Introduction

From Equations to Inequalities

In earlier classes, you learned how to set up and solve equations — statements of equality such as $2x + 3 = 7$ or $x + y = 5$ — and how to translate real-life word problems into equation form. A natural question arises: can every real-world situation be expressed as an equation? Consider two examples:

• The height of every student in your class is less than 160 cm.
• Your classroom can hold at most 60 tables or chairs (or a combination of both).

Neither statement expresses an equality. The first uses "less than" ($<$); the second uses "at most" ($\leq$). Statements that compare two quantities using the symbols $<$, $>$, $\leq$, or $\geq$ are called inequalities.

Why Study Inequalities

This chapter introduces linear inequalities, studied in one variable and in two variables. Working with inequalities is useful well beyond mathematics — the same reasoning appears in science, statistics, economics, and psychology, wherever a condition is a range or a limit rather than an exact value (a budget that must not be exceeded, a height restriction, a passing threshold, and so on).

The precise definition of an inequality, the different kinds you will encounter (numerical, literal, double), and the exact forms a linear inequality can take — including the strict-vs-slack distinction — are covered next.