Is it an axiom that the inequalities are translation-invariant or can we prove it? I was thinking about the inequalities on the set of real numbers. To me and everyone else, it's been taught that an inequality is translation-invariant, i.e.:
$x < y \implies x + c < y + c \quad \forall c \in \mathbb{R}$
But I've been trying to think why. Is it simply a property we assign, or is there a reason that draws from the inequality's existing properties?

The reals can either be defined axiomatically or constructed from (usually) a model of the rationals, in various ways. Axiomatically, the reals are an ordered field, which means that translation invariance property you refer to is an axiom. So, if your approach to the reals is that they are simply a model (one of many, but all isomorphic) of the axioms of the reals, then this property is an axiom. If instead you prefer to construct the reals from the rationals, then you must first accept the same translation invariance property for the rationals (and again, this is either taken axiomatically, or proven for a construction of the rationals, say from the integers) and then prove it for the particular construction giving the reals. For most constructions of the reals this part is actually quite easy. 
