Introduction to Inequalities

A statement involving the symbols $<, >, \le, \ge$ is called an inequality (or an inequation). Just as an equation asserts that two expressions are equal, an inequality asserts that one expression is less than, greater than, less than or equal to, or greater than or equal to another.

Strict and slack inequalities. The symbols $<$ (less than) and $>$ (greater than) are called strict inequalities — they never allow equality between the two sides. The symbols $\le$ (less than or equal to) and $\ge$ (greater than or equal to) are called slack (non-strict) inequalities — they allow the boundary value itself to satisfy the statement.

Numerical vs. literal inequalities. An inequality that compares two known numbers, such as $3 < 7$ or $-2 \ge -5$, is a numerical inequality — a plain true-or-false statement about specific numbers, with no variable involved. An inequality that contains at least one variable, such as $2x - 3 < 7$ or $3x + 2y \le 12$, is a literal inequality. Our interest throughout this chapter is in literal inequalities, since these are the ones with a solution set to be found.

Linear inequalities. A literal inequality is called linear when every variable in it occurs only to the first power (degree 1), with no products of variables with each other and no variable in a denominator. So $2x - 3 < 7$ is linear in the one variable $x$, and $3x + 2y \le 12$ is linear in the two variables $x$ and $y$. By contrast, $x^2 - 4 > 0$ is not linear, since $x$ is squared.

Solution of an inequality. A value of the variable that turns the inequality into a true numerical statement is called a solution. For example, $x = 3$ is a solution of $2x - 3 < 7$, because substituting gives $2(3) - 3 = 3 < 7$, which is true; but $x = 6$ is not a solution, since $2(6) - 3 = 9$ is not less than $7$. The complete collection of every value that satisfies the inequality is its solution set. Unlike a linear equation in one variable, which typically has exactly one solution, a linear inequality in one variable typically has infinitely many solutions, forming an interval or ray of the number line rather than a single point.

What this chapter covers. Section 2 solves a linear inequality in one variable algebraically — isolating the variable using legal operations on both sides, with special care taken over what happens when both sides are multiplied or divided by a negative number. Section 3 extends this to inequalities built from the modulus (absolute value) function, such as $|x - 3| < 5$, which describe how close a value must lie to a fixed point. Section 4 shows how to represent any such one-variable solution set visually on the number line, using open and closed circles to mark whether an endpoint is included. Section 5 turns to linear inequalities in two variables, such as $2x + y \le 10$, whose solution is not a segment of a line at all but an entire region of the coordinate plane, found and shown graphically.