Understanding Interval Notation
Interval notation is a concise way of representing a set of numbers along the number line. Instead of writing lengthy inequality expressions, interval notation uses brackets and parentheses to show which endpoints are included or excluded from the set.
A parenthesis ( or ) means the endpoint is not included (open/strict), while a bracket [ or ] means the endpoint is included (closed/inclusive). Infinity is always written with a parenthesis because it is not a specific number.
Conversion Reference
x > a
All numbers strictly greater than a. Open on left, extends to positive infinity.
x ≥ a
All numbers greater than or equal to a. Closed on left, extends to positive infinity.
x < b
All numbers strictly less than b. Extends from negative infinity, open on right.
x ≤ b
All numbers less than or equal to b. Extends from negative infinity, closed on right.
a < x < b
All numbers strictly between a and b. Both endpoints excluded.
a ≤ x ≤ b
All numbers between a and b, inclusive. Both endpoints included.
Types of Intervals
- Open interval (a, b) — Neither endpoint is included.
- Closed interval [a, b] — Both endpoints are included.
- Half-open [a, b) or (a, b] — One endpoint is included, the other is not.
- Infinite intervals — Extend to positive or negative infinity, always open at infinity.
Number Line Visualization
On a number line, an open endpoint is drawn as an empty circle, while a closed endpoint is drawn as a filled circle. The shaded region between the endpoints represents all numbers in the interval. This visual representation makes it easy to understand the solution set of an inequality.
Tips for Converting Inequalities
- Always write the smaller number first in interval notation.
- Use parentheses ( ) for strict inequalities (< or >) and for infinity.
- Use brackets [ ] for inclusive inequalities (≤ or ≥).
- Infinity and negative infinity always use parentheses, never brackets.
- For compound inequalities, make sure the lower bound is less than the upper bound.