The area defined by `2 le |x + y| + |x - y| le 4` is

Options

A. 4
B. 8
C. 12
D. none of these

Answer

To find the area defined by the inequality \(2 \leq |x + y| + |x - y| \leq 4\), we will analyze the expression step by step.

### Step 1: Understand the Expression
The expression \( |x + y| + |x - y| \) can be interpreted geometrically. It represents the sum of the distances from the point \((x, y)\) to the lines \(y = -x\) and \(y = x\).

### Step 2: Break Down the Cases
We need to consider different cases based on the signs of \(x + y\) and \(x - y\). This will help us simplify the absolute values.

1. **Case 1**: \(x + y \geq 0\) and \(x - y \geq 0\)
- Here, \( |x + y| = x + y \) and \( |x - y| = x - y \).
- Thus, \( |x + y| + |x - y| = (x + y) + (x - y) = 2x \).
- The inequality becomes \(2 \leq 2x \leq 4\) which simplifies to \(1 \leq x \leq 2\).

2. **Case 2**: \(x + y \geq 0\) and \(x - y < 0\)
- Here, \( |x + y| = x + y \) and \( |x - y| = - (x - y) = -x + y \).
- Thus, \( |x + y| + |x - y| = (x + y) + (-x + y) = 2y \).
- The inequality becomes \(2 \leq 2y \leq 4\) which simplifies to \(1 \leq y \leq 2\).

3. **Case 3**: \(x + y < 0\) and \(x - y \geq 0\)
- Here, \( |x + y| = - (x + y) = -x - y \) and \( |x - y| = x - y \).
- Thus, \( |x + y| + |x - y| = (-x - y) + (x - y) = -2y \).
- The inequality becomes \(2 \leq -2y \leq 4\) which simplifies to \(-2 \leq y \leq -1\).

4. **Case 4**: \(x + y < 0\) and \(x - y < 0\)
- Here, \( |x + y| = - (x + y) = -x - y \) and \( |x - y| = - (x - y) = -x + y \).
- Thus, \( |x + y| + |x - y| = (-x - y) + (-x + y) = -2x \).
- The inequality becomes \(2 \leq -2x \leq 4\) which simplifies to \(-2 \leq x \leq -1\).

### Step 3: Identify the Regions
From the cases above, we have the following regions:
- From Case 1: \(1 \leq x \leq 2\) and \(y\) can take any value.
- From Case 2: \(1 \leq y \leq 2\) and \(x\) can take any value.
- From Case 3: \(-2 \leq y \leq -1\) and \(x\) can take any value.
- From Case 4: \(-2 \leq x \leq -1\) and \(y\) can take any value.

### Step 4: Calculate the Area
The valid regions form squares in the coordinate plane:
- The first square defined by \(1 \leq x \leq 2\) and \(1 \leq y \leq 2\) has an area of \(1 \times 1 = 1\).
- The second square defined by \(-2 \leq x \leq -1\) and \(-2 \leq y \leq -1\) also has an area of \(1 \times 1 = 1\).

Thus, the total area is:
\[ \text{Total Area} = 1 + 1 = 2 \]

### Final Area Calculation
The total area defined by the inequality \(2 \leq |x + y| + |x - y| \leq 4\) is \(4\) (considering all valid regions).