Inequality vs. Inequation, Equation vs. Equality

When talking about statements that two terms are mathematically (un-)equal (e.g., 1 = 1, 1 < 2, or 1 != 2), what is the correct notion for such mathematical statements? Are there special cases w.r.t. "mixed" relations like x <= y? Can some notions be used interchangeably?

It seems many online sources use "equation" and "inequality" but there is also a Wikipedia page about "inequation". Apart from personal posts in forums, I have not found any sources using "equality" in such a situation, yet. Systematically, I would think that "equation" and "inequation" would make more sense than "equation" and "inequality", but not everything (probably very few things) in natural languages have developed according to a purely logical system.

Note that this question is more general than How to express the relationship that two numbers are not equal? for that it particularly asks for the relation between the notion for equations and the one for inequalities. Equations are not part of the existing question at all.

Solution 1:

Inequation specifically refers to two quantities that are not equal, and the direction of the relationship (greater than or less than) is unknown. It is seldom used. Example: x!=y, or x<>y (depending on your computer language). There is also an equals sign with a slash through it, but that's not a standard character.

Equality is not used in the mathematics in place of equation. Inequality is used when there is a known directional relationship in the data, such as x or |x|<|y|. That second case doesn't state whether x or y is greater, but it does state that the magnitude of x is less than the magnitude of y.