Introduction and Summary
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 unequal 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.