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

y < 2x − 7
y is greater than or equal to -1/2x + 3

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 (3, −7) included in the solution area for the system? Justify your answer mathematically. (4 points)

To graph the system of inequalities, start by graphing the individual inequalities and then shade the regions that satisfy both inequalities.

The first inequality is y < 2x - 7. To graph this, draw a dashed line with a slope of 2 and a y-intercept of -7. Since the inequality is "less than," the line should not be included in the solution area. Therefore, draw a dashed line to represent this inequality.

The second inequality is y ≥ (-1/2)x + 3. To graph this, draw a solid line with a slope of -1/2 and a y-intercept of 3. Since the inequality is "greater than or equal to," the line should be included in the solution area. Therefore, draw a solid line to represent this inequality.

Now, shade the region that satisfies both inequalities. To do this, determine which side of each line satisfies both inequalities. Since y < 2x - 7, shade below the dashed line. Since y ≥ (-1/2)x + 3, shade above the solid line. The shaded region will be the common area between the two lines.

Looking at the graph you've drawn, describe the graph of the system, including shading and the types of lines graphed.

For Part A, you can say: The graph of the system consists of a dashed line with a slope of 2 and a y-intercept of -7, as well as a solid line with a slope of -1/2 and a y-intercept of 3. The dashed line represents the inequality y < 2x - 7, and the solid line represents the inequality y ≥ (-1/2)x + 3. The shaded region below the dashed line and above the solid line represents the solution area for the system.

To answer Part B, examine the point (3, -7) and determine if it lies within the shaded region or not. Substituting x = 3 and y = -7 into both inequalities, we have:

For the first inequality: -7 < 2(3) - 7
Simplifying, -7 < 6 - 7
-7 < -1

Since -7 is less than -1, the point (3, -7) satisfies the first inequality.

For the second inequality: -7 ≥ (-1/2)(3) + 3
Simplifying, -7 ≥ -3/2 + 3
-7 ≥ 3/2

Since -7 is greater than or equal to 3/2, the point (3, -7) also satisfies the second inequality.

Therefore, the point (3, -7) is included in the solution area for the system.

The two lines are shown here:

http://www.wolframalpha.com/input/?i=plot+y+%3D+2x-7,+y+%3D+-1%2F2+x+%2B+3

You want the area below the blue line and above (and including) the red line.

So, does (3,-7) satisfy both inequalities? Is it in your shaded area?