# Interval Notation Overview

**What is interval notation?** Interval notation is simply a shorthand method for writing sets of numbers, namely continuous ranges of real numbers.

**Interval Notation Sample Questions**

**When do we use it?** Perhaps you are familiar with how we write a finite set of numbers in mathematics. Consider the following example as a refresher:

Q: Write as a set all the possible outcomes for rolling a standard die.

A: {1, 2, 3, 4, 5, 6}

In mathematics, we use curly brackets around finite, or countable, sets of numbers. But how might we answer the following question?

Q: If Charlie’s gas tank can hold ten gallons of gasoline, how much gas might he get next time he stops at a station? List all the possible amounts as a set of numbers.

This time, the answer isn’t as simple as writing out a list of numbers separated by commas. Charlie may need anywhere between zero and ten gallons of gas! For example, he may need 7, 4.5, or 3.1415… It’s impossible to write out every single real number between zero and ten, so that’s where interval notation comes in.

With interval notation, we write the leftmost number of the set, followed by a comma, and then the rightmost number of the set. Then we put parentheses or square brackets around the pair, depending on whether those two numbers are included in the set (sometimes we use one parenthesis and one bracket!). A parenthesis means the number is not included in the set but every number higher than it is included in the set, and a square bracket means that the number is included in the set.

A: [0, 10]

Since there’s a possibility that Charlie may not need any gas at all when he stops, we have a 0 as the lowest number in our set, and we place it as the first value. And since Charlie’s gas tank can hold ten gallons, 10 is the highest number in our set, and we put it as the second value. Interval notation automatically accounts for every number in between! Since 0 and 10 are both included in our set, we used square brackets for both the left and the right.

Whenever we do not wish to include the highest or lowest points in an interval, we use parentheses. Note that the following are equivalent:

Set of numbers expressed as an inequality | Set of numbers expressed using interval notation | |

All real numbers between -1 and 1, including both -1 and 1 | -1 ≤ x ≤ 1 | [-1, 1] |

All real numbers between 3 and 300, not including 3 or 300 | 3< x < 300 | (3, 300) |

All numbers between 5 and 6, including 5 but not 6 | 5 ≤ x < 6 | [5, 6) |

If you need help remembering when to use parentheses versus square brackets, consider the following mnemonic: you have a box and a bowl. When you drop a ball into the box, it stays there, but when you drop the ball into the bowl, it rolls out. We use brackets (which are shaped like the box) to show that a number is included (the ball doesn’t fall out). And we use parentheses (which are shaped like the bowl) when a number is not included (the ball falls out).

**Interval Notation Sample Questions**

Here are a few sample questions going over interval notation.

**Question #1:**

Use interval notation to write the set of all possible real numbers between 4 and 5, including both 4 and 5.

[4, 5]

(4, 5]

(4, 5)

[4, 5)

**Answer:**

The correct answer is [4, 5] because we use square brackets to indicate that both 4 and 5 are included in the set.

**Question #2:**

Write the following inequality using interval notation: 0 < x < 3.5

[0, 3.5]

(3.5, 0)

(0, 3.5)

[0, 3.5)

**Answer:**

The correct answer is (0, 3.5) because we put the lowest number, 0, on the left, and the highest number, 3.5, on the right. We use two parentheses to indicate that neither of the two numbers are included in the set.

**Question #3:**

Jessica is trying to reach her goal of drinking 80 fl. oz. of water today, but she hasn’t reached her goal yet. Use interval notation to write out the set of all possible numbers describing how much water she may have consumed so far today.

(0, 80]

[0, 80]

(0, 80)

[0, 80)

**Answer:**

The correct answer is [0, 80) because Jessica could have consumed 0 ounces of water, or any value less than 80 ounces, but not 80 exactly (remember, she hasn’t met her goal yet!). Therefore, we include 0 but do not include 80 in our set.