Interval Notation

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Understanding Interval Notation

Today we’re going to learn about interval notation. This is a method used to describe a set of numbers that are between two endpoints. Interval notation includes the endpoints unless there is a reason not to include them (such as the endpoint being infinity or the set not including the endpoint number itself).

Types of Intervals

There are four main types of intervals we’ll cover:

• Closed intervals: These include the endpoints. We use square brackets [ ] to show this. For example, [1, 5] includes numbers 1, 2, 3, 4, and 5.
• Open intervals: These do not include the endpoints. We use parentheses ( ) to show this. For example, (1, 5) includes numbers greater than 1 and less than 5, but not 1 or 5 themselves.
• Half-open (or half-closed) intervals: These include only one of the endpoints. For example, [1, 5) includes 1 and all numbers up to but not including 5.
• Infinite intervals: These extend indefinitely in one or both directions. For example, (1, ∞) includes all numbers greater than 1.

Examples of Interval Notation

Let’s look at some examples to clarify these concepts:

• The set of all real numbers between 3 and 7, including 3 and 7, is written as [3, 7].
• The set of all real numbers greater than 3 and less than 7, not including 3 and 7, is written as (3, 7).
• The set of all real numbers greater than or equal to 0 is written as [0, ∞).

Common Questions

Students often have questions about interval notation, such as:

Q: What if there’s an infinite boundary?

A: We use infinity ( ∞ ) to represent an endless boundary. Since we can never reach infinity, we always use a parenthesis with it. For example, (−∞, 5) includes all numbers less than 5.

Q: How do we know if a number is included or not?

A: If the interval uses brackets [ ], the number is included. If it uses parentheses ( ), the number is not included.

Q: Can intervals be used for all numbers?

A: Intervals are most commonly used for real numbers, not for individual integers or other sets of numbers that are not continuous.

Practice Problems

Now, let’s try some problems to practice what we’ve learned. Here are a few for you to solve:

1. Write the interval notation for all numbers greater than -2 and less than or equal to 3.
2. Express the following in interval notation: x is a real number greater than 6.
3. Express the set {x | x ≤ -1 or x > 4} using interval notation.

I’ll give you a few minutes to work on these, and then we’ll go over them together.

Solutions to Practice Problems

Here’s how we solve the problems:

1. For all numbers greater than -2 and less than or equal to 3, we use a parenthesis for -2 and a bracket for 3. So it’s (-2, 3].
2. For x greater than 6, we write (6, ∞).
3. For x ≤ -1 or x > 4, we have two separate intervals. We combine them using the union symbol ∪. So it’s (-∞, -1] ∪ (4, ∞).

Remember, practice is key to understanding interval notation. Don’t hesitate to reach out if you have any questions as you work through these problems!

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