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 negative 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)
Cannot graph on these posts.
To graph the system of inequalities, start by graphing each inequality separately:
First, graph the inequality y < 2x - 7. This is a linear inequality, so we can graph the corresponding line, y = 2x - 7, and shade the area below the line. To do this, we need two points to plot the line. We can start by substituting x = 0 into the equation to find the y-intercept: y = 2(0) - 7 = -7. Next, we can substitute y = 0 to find the x-intercept: 0 = 2x - 7 → 2x = 7 → x = 7/2 = 3.5. So we have the points (0, -7) and (3.5, 0) to plot the line. Draw a dashed line through these points and shade the area below it, not including the line itself.
Next, graph the inequality y ≥ -1/2x + 3. Again, this is a linear inequality, so we can graph the corresponding line, y = -1/2x + 3, and shade the area above the line. Start by finding the y-intercept: y = -1/2(0) + 3 = 3. Next, find the x-intercept: 0 = -1/2x + 3 → 1/2x = 3 → x = 6. So we have the points (0, 3) and (6, 0) to plot the line. Draw a solid line through these points and shade the area above it, including the line itself.
Now, combine the graphs of the two inequalities on the same set of axes. The solution area is the region where the shaded areas overlap.
Part A: The graph of the system will have two lines: a dashed line y = 2x - 7, with the area below shaded, and a solid line y = -1/2x + 3, with the area above shaded. The solution area is the overlap of these shaded regions.
Part B: To determine if the point (3, -7) is included in the solution area, we need to check if it satisfies both inequalities. Substitute the x and y coordinates of the given point into both inequalities:
For y < 2x - 7:
-7 < 2(3) - 7
-7 < 6 - 7
-7 < -1
The inequality is true, so the point (3, -7) satisfies the first inequality.
For y ≥ -1/2x + 3:
-7 ≥ -1/2(3) + 3
-7 ≥ -3/2 + 3
-7 ≥ -3/2 + 6/2
-7 ≥ 3/2
The inequality is false, so the point (3, -7) does not satisfy the second inequality.
Therefore, the point (3, -7) is not included in the solution area for the system of inequalities.