The same general set of rules holds for inequalities except that
if one multiples both sides of an inequality by a negative number (or
perform any other transformation with a similar effect) one must change the direction of the inequality as well. That is, If ,
That's pretty much it. Most solutions to almost any algebraically formulated problem involve taking small steps selected from the list above. It's easy once you get the hang of it! In fact, you can actually learn to ``visualize'' algebra, moving this symbol or that from one side of an equation to the other with your eyes alone even before you put the step down on paper.
Mind you, the list above left several things undone! For one thing, we haven't begun to talk about or define a number of words that we used in our listing above - words like ``function'', ``power'', ``term'', ``domain'', ``range''. Once again we were faced with a bootstrapping issue and chose to begin with the rules themselves even if the rules contained some ``forward references'' to some things we'd rather discuss with the rules (at least some of them) in hand.
Let's get started cleaning all this up. First we'll define some very common algebraic operations, such as forming powers of a symbol. Then we'll talk about a special class of equations called functions - maps from one symbol (or set of symbols) onto another. Functions often describe relationships between quantities, which in turn have meaning, so this is a good step to have underway.