If I graph the single equation in one variable $x^2 - 1 = 10$ in $\mathbb R$ I get two distinct points on the real line (obviously I could say the solutions don't overlap).
I'm trying to think of a single equation in two variables that when graphed in $\mathbb R^2$ gives me two distinct curves that also do not overlap.
Is this possible? I can't think of one off the off the top of my head. May be a dumb question.
On
It is pretty hard to interpret, what you mean by "gives me", if you mean a surface in $\mathbb{R}^3$ such that its intersections with the plane $z=0$ are two disjoint curves, then take a look at $z=\frac{1}{x}-y$, when $z=0$, it has
$$\frac{1}{x}-y=0 \Rightarrow y=\frac{1}{x},$$
Which are two "curves" in the form of a hyperbola.
EDIT: Here you are, it's a "vertical torus" cut at $z=0$:
$$(5-\sqrt{x^2}))^2+y^2=2^2$$
The $2$ is the radius of the circles, and the $5$ is the distance from the origin to the centres of each circle.
It's very possible. A simple example is $xy=1$, which is just the standard hyperbola.
You can even do it with one variable: $|x|=1$.
You can get two concentric circles around $(0, 0)$ with something like $|x^2+y^2-a|=b$.
You can get two circles next to each other with something like: $(|x|-a)^2+y^2=b$.