In the following, shade the region, where the following inequations hold. Also find the vertices of the region so formed:
`xge2,xle8,yge-4,ylex+2,2x+yle14`.

Answer

To solve the problem of shading the region defined by the given inequalities and finding the vertices of the region, we will follow these steps:

### Step 1: Write down the inequalities
The inequalities given are:
1. \( x \geq 2 \)
2. \( x \leq 8 \)
3. \( y \geq -4 \)
4. \( y \leq x + 2 \)
5. \( 2x + y \leq 14 \)

### Step 2: Graph the boundary lines
We will first convert the inequalities into equations to find the boundary lines.

1. **For \( x = 2 \)**: This is a vertical line at \( x = 2 \).
2. **For \( x = 8 \)**: This is a vertical line at \( x = 8 \).
3. **For \( y = -4 \)**: This is a horizontal line at \( y = -4 \).
4. **For \( y = x + 2 \)**: This line has a y-intercept of 2 and a slope of 1.
5. **For \( 2x + y = 14 \)**: Rearranging gives \( y = -2x + 14 \), which has a y-intercept of 14 and a slope of -2.

### Step 3: Find the intersection points
Next, we need to find the points of intersection of the lines to determine the vertices of the region.

1. **Intersection of \( y = x + 2 \) and \( y = -4 \)**:
\[ -4 = x + 2 \implies x = -6 \quad \text{(not in the region)} \]

2. **Intersection of \( y = x + 2 \) and \( 2x + y = 14 \)**:
Substitute \( y = x + 2 \) into \( 2x + y = 14 \):
\[ 2x + (x + 2) = 14 \implies 3x + 2 = 14 \implies 3x = 12 \implies x = 4 \]
Substitute \( x = 4 \) back into \( y = x + 2 \):
\[ y = 4 + 2 = 6 \quad \Rightarrow (4, 6) \]

3. **Intersection of \( y = -4 \) and \( 2x + y = 14 \)**:
Substitute \( y = -4 \):
\[ 2x - 4 = 14 \implies 2x = 18 \implies x = 9 \quad \Rightarrow (9, -4) \]

4. **Intersection of \( y = -4 \) and \( y = x + 2 \)**:
\[ -4 = x + 2 \implies x = -6 \quad \text{(not in the region)} \]

5. **Intersection of \( y = -4 \) and \( x = 2 \)**:
\[ (2, -4) \]

6. **Intersection of \( y = -4 \) and \( x = 8 \)**:
\[ (8, -4) \]

### Step 4: Identify the vertices
The vertices of the region formed by the inequalities are:
1. \( (2, -4) \)
2. \( (4, 6) \)
3. \( (8, -4) \)
4. \( (2, 4) \) (from \( y = x + 2 \) at \( x = 2 \))
5. \( (9, -4) \) (from \( 2x + y = 14 \) at \( y = -4 \))

### Step 5: Shade the region
Now, we will shade the region that satisfies all the inequalities:
- The region is bounded by the lines \( x = 2 \) and \( x = 8 \) on the left and right.
- The region is above the line \( y = -4 \) and below the lines \( y = x + 2 \) and \( 2x + y = 14 \).

### Final Answer
The vertices of the region are:
- \( (2, -4) \)
- \( (4, 6) \)
- \( (8, -4) \)