To solve the problem of finding the region enclosed by the given inequalities and the ordered pairs of the vertices, we will follow these steps:
### Step 1: Write the equations of the lines from the inequalities
The given inequalities are:
1. \( x + y - 2 \leq 0 \)
2. \( 2x + y - 3 \leq 0 \)
3. \( x \geq 0 \)
4. \( y \geq 0 \)
First, we convert the inequalities into equations:
1. \( x + y = 2 \)
2. \( 2x + y = 3 \)
### Step 2: Find the intercepts of the lines
For the first equation \( x + y = 2 \):
- When \( x = 0 \), \( y = 2 \) (Point: \( (0, 2) \))
- When \( y = 0 \), \( x = 2 \) (Point: \( (2, 0) \))
For the second equation \( 2x + y = 3 \):
- When \( x = 0 \), \( y = 3 \) (Point: \( (0, 3) \))
- When \( y = 0 \), \( x = \frac{3}{2} \) (Point: \( \left(\frac{3}{2}, 0\right) \))
### Step 3: Plot the lines on a graph
Now we plot the lines on a coordinate plane:
- The line \( x + y = 2 \) passes through points \( (0, 2) \) and \( (2, 0) \).
- The line \( 2x + y = 3 \) passes through points \( (0, 3) \) and \( \left(\frac{3}{2}, 0\right) \).
### Step 4: Determine the region defined by the inequalities
- For \( x + y \leq 2 \), we shade the region below the line \( x + y = 2 \).
- For \( 2x + y \leq 3 \), we shade the region below the line \( 2x + y = 3 \).
- Since \( x \geq 0 \) and \( y \geq 0 \), we only consider the first quadrant.
### Step 5: Find the intersection points of the lines
To find the intersection of the lines \( x + y = 2 \) and \( 2x + y = 3 \), we can solve the equations simultaneously:
1. From \( x + y = 2 \), we can express \( y = 2 - x \).
2. Substitute \( y \) in the second equation:
\[
2x + (2 - x) = 3 \\
2x + 2 - x = 3 \\
x + 2 = 3 \\
x = 1
\]
3. Substitute \( x = 1 \) back into \( y = 2 - x \):
\[
y = 2 - 1 = 1
\]
Thus, the intersection point is \( (1, 1) \).
### Step 6: Identify the vertices of the enclosed region
The vertices of the region are:
1. \( (0, 0) \) (from \( x \geq 0 \) and \( y \geq 0 \))
2. \( (0, 2) \) (from \( x + y = 2 \))
3. \( (1, 1) \) (intersection point)
4. \( \left(\frac{3}{2}, 0\right) \) (from \( 2x + y = 3 \))
### Final Answer
The ordered pairs of the vertices of the region are:
- \( (0, 0) \)
- \( (0, 2) \)
- \( (1, 1) \)
- \( \left(\frac{3}{2}, 0\right) \)
