**Spoiler:**

It seems to me that whenever an equation or inequality without any boundaries on the variables is found that has no solutions, we make a name for the set of numbers that includes these solutions. However, doing this implies that solutions do exist for any equation. In other words, any operation preformed on numbers will result in numbers. I feel like that is a pretty powerful assumption to make, so I was wondering what the justification for it was.

Before anyone asks, I would not consider dividing by 0 to be a counterexample, because I can use limits to calculate what the result is. Limits are a method of calculation; it does not change what the operation in question actually is.

Bonus questions: Can the first inequality be replaced by one that has a single inequality sign? Is there an equation that can replace the inequalities? I nearly drove myself insane trying to do these two things, but it just feels like there is something really obvious that I am missing..