Question

The graph shows a system of inequalities. Graph of two inequalities. One is a solid line increasing from left to right passing through negative 3 comma 0 and 0 comma 3, and it has shading below the line. The second is a solid upward opening parabola with a vertex at 1 comma negative 4 and x-intercepts at negative 1 comma 0 and 3 comma 0. This parabola is shaded inside the curve. Which system is represented in the graph? y > x2 –2x – 3 y > x + 3 y < x2 – 2x – 3 y < x + 3 y ≥ x2 – 2x – 3 y ≤ x + 3 y > x2 – 2x – 3 y < x + 3

Answer 1

The inequalities in the system of inequalities are y > x² – 2x – 3 and y > 3x + 4

How to determine the system of inequalities?

The graph that completes the question is added as an attachment

The parabola

It has a vertex of (1,  -4)

So, we have:

y = a(x - 1)^2 - 4

It passes through (3, 0).

So, we have:

0 = a(3 - 1)^2 - 4

This gives

4a - 4 = 0

Solve for a

a = 1

Substitute a = 1 in y = a(x - 1)^2 - 4

y = (x - 1)^2 - 4

Expand

y = x^2 - 2x + 1 - 4

y = x^2 - 2x - 3

The inner part is shaded.

So, we have:

y > x^2 - 2x - 3

The linear equation

It passes through (0, 4).

So, we have:

y = mx + 4

It passes through (1, 7).

So, we have:

7 = m + 4

Solve for m

m = 3

Substitute m = 3 in y = mx + 4

y = 3x + 4

The upper part is shaded.

So, we have:

y > 3x + 4

Hence, the inequalities in the system of inequalities are y > x² – 2x – 3 and y > 3x + 4

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