Graph the system of inequalities presented here on your own paper, then use your graph to answer the following questions:
y > 3x + 10
y is less than negative 3 over 4 times x minus 1
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 (8, 10) included in the solution area for the system? Justify your answer mathematically. (4 points)
To graph the system of inequalities, we start by graphing each inequality separately and then determine the overlapping region.
First, let's graph the inequality y > 3x + 10:
1. Start by graphing the line y = 3x + 10. This is a straight line with a slope of 3 and a y-intercept of 10. Connect two points on the line to sketch it.
Next, let's graph the inequality y < -3/4x - 1:
1. Start by graphing the line y = -3/4x - 1. This is a straight line with a slope of -3/4 and a y-intercept of -1. Connect two points on the line to sketch it.
Now, let's determine the overlapping region:
Since y > 3x + 10, we shade the region above the line y = 3x + 10.
Since y < -3/4x - 1, we shade the region below the line y = -3/4x - 1.
The solution area is the region that is shaded in both regions simultaneously.
For Part A: The graph of the system would include two lines: one with a solid line for y = 3x + 10, and the other with a dashed line for y = -3/4x - 1. The shaded region would be above the solid line and below the dashed line. It forms a triangular region.
For Part B: We need to check if the point (8, 10) lies within the shaded region.
Plug in the x and y coordinates of the point into each inequality:
1. For y > 3x + 10:
y > 3(8) + 10
10 > 34
Since 10 is not greater than 34, the point (8, 10) is not a part of the solution area.
Therefore, the point (8, 10) is not included in the solution area for the system.
To graph the system of inequalities, we can follow these steps:
Step 1: Graph the first inequality, y > 3x + 10.
To graph this inequality, start by graphing the line y = 3x + 10. Since the inequality is y > 3x + 10, the line should be dashed to indicate that it is not included in the solution set.
Step 2: Determine which side of the line to shade.
To determine which side to shade, select any point that is not on the line. The easiest point to select is (0,0) since it makes calculations easier, but any point will work.
Substitute the coordinates of the point into the inequality and check if the inequality holds true.
For example, substitute (0,0) into the inequality: 0 > 3(0) + 10. Simplifying this, we get 0 > 10, which is false.
Since the inequality is not satisfied for the side that contains the point (0,0), shade the other side.
Step 3: Graph the second inequality, y < -3/4x - 1.
To graph this inequality, start by graphing the line y = -3/4x - 1. Since the inequality is y < -3/4x - 1, the line should be dashed to indicate that it is not included in the solution set.
Step 4: Determine which side of the line to shade.
To determine which side to shade, again select any point that is not on the line and substitute it into the inequality.
For example, substitute (0,0) into the inequality: 0 < -3/4(0) - 1. Simplifying this, we get 0 < -1, which is true.
Since the inequality is satisfied for the side that contains the point (0,0), shade that side.
Step 5: Analyze the shaded regions to find the solution area.
The solution area is the region where the shaded regions of both inequalities overlap or intersect. Shade the overlapping region to indicate the solution area.
Now, let's move on to answering the questions:
Part A: Description of the graph and solution area.
On your own paper, draw the two dashed lines, one for y > 3x + 10 and the other for y < -3/4x - 1. Shade the appropriate regions according to the steps described above. The overlapping shaded region represents the solution area.
Part B: Determining if the point (8, 10) is included in the solution area.
To determine if the point (8, 10) is in the solution area, we need to check if it satisfies both inequalities.
Substitute the x and y coordinates of the point into each inequality:
For y > 3x + 10:
10 > 3(8) + 10
10 > 34
This inequality is not true, so (8, 10) is not included in the solution for y > 3x + 10.
For y < -3/4x - 1:
10 < -3/4(8) - 1
10 < -7
This inequality is true, so (8, 10) is included in the solution for y < -3/4x - 1.
Since (8, 10) does not satisfy both inequalities, it is not included in the solution area for the system of inequalities.
y > 3x+10
y < 3/4 x - 1
So, graph the two lies. Shade the area above the first line, and below the 2nd line.
Use dotted lines. (why)
Does (8,10) fit both inequalities?