Interval Notation
An interval is a space of value, or a range of numbers. For example, whole numbers between 1 and 10 is an interval.
Interval notation is a way to write all of the values bound by end points. Conceptually this is very similar to writing inequalities and compound inequalities. Read through the notes in the tabs below, watch the video. Try the practice problems. Then, grade the practice problems. Finally, to really test your knowledge and understanding, try the quiz! You’ll get an instant score with feedback.
Learning takes time and patience, but is an investment in yourself. Allow yourself to be confused, you’ll work through it. That’s what’s required … nobody is born knowing this stuff!
Interval notation and set builder notation are just ways of writing a range of values. By range here, we do not mean the set of all possible outputs of a function, but instead a group of numbers with a boundary, similar to range in statistics. To make sense of this, let’s establish some common understanding. In this section we will discuss interval notation. In a future section we will learn about set builder notation.
Consider f(x) = (x – 3)2 + 4. This is a quadratic function with a vertex at (3, 4). Since the leading coefficient is positive, its graph is going up.
Specifically, the graph is decreasing from negative infinity up to the value of x = 3. However, at x = 3, the graph is not increasing or decreasing! If we were to write that range of values with inequalities, it would be – ∞ < x < 3. Notice that neither negative infinity or the 3 are included. Infinity is not a number and therefore cannot be included. This compound inequality is exactly the same as x < 3, but we used the compound version to help us better understand Interval Notation.
Let’s see how this compound inequality, – ∞ < x < 3, looks in interval notation. (- ∞, 3).
The most confusing part about interval notation is that it can often look like an ordered pair, (x, y). This is because Interval Notation has two numbers inside of a set of parenthesis, and the numbers are separated by a comma. The difference is that each number is a boundary for the interval described. The left number is the lower boundary, and larger boundary is the right number.
So, as an ordered pair, (5, 8) means that x = 5 and y = 8. But as interval notation, (5, 8) means 5 < x < 8.
What if the compound inequality included one or the boundaries, like 5 ≤ x < 8. The way this is written in interval notation is: [5, 8). So much like a solid dot on a graph means the value is included, and an open circle means the value is NOT included, the square bracket indicates the value is included, while the curved parenthesis indicates the value is not included. The diagram below ties all of that information together for you.
Let’s see one more example of Interval Notation.
The domain of the graph to the left is from -1 to 4, with the -1 included, while the 4 is not included. As an inequality that would be as follows.
-1 ≤ x < 4
Written in interval notation, it would be as follows.
[-1, 4)
Now, if the -1 was an open circle and the 4 had a solid dot, then it would be: (-1, 4].
Let’s see another chart that will help you translate between the verbal description, inequality notation, and interval notation.
Last issue is called a Union: If there is a separation in an interval, we use the union symbol, which is : 
For example, say a function decreased from -2 to 4 and then from 8 to 11. (-2, 4) (8, 11).
Let’s see how to write a union from the graph of an inequality in one variable.
x ≤ -4 or x > 12
On the number line there would be a solid dot on -4, because the symbol says, “less than or equal to.” In interval notation, the “or equal to,” portion is noted with a square bracket. However, this is an upper bound. For example, x = -817 satisfies the inequality because – 817 < -4. Anything smaller than -4 satisfies the inequality. There are infinitely many numbers less than -4, so we show the lower value as “negative infinity.” Since infinity is NOT a number, we cannot include it in the interval notation. We always will use parenthesis with infinity.
The other side of the compound inequality says that x is greater than 12. In this case x cannot equal 12 because 12 is not bigger than itself. (It’s ego is in check, it will never be a wide receiver in the NFL.) To show this with interval notation we used a parenthesis. Any number larger than 12 also satisfies the inequality. Here will use a positive infinity.
Now, consider x = 2. Does x = 2 satisfy the inequality?
It does not because it is not less than (or equal to) negative four, and it is also not larger than 12. No number between -4 and 12 is a solution. So, here is this compound inequality written in interval notation.
Summary: The curved parenthesis means that the value next to it is NOT included, but is a boundary. This is the same concept as a strict inequality, < or >, which is graphed with an open circle or dotted line. The square bracket is used to show that the value is included. This is the same concept as a non-strict inequality, like less than or equal to. When graphed, this is a solid dot.
Interval notation is written from smallest to largest.
The union symbol, which looks like a capital U, is used to show that there are missing values in an interval.
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