Linear Inequalities: Intervals and Sets

Table of Contents

1. What is an Interval?
2. Types of Intervals
3. Intervals for Inequality Solutions
4. Set-Builder Notation
5. Conversion Table
6. Special Cases
7. Worked Examples
8. Common Mistakes
9. Practice Exercises
10. Related Articles

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1. What is an Interval?

An interval is a way of describing a continuous set of numbers between two endpoints. Instead of listing individual solutions, we write a compact notation that captures all the numbers at once.

Key idea: Every linear inequality in one variable has a solution that can be expressed as an interval (or as the empty set, or as all real numbers).

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2. Types of Intervals

Open Interval

An open interval $(a,b)$ contains all numbers between $a$ and $b$, but not the endpoints themselves.

Closed Interval

A closed interval $[a,b]$ contains all numbers between $a$ and $b$, including both endpoints.

Half-Open (Half-Closed) Intervals

These include one endpoint but not the other:

Unbounded Intervals

When the solution extends to infinity, we use $\infty$ or $-\infty$:

Important: Infinity is not a real number, so we always use a round bracket next to $\infty$ or $-\infty$. Never write $[\infty$ or $\infty ]$.

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3. Intervals for Inequality Solutions

Linear inequalities with one variable always produce unbounded intervals:

The rule is straightforward:

• Strict inequality ($<$, $>$) uses a round bracket --- the boundary is excluded
• Non-strict inequality ($\le$, $\ge$) uses a square bracket --- the boundary is included

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4. Set-Builder Notation

Another way to write solutions is set-builder notation:

This reads: "the set of all real numbers $x$ such that $x$ is greater than 3."

Note: In most school contexts, interval notation is preferred because it is shorter. Set-builder notation is useful when the condition is more complex.

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5. Conversion Table

Here is a complete reference for converting between the three notations:

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6. Special Cases

Empty Set (No Solution)

When an inequality has no solution, we write:

This happens when simplifying leads to a false statement like $3>7$.

All Real Numbers

When every number is a solution, we write:

This happens when simplifying leads to a true statement like $3<7$.

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7. Worked Examples

Example 1

Solve $2x-4>6$ and express the solution in all three notations.

• Inequality: $x>5$
• Interval: $(5,\infty )$
• Set-builder: $\{x\in \mathbb{R}\mid x>5\}$

Example 2

Solve $-x+3\ge 7$ and express the solution in all three notations.

• Inequality: $x\le -4$
• Interval: $(-\infty ,-4]$
• Set-builder: $\{x\in \mathbb{R}\mid x\le -4\}$

Example 3

Solve $4x+1<4x+9$ and express the solution.

This is always true.

• Inequality: all $x$
• Interval: $(-\infty ,\infty )$
• Set-builder: $\mathbb{R}$

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8. Common Mistakes

Mistake 1: Using a Square Bracket with Infinity

Wrong: $[3,\infty ]$

Correct: $[3,\infty )$ --- always a round bracket at infinity

Mistake 2: Confusing Round and Square Brackets

For $x>5$:

Wrong: $[5,\infty )$ --- this includes 5

Correct: $(5,\infty )$ --- this excludes 5

Mistake 3: Reversed Interval Endpoints

Wrong: $(\infty ,3)$

Correct: $(3,\infty )$ --- the smaller value always comes first

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Practice Exercises

• Inequalities - Intervals - Practice writing intervals
• Inequalities - Number Line - Convert between diagrams and notation
• Inequalities - Basic - Solve and write solutions

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Related Articles

• Linear Inequalities - Comprehensive Guide - Full introduction to linear inequalities
• Linear Inequalities - Number Line - Graphical representation of solutions
• Linear Inequalities - Rules and Formulas - Quick reference for rules
• Linear Inequalities - Special Cases - Empty set and all real numbers