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

The correct answer is 8.

To find the number of distinct integer solutions $(x,y)$ of the equation:

$$|x+y|+|x-y|=2$$

we can use a substitution method to simplify the equation. Let us define two new variables:

$$u=x+y$$

$$v=x-y$$

Since $x$ and $y$ must be integers, their sum $u$ and difference $v$ must also be integers.

Adding the two equations gives:

$$u+v=2x$$

This implies that $u+v$ must be an even integer. Therefore, $u$ and $v$ must share the same parity (either both are even, or both are odd).

Substituting $u$ and $v$ into our original equation gives:

$$\left|u\right|+\left|v\right|=2$$

Since $u$ and $v$ are integers, we can analyze the possible non-negative integer pairs for $\left(\right|u|,|v\left|\right)$ that sum to 2, ensuring they share the same parity:

Case 1: $\left|u\right|=2$ and $\left|v\right|=0$
This gives the integer pairs $(u,v)=(2,0)$ or $(-2,0)$. Note that in both cases, $u$ and $v$ are even (same parity).
Solving for $x=\frac{u+v}{2}$ and $y=\frac{u-v}{2}$, we find the corresponding solutions for $(x,y)$:
• For $(2,0)$: $x=1,y=1$, yielding the coordinate $(1,1)$.
• For $(-2,0)$: $x=-1,y=-1$, yielding the coordinate $(-1,-1)$.

Case 2: $\left|u\right|=0$ and $\left|v\right|=2$
This gives the integer pairs $(u,v)=(0,2)$ or $(0,-2)$. Here, both $u$ and $v$ are even (same parity).
• For $(0,2)$: $x=1,y=-1$, yielding the coordinate $(1,-1)$.
• For $(0,-2)$: $x=-1,y=1$, yielding the coordinate $(-1,1)$.

Case 3: $\left|u\right|=1$ and $\left|v\right|=1$
This gives the integer pairs $(u,v)=(1,1)$, $(1,-1)$, $(-1,1)$, or $(-1,-1)$. In all these cases, both $u$ and $v$ are odd, so they share the same parity.
• For $(1,1)$: $x=1,y=0$, yielding the coordinate $(1,0)$.
• For $(1,-1)$: $x=0,y=1$, yielding the coordinate $(0,1)$.
• For $(-1,1)$: $x=0,y=-1$, yielding the coordinate $(0,-1)$.
• For $(-1,-1)$: $x=-1,y=0$, yielding the coordinate $(-1,0)$.

Combining all three cases, we find the distinct integer solutions for $(x,y)$ are:

$$(1,1),(-1,-1),(1,-1),(-1,1),(1,0),(0,1),(0,-1),(-1,0)$$

Counting these ordered pairs, we find there are exactly 8 distinct integer solutions.