Inequalities help us impose limits in a world in which perfect precision is not possible.
For example, let's say that we want to get as many donuts as possible while not going over $10. Let's say each donut costs $.79. Then we want to find an positive integer {$$}n{/$$}such that {$$}n*0.79 < 10{/$$}. Solving the for {$$}n{/$$}. We will go over such problems and expand on them in some depth in the first two sections.
We explore a classic inequality called the triangle inequality. It is the algebraic expression of the basic idea that going along the other two sides of a triangle takes longer than going along the third side.
We spend a few sections on absolute values. Absolute values seem fairly harmless at first as it seems we are just erasing negative signs. But it quickly gets difficult with variables involved.
We end with extending the idea of absolute value to that of a more general notion of distance
Inequalities and absolute values are notorious for being confusing. Building on our notion of distance, we hopefully will convey some interesting ideas about it. We will explore some stuff using polynomials, but the majority of an introduction to this material is with lines.
With our core example set in place, we then move on into exploring inequalities fully, including the extremely useful but confusing topic of abslute values.