Interval notations are used to express the set of inequalities. There are 3 types of interval notation – open interval, closed interval, and half-open interval. An interval with no infinity symbol is called a bounded interval. An interval containing the infinity symbol is called an unbounded interval. We get intervals in movies. It indicates that half of the movie is over. We also use intervals while referring to time Ex: The task was finished at a particular interval of 2 hours. Similarly, in Maths we use intervals to bound the numbers. Ex: All the numbers between 2 to 8 are an interval.
Interval notation is used to represent subsets of the real numbers. Example: The set of numbers x satisfying 1 ≤ x ≤ 6 is an interval that contains 1, 6, and all numbers between 1 and 6.
Types of Intervals:
Based on the endpoints, intervals are broadly classified into 3 groups. They are
Closed interval: These intervals include endpoints, and are denoted with square brackets. The square bracket symbols are used to describe sets with a “less than or equal to” or a “greater than or equal to” element, respectively. Ex: The set {x | 4 ≤ x ≤ 9} include the endpoints, -4 and 9. This is expressed using closed interval notation: [4, 9].
Open interval: In this type of interval both the endpoints are excluded from the interval. These are denoted by common brackets. The parentheses are used to describe sets with a “less than” or “greater than” element, respectively. Ex: The set {x | 4< x < 9} does not include the endpoints, 4 and 9. This is expressed using open interval notation: (4, 9).
Half-open interval: In this interval, one of the endpoints of inequality is excluded and the other one is included. Combinations of square and common brackets are used to describe these sets. Ex: The set {x | 4 ≤ x < 9} include the endpoint 4. This is expressed using half-open interval notation: [4,9)
-∞ and +∞ are used to show that an interval is unbounded or extends indefinitely to the left or to the right respectively.
Number Line of Intervals
Interval notations can be represented on a number line.
An open interval notation can be represented by showing a continuation of the line on both the ends. Ex: The inequality A< x < B can be represented as follows.
The interval notation = (A,B)
| A |
| B |
A closed interval notation can be represented by showing the end of the line on both ends. Ex: The inequality A x B can be represented as follows. The interval notation = [A,B]
| A |
| B |
A half-open interval notation can be represented by showing a continuation of line on one end and the end of the line on the other end. Ex: The inequality x A can be represented as follows.
The interval notation = (-∞, A]
| A |
| B |
Solved Examples
- Using interval notation indicates all the real numbers from -2 to 5.
Solution: Interval notation for inequality -2 x 5 can be written as [-2,5]. Because from -2 means -2 is the starting point and it is up to 5, that means 5 is included.
- Using interval notation indicates all the real numbers less than or equal to -2.
Solution: Interval notation for inequality x -2 can be written as (-∞, -2]. Because all the real numbers less than or equal to -2 are included.
Similarly, you can try writing interval notations for various conditions. Recently the development of online math classes makes it easier for all the students to clarify their doubts quickly. I recommend Cuemath for the best math classes since they adopt active learning techniques. For more details about math classes log on to the cuemath website.
