This chapter is about inequalities -- statements that show the
relationship between two (or more) expressions with one of the
following five signs: <, ≤, >, ≥, ≠. These are
similar in form to the equations learned in
chapter one, with one key difference: here, we are dealing with
**un**equal quantities instead of equal quantities.

The first section explains the meaning of an inequality. It
introduces the concept of inequalities with variables, and shows how
to find a solution set for an
inequality, given a replacement
set.

The second section introduces the formal properties of inequalities.
These properties are similar to the properties of
equality properties of
operations and identities, with
a few key differences, particularly in the Multiplication and
Division Properties of Inequalities. It is necessary to know all
the properties in this section in order to solve inequalities. In
addition, these properties teach us more about the real numbers and
how they interact with each other.

The third section explains how to solve inequalities using inverse
operations. This section presents easy-to-follow steps for
solving inequalities.

Certain inequalities can be confusing to solve merely by using inverse
operations. Thus, the next section presents an alternative way of
solving them: graphing. It first explains how to graph any inequality
on a number line, and then shows how
to use the number line to solve an inequality.

The final section highlights one application of inequalities to
geometry: classifying angles. Here, we will learn how to use
inequalities to classify angles as right angles, acute angles,
or obtuse angles. This provides practice working with
inequalities, as well as an introduction to material which will be
covered in great depth in geometry.

Though this may be your first formal exposure to inequalities, you
have probably been working with them all of your life. Any statement
which includes the words "at least," "at most," "more than," or "less
than" is an inequality. This chapter makes working with such
statements easier, explaining what they mean in a mathematical sense,
as well as how to figure out which numbers satisfy them and how to
graph them. Inequalities appear in a variety fields -- math, physics,
chemistry, biology, economics, business -- as well as in everyday
tasks like cooking, spending money, and driving, for instance. Thus,
it is useful to understand them and to know how to work with them.