Let `x` be a real number between two values, let `y` be a real number between two other values. What's the range of `x/y`?

Question

Like the title says:

$$ x \in \Bbb R : x \in [x_{min}, x_{max}] $$ $$ y \in \Bbb R : y \in [y_{min}, y_{max}] $$

What is the range of:

$$ x \over y $$

?

I find this hard to reason about because the division is discontinuous.

I can identify up to 49 different combinations of possibilities that I would need to analyze one by one... 7 for each variable: the cases where the variable is entirely negative, negative or 0, exactly 0, 0 or positive, entirely positive, from negative to positive.

For instance let's take the four cases where neither $x$ nor $y$ pass by 0, are somewhat simple:

1. $$ x_{max} \lt 0 \wedge y_{max} \lt 0 \Rightarrow [{x_{min} \over y_{max}}, {x_{max} \over y_{min}}] $$

2. $$ x_{min} \gt 0 \wedge y_{min} \gt 0 \Rightarrow [{x_{max} \over y_{min}}, {x_{min} \over y_{max}}] $$

3. $$ x_{max} \lt 0 \wedge y_{min} \gt 0 \Rightarrow [{x_{min} \over y_{min}}, {x_{max} \over y_{max}}] $$

4. $$ x_{min} \gt 0 \wedge y_{max} \lt 0 \Rightarrow [{x_{max} \over y_{max}}, {x_{min} \over y_{min}}] $$

These are probably the simplest cases, since they're away from the discontinuity of the division. Yet I'm getting lost with the signs. And many many more cases are still missing.

I assumed this might have been a standard question in interval arithmetic. So I was hoping there could be a simple precooked solution.