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 ,
then:

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.