Equality and equivalence get murky sometimes. Often there is some sense of equivalence that's 'good enough'. Like spaces that are isomorphic or functions that are equal 'almost everywhere'.
Now in by far the majority of cases inequality is defined for partial orders, which have the property that a<=b and b<=a imply a=b, so in that case a<=b and not b<=a implies b=/=a, but (non-strict) weak orders can have a<=b and b<=a without necessarily a=b.
I'm pretty sure there are also a few cases where they just prove reflexivity and transitivity and just make a and b where a<=b and b<=a equal by definition. I think you could use this in the definition for real numbers, but usually the equivalence classes are defined first and inequality is defined afterwards (which might be more work).
Now in by far the majority of cases inequality is defined for partial orders, which have the property that a<=b and b<=a imply a=b, so in that case a<=b and not b<=a implies b=/=a, but (non-strict) weak orders can have a<=b and b<=a without necessarily a=b.
I'm pretty sure there are also a few cases where they just prove reflexivity and transitivity and just make a and b where a<=b and b<=a equal by definition. I think you could use this in the definition for real numbers, but usually the equivalence classes are defined first and inequality is defined afterwards (which might be more work).