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.

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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.

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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.

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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.

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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.