How to Write Interval Notation: A Step-by-Step Guide

Interval notation is a concise way to represent a range of real numbers using brackets and parentheses. While the Interval Notation Calculator automates this process, understanding how to write it by hand builds a strong foundation for algebra, calculus, and beyond. This guide walks you through the manual steps, complete with examples and common pitfalls.

First, make sure you're familiar with the basics. If you need a refresher, check out What Is Interval Notation? Definition & Examples (2026).

You'll need

• Paper and pencil
• A number line sketch (helpful for visualizing endpoints)
• Knowledge of inequality symbols: <, ≤, >, ≥
• Understanding of infinity (∞) and that it always gets a parenthesis

Step-by-step guide to writing interval notation

1. Identify the endpoints – Determine the smallest and largest numbers in the set. For example, for the inequality 3 ≤ x < 7, the endpoints are 3 and 7.
2. Check whether each endpoint is included – If the inequality uses ≤ or ≥, the endpoint is included (closed). If it uses < or >, the endpoint is excluded (open). In our example, 3 is included, 7 is excluded.
3. Choose the correct bracket – Use a bracket [ for included endpoints and a parenthesis ( for excluded endpoints. For the left endpoint, if included: [3; for the right endpoint, if excluded: 7).
4. Handle infinity – If the interval extends to infinity (positive or negative), always use a parenthesis next to ∞ or –∞. For instance, x > –2 becomes (–2, ∞).
5. Write the interval – Place the left endpoint first, then a comma, then the right endpoint. Enclose everything with the appropriate brackets/parentheses. Example: [3, 7).
6. Double-check your interpretation – Read the interval aloud: “from 3 to 7, including 3 but not 7.” Ensure it matches the original inequality. For more advanced rules, see Mathematical Rules of Interval Notation Explained 2026.

Fully worked examples

Example 1: Writing interval notation for an inequality

Problem: Write interval notation for the set of numbers x such that –1 ≤ x ≤ 4.

Solution:

• Endpoints: –1 and 4 both included (≤).
• Left bracket: [–1
• Right bracket: 4]
• Final notation: [–1, 4]

This represents all numbers from –1 to 4, including both endpoints.

Example 2: Writing interval notation with infinity

Problem: Write interval notation for x > 5.

Solution:

• Left endpoint: 5, but it is excluded (>>). So use parenthesis: (5
• Right endpoint: positive infinity – always parenthesis: ∞)
• Final notation: (5, ∞)

This means all numbers greater than 5, extending forever. For more on infinite intervals, visit Interval Notation Ranges: What Different Intervals Mean (2026).

Common pitfalls to avoid

• Mixing up brackets: Remember: bracket = included, parenthesis = excluded. A common mistake is writing [5, ∞) – that would incorrectly include 5 if the inequality was x > 5.
• Using brackets with infinity: Infinity is not a real number, so it can never be included. Always use parentheses: (–∞, ∞), never [–∞, ∞].
• Reversing order: The left endpoint must be smaller than the right endpoint. For example, [7, 3) is invalid because 7 > 3.
• Forgetting the comma: Interval notation always uses a comma between the two endpoints, e.g., (–2, 3], not (–2 3].

If you ever get stuck, the Interval Notation Calculator can instantly convert between notation types and check your work. But knowing how to write it by hand gives you confidence and a deeper grasp of mathematical sets.