Linear Inequalities in One Variable

Solve linear inequalities in one variable. Grade 11 Math 20-1 curriculum.

Lesson 2.5 of Linear Functions & Systems of Equations in Math 20-1 — Alberta curriculum lessons.

What you'll learn in this lesson

• Three Ways to Express an Inequality. Learn set notation, interval notation, and number lines — three equivalent ways to describe the same solution set — and practice converting between all three.
• Solving Inequalities. Solve linear inequalities using the same algebraic steps as equations, with one critical rule: flip the symbol whenever you multiply or divide by a negative number.
• Summary and Common Mistakes. Compare equations and inequalities side by side, review the five-step solving strategy, and avoid the four most common errors students make on tests.

Inside the lesson — a free preview

What is an Inequality?

Definition. An inequality is a mathematical statement that says two expressions are not necessarily equal. Instead of an equals sign, it uses one of four comparison symbols.

Key Idea — Infinitely Many Solutions. Unlike an equation like x = 5 (which has exactly one solution), an inequality like x < 5 represents infinitely many values — every real number less than 5. Inequalities describe a set of numbers along a number line, not just a single point.

The Three Notations

Three Equivalent Ways to Describe the Same Set. Set notation — write the inequality directly using the variable and a comparison symbol. Example: x ≥ -2.

Interval notation — use brackets and numbers to describe the range. Example: [-2, ∞).

Number line — draw a visual with dots and shading. Each notation carries the same information in a different form.

Round vs. Square Brackets. The bracket type tells you whether an endpoint is included or excluded.

Infinity Always Gets a Round Bracket. Infinity (∞ or -∞) is not a real number — you can never reach it, so it can never be included. This means infinity always uses a round bracket, without exception: (-∞, 5] or (-3, ∞).

Worked example: All Three Notations Together

Problem: Describe all numbers ≥ -2 AND ≤ 4. Set notation: -2 ≤ x ≤ 4

Interval notation: [-2,\ 4]

Both endpoints use square brackets because both -2 and 4 are included (the inequality uses ≤).

Vocabulary: the value -2 is the lower bound and 4 is the upper bound of the interval.

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