LESSON 14

INEQUALITIES

When Two Sides Are Not Equal

What is an Inequality?

An inequality is a mathematical statement that compares two expressions using one of four symbols: less than (<), greater than (>), less than or equal to (≤), or greater than or equal to (≥). Unlike an equation, an inequality has a range of solutions rather than a single answer.

Inequalities are used constantly in real life — speed limits, age requirements, weight limits, and budget constraints are all examples of inequalities. Solving them follows rules very similar to solving equations, with one critical exception.

Solving One-Step Inequalities

Solve an inequality the same way you solve an equation — perform inverse operations to isolate the variable. The critical rule: when you multiply or divide both sides by a negative number, you must flip the inequality symbol. Adding or subtracting never changes the direction of the symbol.

Example:x + 4 > 9 → x > 5 | −2x < 8 → x > −4 (flip! divided by −2)

Graphing on a Number Line

The solution to an inequality is a set of numbers, which we show by graphing on a number line. Use an open circle (○) for strict inequalities (< or >) to show the endpoint is NOT included. Use a closed circle (●) for ≤ or ≥ to show the endpoint IS included. Shade the line in the direction of the solutions.

Example:x > 3: open circle at 3, shade right | x ≤ −1: closed circle at −1, shade left

Two-Step Inequalities

Two-step inequalities are solved just like two-step equations — undo addition or subtraction first, then undo multiplication or division. Remember to flip the inequality symbol any time you multiply or divide by a negative number.

Example:3x − 5 ≥ 10 Add 5: 3x ≥ 15 Divide by 3: x ≥ 5

Compound Inequalities

A compound inequality combines two inequalities. An 'and' compound inequality (also written as a three-part inequality) requires both conditions to be true — the solution is the overlap. An 'or' compound inequality requires at least one condition to be true — the solution is the union of both sets.

Example:−2 < x ≤ 5 (and: x is between −2 and 5, including 5) x < 0 or x > 4 (or: x is less than 0 or greater than 4)

Key Facts

The symbols < and > were introduced by English mathematician Thomas Harriot in 1631, more than 70 years after the equals sign was invented.

Flipping the inequality when multiplying or dividing by a negative is one of the most commonly forgotten rules in algebra. Always check the sign of what you are dividing by.

Linear programming — used in business, logistics, and AI — is built entirely on systems of inequalities. It finds the best solution within a set of constraints.

The solution set of an inequality is infinite. x > 3 includes 3.1, 4, 100, 1,000,000 — every number greater than 3 is a valid solution.

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