An inequality is a relation which makes a non-equal comparison between two numbers or other mathematical expressions. The "inequality" has a plural form by "inequalities". I would like to ask whether "equality" has plural form "equalities".
In my paper, I would like to write the following sentence.
We are concerned with the conditions under which two inequalities come to equality in the proof.
I'm not sure if that's right. Maybe the following is true
We are concerned with the conditions under which two inequalities come to equalities in the proof.
For another example, here are two inequalities.
$$2x\le 1$$ Equality occurs if and only if $x=\frac{1}{2}$.
$$5x\le 2$$ Equality occurs if and only if $x=\frac{2}{5}$.
I would like to say:
These two inequalities come to equalities(or equality?) when $x=\frac{1}{2}$ and $x=\frac{2}{5}$, respectively.
I want to replace "equality" with "equations", but I feel that equations and equality don't seem quite the same.