Question

Graph the system of inequalities presented here on your own paper, then use your graph to answer the following questions:

y > −2x + 3
y is less than 1 over 2 times x minus 2

Part A: Describe the graph of the system, including shading and the types of lines graphed. Provide a description of the solution area. (6 points)

Part B: Is the point (−2, 3) included in the solution area for the system? Justify your answer mathematically. (4 points)

Answer 1

A: for the first problem, the line is shades above a dotted line with a y intercept of 3, and a slope of -2. for the second equation, it is shaded below the dotted line, with a y intercept of -2 and a slope of 1/2. they intersect at (2,-1). the region has an x intercept of 4, and it does not have a y intercept, the double shaded region, it completely on the right side of the graph.

B: if you were to substitute the (-2,3) into both equations, it would not be true. 3>7, and 3< -3.5

Explanation:

Answer 2

Part A) The description of the solution area in the procedure

Part B) The ordered pair (−2, 3) is not included in the solution area for the system

Explanation:

Part A) we have

y > -2x+3 -----> inequality A

The solution of the inequality A is the shaded area above the dashed line

y=-2x+3

The slope of the dashed line is negative (m=-2)

The y-intercept of the dashed line is (0,3)

The x-intercept of the dashed line is (1.5,0)

y < (1/2)x-2 -----> inequality B

The solution of the inequality B is the shaded area beow the dashed line

y=(1/2)x-2

The slope of the dashed line is positive (m=1/2)

The y-intercept of the dashed line is (0,-2)

The x-intercept of the dashed line is (4,0)

using a graphing tool

The solution of the system of inequalities is the shaded area between the two dashed lines

see the attached figure

Part B) Is the point (−2, 3) included in the solution area for the system?

we know that

If a ordered pair is a solution of the system of inequalities, then the ordered pair must satisfy both inequalities of the system

Verify inequality A

Substitute the value of x=-2 and y=3 in the inequality A

3 > -2(-2)+3

3 > 7 -----> is not true

therefore

The ordered pair (−2, 3) is not included in the solution area for the system