To find the area of the region defined by the inequalities \(3 \leq |x| + |y| \leq 5\), we can break this down into manageable steps.
### Step 1: Understand the inequalities
The expression \( |x| + |y| \) represents the Manhattan distance from the origin to the point \((x, y)\). The inequalities define a region between two diamond shapes (or squares rotated 45 degrees) centered at the origin.
### Step 2: Analyze the outer inequality \( |x| + |y| \leq 5 \)
This inequality can be expressed in four cases based on the signs of \(x\) and \(y\):
1. **Case 1**: \(x \geq 0, y \geq 0\)
\[
x + y \leq 5
\]
- Intercepts: (5, 0) and (0, 5)
2. **Case 2**: \(x \geq 0, y < 0\)
\[
x - y \leq 5
\]
- Intercepts: (5, 0) and (0, -5)
3. **Case 3**: \(x < 0, y < 0\)
\[
-x - y \leq 5 \implies x + y \geq -5
\]
- Intercepts: (-5, 0) and (0, -5)
4. **Case 4**: \(x < 0, y \geq 0\)
\[
-x + y \leq 5 \implies x + y \leq 5
\]
- Intercepts: (-5, 0) and (0, 5)
Plotting these lines gives us a diamond shape with vertices at (5, 0), (0, 5), (-5, 0), and (0, -5).
### Step 3: Analyze the inner inequality \( |x| + |y| \geq 3 \)
Similarly, we analyze this inequality in four cases:
1. **Case 1**: \(x \geq 0, y \geq 0\)
\[
x + y \geq 3
\]
- Intercepts: (3, 0) and (0, 3)
2. **Case 2**: \(x \geq 0, y < 0\)
\[
x - y \geq 3
\]
- Intercepts: (3, 0) and (0, -3)
3. **Case 3**: \(x < 0, y < 0\)
\[
-x - y \geq 3 \implies x + y \leq -3
\]
- Intercepts: (-3, 0) and (0, -3)
4. **Case 4**: \(x < 0, y \geq 0\)
\[
-x + y \geq 3 \implies x + y \geq 3
\]
- Intercepts: (-3, 0) and (0, 3)
Plotting these lines gives us another diamond shape with vertices at (3, 0), (0, 3), (-3, 0), and (0, -3).
### Step 4: Determine the area between the two diamonds
The area of the outer diamond (with vertices at (5, 0), (0, 5), (-5, 0), and (0, -5)) can be calculated as follows:
- The side length of the diamond is \(5\sqrt{2}\) (the distance from the center to a vertex).
- The area of the outer diamond is:
\[
\text{Area}_{outer} = 2 \times \text{base} \times \text{height} = 2 \times 5 \times 5 = 50 \text{ square units}
\]
The area of the inner diamond (with vertices at (3, 0), (0, 3), (-3, 0), and (0, -3)) is:
\[
\text{Area}_{inner} = 2 \times 3 \times 3 = 18 \text{ square units}
\]
### Step 5: Calculate the area of the region between the two diamonds
The area of the region satisfying \(3 \leq |x| + |y| \leq 5\) is:
\[
\text{Area}_{between} = \text{Area}_{outer} - \text{Area}_{inner} = 50 - 18 = 32 \text{ square units}
\]
### Final Answer
The area of the region in which points satisfy \(3 \leq |x| + |y| \leq 5\) is \(32\) square units.
