We are taking a course in Real Analysis with the text: "Elementary Analysis, the theory of Calculus", by Kenneth Ross. I have also been reading a little bit of Beckenbach and Bellman's book, "Introduction to Inequalities."
In the Real Analysis book five order axioms are stated for the real numbers. In the Inequality book there are only two axioms and things like transitivity are derived from these two axioms.
My question is, why is it necessary to use five order axioms in Real analysis, instead of just the two inequality axioms?
I think you're not asking the right question. For a typical analysis textbook, it is not about whether some axioms are necessary or what the minimal set of axioms should be. A good analysis textbook should pick a manageable set of axioms that are intuitive and from there proceed with a good exposition of the subject. You'll have forgotten the details of the axiom system very soon anyway.
If you're into questions like whether certain axioms are necessary or how to minimize axiom systems (or whether two sets of axioms are equivalent), then foundations is the way to go.