Let's start by talking about our solving equations work:
there are several different techniques for us to use when solving these. substitution, elimination, and graphing are usually our best three. In fact, graphing or using a computer to help us is going to be the best strategy almost all of the time, but I want us to think about WHY the other two work so well.
But these things work really well when we are just talking about when two sides of something are equal. That isn't always the case. Sometimes, something in MORE than something else, or BETWEEN something else. This can lead to complications--like a whole lot of possible answers. This is called an inequality, and it's trying to let us know that one is more than the other -- possibly a lot more. By working with these, we can sometimes answer some interesting questions -- this is called linear programming.