The area defined by `1 le |x - 2| + |y + 1| le 2` is

Options

A. 2
B. 4
C. 6
D. none of these

Answer

To find the area defined by the inequality \(1 \leq |x - 2| + |y + 1| \leq 2\), we will break it down into steps.

### Step 1: Understand the Inequality
The expression \( |x - 2| + |y + 1| \) represents a diamond (or rhombus) shape in the coordinate plane. The inequality \(1 \leq |x - 2| + |y + 1| \leq 2\) indicates that we are interested in the area between two diamonds: one with a "radius" of 1 and the other with a "radius" of 2.

### Step 2: Find the Vertices of the Diamonds
1. **For the outer diamond** defined by \( |x - 2| + |y + 1| = 2 \):
- The vertices can be found by setting \( |x - 2| + |y + 1| = 2 \):
- \(x - 2 + y + 1 = 2 \Rightarrow y = -x + 3\) (1st quadrant)
- \(x - 2 - (y + 1) = 2 \Rightarrow y = x - 5\) (4th quadrant)
- \(-(x - 2) + (y + 1) = 2 \Rightarrow y = x + 1\) (2nd quadrant)
- \(-(x - 2) - (y + 1) = 2 \Rightarrow y = -x - 1\) (3rd quadrant)

The vertices are:
- \( (4, -1) \)
- \( (0, -1) \)
- \( (2, 1) \)
- \( (2, -3) \)

2. **For the inner diamond** defined by \( |x - 2| + |y + 1| = 1 \):
- Following a similar process:
- The vertices are:
- \( (3, -1) \)
- \( (1, -1) \)
- \( (2, 0) \)
- \( (2, -2) \)

### Step 3: Calculate the Area of Each Diamond
1. **Area of the outer diamond**:
- The formula for the area of a diamond (rhombus) is given by:
\[ \text{Area} = \frac{1}{2} \times d_1 \times d_2 \]
- The diagonals of the outer diamond are 4 (vertical) and 4 (horizontal), so:
\[ \text{Area}_{\text{outer}} = \frac{1}{2} \times 4 \times 4 = 8 \]

2. **Area of the inner diamond**:
- The diagonals of the inner diamond are 2 (vertical) and 2 (horizontal), so:
\[ \text{Area}_{\text{inner}} = \frac{1}{2} \times 2 \times 2 = 2 \]

### Step 4: Find the Required Area
To find the area defined by the original inequality, we subtract the area of the inner diamond from the area of the outer diamond:
\[ \text{Required Area} = \text{Area}_{\text{outer}} - \text{Area}_{\text{inner}} = 8 - 2 = 6 \]

### Final Answer
The area defined by \(1 \leq |x - 2| + |y + 1| \leq 2\) is \(6\) square units.

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