > -¼(x)

y + 3 > -¼x

[Multiply through by 4]

4y + 12 > -x

[Add x to both sides]

x + 4y + 12 > x - x

x + 4y + 12 > 0

[Subtract 12 from both sides.]

x + 4y + 12 -12 >0 -12

x + 4y > -12

[We do the same operations for line B]

First, we need to determine the slope (m)

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

Where

(x1,y1) = (0,-2) and (x2,y2) = (2,2)

[The equation becomes]

m = (2 -(-2))/(2 - 0)

m = (2 + 2)/(2-0)

m = 4/2

m = 2

[Next, we calculate the equation of B using]

y - y1 > m(x - x1)

[We used > for the same reason as line A]

So, the inequality becomes.

y - (-2) > 2(x - 0)

y + 2 > 2(x)

y + 2 > 2x

[Subtract 2 from both sides]

y > 2x - 2

[Hence, the complete linear inequalities are:]

y > 2x - 2 and x + 4y > -12