2x − y > 4
x + y < −1
In each graph, the area for f(x) is shaded and labeled A, the area for g(x) is shaded and labeled B, and the area where they have shading in common is labeled AB.
Unfortunately, as an AI language model, I am not able to create and display graphs. However, I can provide some guidance on how to graph these two inequalities.
To graph 2x - y > 4:
1. Start by graphing the line 2x - y = 4. This can be done by finding the x and y intercepts. When x = 0, y = -4. When y = 0, 2x = 4, so x = 2. Plot these two points and draw a line through them.
2. Since the inequality is greater than, shade the region above the line.
To graph x + y < -1:
1. Start by graphing the line x + y = -1. Again, find the x and y intercepts. When x = 0, y = -1. When y = 0, x = -1. Plot these two points and draw a line through them.
2. Since the inequality is less than, shade the region below the line.
To find the area of AB (where the two shaded regions overlap), find the intersection of the two lines by solving the system:
2x - y = 4
x + y = -1
One way to do this is to solve for y in the second equation (y = -x - 1) and substitute into the first equation:
2x - (-x - 1) > 4
Simplify and solve for x:
3x > 5
x > 5/3
Substitute this value back into either equation to find the corresponding y value:
(y = -x - 1)
y = -(5/3) - 1
y = -8/3
Therefore, the point of intersection is (5/3, -8/3). Shade this region as well and label it AB.
To graph the inequalities and shade the appropriate areas, we will first graph the equations:
1. Graph the equation 2x − y > 4.
To graph this equation, we can rewrite it as 2x − y = 4 and find two points that satisfy this equation.
Let's assume x = 0. Substituting this into the equation, we have:
2(0) - y = 4
0 - y = 4
-y = 4
y = -4
So, one point on the line is (0, -4).
Now, let's assume y = 0. Substituting this into the equation, we have:
2x - 0 = 4
2x = 4
x = 2
So, another point on the line is (2, 0).
Plotting these two points and drawing a line passing through them, we get:
Diagram 1: Line for equation 2x - y = 4
|
A | .
| .
| .
------ + ------ + ------
|
-4 2
2. Graph the equation x + y < -1.
To graph this equation, we can rewrite it as x + y = -1 and find two points that satisfy this equation.
Let's assume x = 0. Substituting this into the equation, we have:
0 + y = -1
y = -1
So, one point on the line is (0, -1).
Now, let's assume y = 0. Substituting this into the equation, we have:
x + 0 = -1
x = -1
So, another point on the line is (-1, 0).
Plotting these two points and drawing a line passing through them, we get:
Diagram 2: Line for equation x + y = -1
|
B | .
| .
| .
------ + ------ + ------
|
-1 0
3. Shade the appropriate areas:
To find the shaded areas, we need to determine the regions that satisfy each inequality.
In Diagram 1, we want to shade the region where 2x - y is greater than 4. Since we are interested in the area above the line, we shade the region above the line. Let's label this area as A.
Diagram 1: Line for equation 2x - y = 4
|
A |*********.
|****.
|*******.
------ + ------ + ------
|
-4 2
In Diagram 2, we want to shade the region where x + y is less than -1. Since we are interested in the area below the line, we shade the region below the line. Let's label this area as B.
Diagram 2: Line for equation x + y = -1
|
B | .
|*******.
|****.
------ + ------ + ------
|
-1 0
Lastly, we want to find the region where both equations have shading, which is the overlapping region of A and B. Let's label this area as AB.
Diagram 3: Shaded areas
|
AB | .
| .
| .
------ + ------ + ------
|
-4 2
This is the final graph with the shaded areas labeled A, B, and AB.