Directions: Solve the following system of inequalities.
1.) 3x + 2y < 6
x > 0
y > 0
2.) 2x^2 + y ≥ 2
x ≤ 2
y ≤ 1
Sorry, the x's and y's got put in there by mistake. I only want the top equations solved.
1.) 3x + 2y < 6
2.) 2x^2 + y ≥ 2
To solve a system of inequalities, we need to find the intersection of the solution regions for each inequality.
For the first system of inequalities:
1.) 3x + 2y < 6
To graph this inequality, we can start by graphing the line 3x + 2y = 6. To do this, we can find the x and y intercepts:
When x = 0, 3(0) + 2y = 6
2y = 6
y = 3
So, the y-intercept is (0, 3).
When y = 0, 3x + 2(0) = 6
3x = 6
x = 2
So, the x-intercept is (2, 0).
Plotting these intercepts and drawing a line through them, we get:
```
|
|
| /
| /
| /
| /
|/_________
```
Since the inequality is less than (<), the solution region is below the line. However, we also have the additional inequalities x > 0 and y > 0.
For x > 0, we shade the region to the right of the y-axis:
```
|
|
|_______/
| /
| /
| /
|/_________
```
For y > 0, we shade the region above the x-axis:
```
| _____
| /
| /
| /
|/_________
```
The intersection of the shaded regions represents the solution region for the first inequality.
Now let's move on to the second inequality:
2.) x > 0
Since this inequality only involves x, it tells us that the solution is to the right of the y-axis:
```
| ____
| /
| /
| /
|/_________
```
Finally, let's consider the third inequality:
3.) y > 0
This inequality only involves y, so the solution is above the x-axis:
```
| _____
|
|
|
|
|
|
|/_________
```
Now we can see that the overlapping region of all three shaded regions represents the solution to the system of inequalities:
```
| _____
|
|
| /
| /
| /
|/_________
```
So the solution to the system of inequalities is the region above the x-axis and to the right of the y-axis.
Now let's move on to the second system of inequalities:
1.) 2x^2 + y ≥ 2
To graph this inequality, we can start by graphing the curve 2x^2 + y = 2.
Since it is a curved line, it may be helpful to find a few points to plot. We can choose x = -1, x = 0, and x = 1:
For x = -1, 2(-1)^2 + y = 2
2 + y = 2
y = 0
So we have the point (-1, 0).
For x = 0, 2(0)^2 + y = 2
y = 2
So we have the point (0, 2).
For x = 1, 2(1)^2 + y = 2
2 + y = 2
y = 0
So we have the point (1, 0).
Plotting these points and drawing a curved line through them, we get:
```
|___________________
|
| / .
| .
| .
| /
|/_________
```
Since the inequality is greater than or equal to (≥), the solution region is above the curved line. However, we also have the additional inequalities x ≤ 2 and y ≤ 1.
For x ≤ 2, we shade the region to the left of or on the vertical line x = 2:
```
| |
| |
| / .
| .
| .
| /
|/_________
```
For y ≤ 1, we shade the region below or on the horizontal line y = 1:
```
|_______________
| |
| / .
| .
| .
| /
|/_________
```
The intersection of the shaded regions represents the solution region for the second inequality.
Now we can see that the overlapping region of the shaded regions represents the solution to the system of inequalities:
```
|________
| |
| |
| |
| _
| |
|________|
```
So the solution to the second system of inequalities is the region below or on the line y = 1, to the left of or on the line x = 2, and above the curved line 2x^2 + y = 2.