Question

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

Part B: Is the point (-6, 3) included in the solution area for the system? Justify your answer mathematically.

To answer Part A, you’ll need to graph the system of inequalities on paper and describe the resulting graph, including the types of lines used and the shading to represent the solution area. For Part B, you should determine whether the point (-6, 3) falls within the solution area based on your graph and provide a mathematical justification for your answer.

I recommend graphing the two inequalities separately and then identifying the overlapping region as the solution area for the system. You can use the slope-intercept form (y = mx + b) to graph each line and shade the appropriate regions to represent the solution. Then, check if the point (-6, 3) falls within the shaded region.

Answer 1

To graph the system of inequalities, we can graph each inequality separately and shade the appropriate regions. The solution area is the overlapping region of the shaded regions. To determine if a point is in the solution area, we can plug in its coordinates into the inequalities.

Step-by-step explanation:

Part A:To graph the system of inequalities, let's start by graphing each inequality separately. First, graph the inequality y > 2x + 3. We can do this by drawing a dashed line with a slope of 2 and a y-intercept of 3. Since the inequality is y > 2x + 3, we should shade the region above the line to represent all the y-values greater than the line. Next, graph the inequality y ≤ -x - 2. This time, draw a solid line with a slope of -1 and a y-intercept of -2. Since the inequality is y ≤ -x - 2, we should shade the region below the line to represent all the y-values less than or equal to the line. The solution area is the overlapping region of the shaded regions.

Part B:To determine if the point (-6, 3) falls within the solution area, we can plug in the x-coordinate and y-coordinate of the point into both inequalities. For y > 2x + 3, we have 3 > 2(-6) + 3, which simplifies to 3 > -12 + 3, and further simplifies to 3 > -9. This is true, so the point satisfies the first inequality. For y ≤ -x - 2, we have 3 ≤ -(-6) - 2, which simplifies to 3 ≤ 6 - 2, and further simplifies to 3 ≤ 4. This is also true, so the point satisfies the second inequality. Therefore, the point (-6, 3) is included in the solution area for the system of inequalities.