Interval notation is a way to describe a set of numbers. In elementary school, students are introduced to number lines and inequalities. Number lines are a wonderful visual tool for students to make sense of numbers and to process whether a value is greater than or less than another value. Eventually, these ideas morph into a solution set when they reach pre-algebra. For example, x is greater than 3 has infinitely many solutions. Its solution set could be displayed on a number line with an open ended ray.
Students first learn to write this analytically as an inequality, x > 3. We can write this same interval of values with interval notation such as (3,∞). In algebra, we start to study intervals of numbers such as the domain and range. Instead of using an inequality to represent an interval of real numbers, interval notation is often used instead. When a set of numbers does not include the endpoint, as shown above, a parenthesis is used to indicate that the interval approaches that number, but does not include it in the interval.
Suppose we would like to include the endpoint of an interval. Let’s look at the interval when x is less than or equal to 2.
As an inequality we would write this as x ≤ 2. However as an interval of values, we would use a square bracket to show that the endpoint 2 is included in the interval such as (-∞, 2]. Note that one can never actually reach infinity, so infinity will always have a parenthesis and not a square bracket.
Sometimes an interval of numbers has a starting and ending point on the number line. This is sometimes referred to as an “and” compound inequality.
As an inequality we would write -3<x<1. And then for interval notation it would look like (-3,1).
We can make a union of intervals when the rays go in opposite directions. This would be an “or” compound inequality.
The set of all real numbers can also be written with interval notation.
A common mistake I students make when using interval notation is to write the larger value first and then the smaller value. So, make sure that when you introduce interval notation to your students that you remind them it looks like (lower bound, upper bound).
You can grab my free interval notation graphic organizer here. I also have a full version Interval Notation Lesson available to purchase in my TPT store that is part of my Key Features Unit.