Logical deduction from being able to solve inequalities

I was reading my lecture notes a few moment ago and I thought the logical deduction of the fact that since we are able to solve inequalities, then it also means we can solve equalities and negation.

Suppose we are able to solve the below two inequalities,

1. α ≤ r
2. β ≥ r

then, we have

• α ≤ r ≤ β

With that, we should be able to solve r = α since we know α ≤ r ≤ α

Hence,

• r = α ≡ α ≤ r ≤ α

And,

• α < r < β ≡ α + 1 ≤ r ≤ β - 1

And also,

• α ≠ r ≡ (L < r < α) ∪ (α < r < R)