Area bouinded by `|x| + |y|=1` is 2 sq. units.

Answer

To find the area bounded by the equation \(|x| + |y| = 1\), we will analyze the equation in different quadrants and then calculate the area.

### Step 1: Understanding the Equation
The equation \(|x| + |y| = 1\) represents a diamond shape (or rhombus) centered at the origin in the Cartesian coordinate system. It opens in all four quadrants.

### Step 2: Finding the Lines in Each Quadrant
1. **First Quadrant**: Here, both \(x\) and \(y\) are positive.
- The equation simplifies to \(x + y = 1\).
- When \(x = 0\), \(y = 1\) (point (0, 1)).
- When \(y = 0\), \(x = 1\) (point (1, 0)).
- This line segment connects points (0, 1) and (1, 0).

2. **Second Quadrant**: Here, \(x\) is negative and \(y\) is positive.
- The equation becomes \(-x + y = 1\) or \(y = x + 1\).
- When \(y = 0\), \(x = -1\) (point (-1, 0)).
- When \(x = 0\), \(y = 1\) (point (0, 1)).
- This line segment connects points (0, 1) and (-1, 0).

3. **Third Quadrant**: Here, both \(x\) and \(y\) are negative.
- The equation simplifies to \(-x - y = 1\) or \(x + y = -1\).
- When \(x = -1\), \(y = 0\) (point (-1, 0)).
- When \(y = -1\), \(x = 0\) (point (0, -1)).
- This line segment connects points (-1, 0) and (0, -1).

4. **Fourth Quadrant**: Here, \(x\) is positive and \(y\) is negative.
- The equation becomes \(x - y = 1\) or \(y = x - 1\).
- When \(x = 1\), \(y = 0\) (point (1, 0)).
- When \(y = -1\), \(x = 0\) (point (0, -1)).
- This line segment connects points (1, 0) and (0, -1).

### Step 3: Area Calculation
The shape formed is a rhombus with vertices at the points (1, 0), (0, 1), (-1, 0), and (0, -1).

To calculate the area of the rhombus:
- The diagonals of the rhombus are the segments connecting (1, 0) to (-1, 0) and (0, 1) to (0, -1).
- The length of the horizontal diagonal is \(2\) (from (-1, 0) to (1, 0)).
- The length of the vertical diagonal is \(2\) (from (0, -1) to (0, 1)).

The area \(A\) of a rhombus can be calculated using the formula:
\[ A = \frac{1}{2} \times d_1 \times d_2 \]
where \(d_1\) and \(d_2\) are the lengths of the diagonals.

Substituting the values:
\[ A = \frac{1}{2} \times 2 \times 2 = 2 \text{ square units} \]

### Conclusion
The area bounded by the equation \(|x| + |y| = 1\) is indeed \(2\) square units, confirming the statement is true.

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