The number of distinct integer solutions (x,y)
of the equation |x+y|+|x−y|=2
, is

Question

The number of distinct integer solutions (x,y)
 of the equation |x+y|+|x−y|=2
, is

Accepted Answer

To solve the equation $$|x+y| + |x-y| = 2$$, we analyze the possible integer solutions by considering different cases based on the properties of absolute values.

### Case Analysis:
1. **Case 1: $$x \geq y$$ and $$x \geq -y$$:**
   - Here, $$|x+y| = x+y$$ and $$|x-y| = x-y$$.
   - Substituting into the equation:  
     $$
     x + y + x - y = 2 \quad \implies \quad 2x = 2 \quad \implies \quad x = 1
     $$
   - For $$x = 1$$, possible values for $$y$$ are:
     - $$y = 1$$ → Solution: (1, 1)
     - $$y = 0$$ → Solution: (1, 0)
     - $$y = -1$$ → Solution: (1, -1)

2. **Case 2: $$x  y$$ and $$x < -y$$:**
   - Here, no solutions exist since this case is contradictory.

### Total Distinct Solutions:
Combining all valid solutions from the above cases:
- (1, 1), (1, 0), (1, -1), (0, 1), (0, -1), (-1, 1), (-1, 0), (-1, -1).

Thus, the number of distinct integer solutions is **8**.