If $x-y=8$, then which of the following must be true?

1. Both x and y must be positive for any value of x and y.
2. If x is positive, y must be negative for any value of x and y.
3. If x is negative, y must be positive for any value of x and y.

Select the correct answer using the code given below.

The correct option is Neither 1 nor 2 nor 3.

To understand why none of the given statements must be true, let us analyze the given equation step-by-step:

$$x-y=8$$

We can rewrite this equation to express $x$ in terms of $y$ or vice versa:

$$x=y+8$$

Now, let us evaluate each of the three statements by finding counterexamples (values of $x$ and $y$ that satisfy the equation $x-y=8$ but make the statements false):

Analysis of Statement 1: "Both x and y must be positive for any value of x and y."
Let us choose $y=-2$ (which is negative).
Then, $x=-2+8=6$ (which is positive).
Here, the pair $(x,y)=(6,-2)$ satisfies $6-(-2)=8$, but $y$ is negative. Thus, both do not have to be positive. Statement 1 is false.

Analysis of Statement 2: "If x is positive, y must be negative for any value of x and y."
Let us choose $y=2$ (which is positive).
Then, $x=2+8=10$ (which is also positive).
Here, the pair $(x,y)=(10,2)$ satisfies $10-2=8$. In this case, $x$ is positive, yet $y$ is positive (not negative). Statement 2 is false.

Analysis of Statement 3: "If x is negative, y must be positive for any value of x and y."
Let us choose $x=-3$ (which is negative).
Using our equation:
$$-3-y=8$$
Solving for $y$ gives:
$$y=-3-8=-11$$
Here, the pair $(x,y)=(-3,-11)$ satisfies $-3-(-11)=8$. In this case, $x$ is negative, yet $y$ is negative (not positive). Statement 3 is false.

Since none of the statements 1, 2, or 3 must be true, the correct answer is indeed "Neither 1 nor 2 nor 3".