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I’ve struggled with the WHY on this myself. I think Jon had some great thoughts about looking at patterns with the numbers / equations. For example, I used to start with equations: if x = 4, then it makes sense that x = 4 (mult. prop. equality). BUT, a true statement like 1<3 is not true when using that property: 1 </ 3. Gives an opportunity to discuss how negative numbers are inverses of positive numbers so using an inverse inequality when a sign change occurs makes some sense.
This Dr. Math website explanation also looks pretty clear to me on first glance: http://mathforum.org/library/drmath/view/53142.html
Good luck!