Just as axioms exist for equality, similar axioms exist for inequality. The only axiom of equality that has no counterpart for inequality is the reflexive axiom. The other seven are as follows.
The Transitive Axiom
PARGRAPH The transitive axiom of inequality is states that if one quantity is greater than the second and the second quantity is greater than the third, then the first quantity is greater than the third.
The Substitution Axiom
The substitution axiom works the same way for inequalities as it does for equalities. If two quantities are equal, they can replace each other in any inequality. So if two triangles are congruent, and a segment is greater than a side in one triangle, that segment is greater than the corresponding side of the other triangle as well.
The Partition Axiom
The partition axiom for inequalities is as follows: A whole quantity is greater than any one of its parts. We have seen this at work with the exterior angle of a triangle and the remote interior angles. The exterior angle is equal to the sum of the remote interior angles, and greater than either remote interior angle.
The Addition, Subtraction, Multiplication, and Division Axioms
The addition, subtraction, multiplication, and division axioms for equality work the same for inequalities. The difference is that the inequality axioms state that if unequal quantities are added, subtracted, etc. from equal quantities, then their sums, differences, etc., will be unequal.