Graph the solution of the system of linear inequalities.
y ≥ - x + 3
y ≤ 2x + 5
To graph the solution of the system of linear inequalities, we start by graphing each individual inequality and shading the appropriate region.
First, let's graph the inequality y ≥ -x + 3. To do this, we can start by graphing the line y = -x + 3. This line has a y-intercept of 3 and a slope of -1 (since the coefficient of x is -1). We can plot two points on this line, for example when x = 0, y = 3 and when x = 1, y = 2. Then we can draw a line through these points.
Graphing y = -x + 3:
```
| /
| /
| /
3 | /
|/
-------------
0 1 2
```
Now, we need to shade the region above the line. Since the inequality is y ≥ -x + 3, any point above the line (including the line itself) satisfies the inequality. We can shade the region above the line.
Shaded region for y ≥ -x + 3:
```
----------------
| / /
| / /
|/
3 -------------
|/
-------------
0 1 2
```
Next, let's graph the inequality y ≤ 2x + 5. To do this, we can start by graphing the line y = 2x + 5. This line has a y-intercept of 5 and a slope of 2 (since the coefficient of x is 2). We can plot two points on this line, for example when x = 0, y = 5, and when x = 1, y = 7. Then we can draw a line through these points.
Graphing y = 2x + 5:
```
|
| /
| /
7 | /
|/
-------------
0 1 2
```
Now, we need to shade the region below the line. Since the inequality is y ≤ 2x + 5, any point below the line (including the line itself) satisfies the inequality. We can shade the region below the line.
Shaded region for y ≤ 2x + 5:
```
-------------
|/
--------
|/
| /
| /
7 | /
|/
-------------
0 1 2
```
Finally, we can combine the shaded regions for both inequalities to find the solution.
Solution graph:
```
-------------
| / /
| / /
|/
3 -------------
|/
--------
|/
| /
| /
7 | /
|/
-------------
0 1 2
```
The shaded region above the line y = -x + 3 and below the line y = 2x + 5 represents the solution of the system of linear inequalities.