The number of solution of `|x| + |y| le 0,` for `(x , y) in R`, is :

Options

A. 0
B. 1
C. 2
D. infinitely many

Answer

To solve the problem of finding the number of solutions for the inequality \( |x| + |y| \leq 0 \), we can follow these steps:

### Step 1: Understand the properties of absolute values
The absolute value of any real number is always non-negative. This means that \( |x| \geq 0 \) and \( |y| \geq 0 \) for all \( x, y \in \mathbb{R} \).

**Hint:** Remember that the absolute value function outputs a value that is never negative.

### Step 2: Analyze the inequality
The expression \( |x| + |y| \) represents the sum of two non-negative values. Therefore, the minimum value of \( |x| + |y| \) is 0, which occurs only when both \( |x| = 0 \) and \( |y| = 0 \).

**Hint:** Consider what values \( |x| \) and \( |y| \) can take to satisfy the inequality.

### Step 3: Determine when the inequality holds
The inequality \( |x| + |y| \leq 0 \) can only hold true if \( |x| + |y| = 0 \). Since both \( |x| \) and \( |y| \) are non-negative, the only way for their sum to equal zero is if both are individually zero. This leads us to the equations:
- \( |x| = 0 \) implies \( x = 0 \)
- \( |y| = 0 \) implies \( y = 0 \)

Thus, the only solution to the inequality is the point \( (0, 0) \).

**Hint:** Think about what values make the sum of two non-negative numbers equal to zero.

### Step 4: Conclusion
Since the only solution to the inequality \( |x| + |y| \leq 0 \) is the single point \( (0, 0) \), we conclude that there is exactly **one solution**.

**Final Answer:** The number of solutions is **1**.