Interval Notation
- What is Interval Notation?
- Visual Representation of Interval Notation
- Types of Intervals
- Types of Parentheses Used for Different Types of Interval
- Representation of Intervals on Number Line
- Solved Examples
- Practice Questions
- Frequently Asked Questions
What is Interval Notation?
Interval notation is a technique used to illustrate an interval of numbers on a number line. Interval is referred to as a group of numbers lying between two certain numbers. For example: `3 ≤ x ≤ 8` refers to a group of numbers `x` that are greater than or equal to `3` and less than or equal to `8`.
Visual Representation of Interval Notation
To represent a set of real numbers that are greater than `-1` and less than `10` can be represented on a number line as shown below:
Types of Intervals
Based on the nature of endpoints on the number line, intervals are classified into three different types as given below:
- Open Interval: In this type of interval the endpoints i.e. the numbers lying on either side of inequality are not considered to be in the range. For example: Interval `13 < x < 17` consists of all real numbers between `13` and `17`, however, it does not include the endpoints which are the numbers `13` and `17`.
Open interval is represented with round parentheses like `(13, 17)`.
Hollow circles on the number line indicate the open interval as shown below:
- Closed Interval: In this type of interval both the endpoints are considered to be in the range. For example: `3 ≤ x ≤ 8` consists of all real numbers between `3` and `8` including endpoints which are the numbers `3` and `8`.
Closed interval is represented by square parentheses like `[3, 8]`.
Filled circles on the number line indicate the closed interval as shown below:
- Half-Open Interval: This type of interval includes only one of the endpoints on either side of the inequality. This interval is represented by both round parentheses and square parentheses. For example: `5 < x ≤ 10` consists of all real numbers between `5` and `10` including `10` but not `5`.
Half-open interval is represented by both types of parentheses (round and square) like `(5, 10]`.
A hollow circle at `5` and a filled circle at `10` are used to indicate a half-open interval on the number line as shown below:
Types of Parentheses Used for Different Types of Intervals
There are different types of parentheses used for different types of intervals:
- [ ]: A pair of square brackets is used to include both the endpoints of the inequality.
- ( ): A pair of round brackets is used to exclude both the endpoints of the inequality.
- ( ]: One round parenthesis on the left side means that the left endpoint of the inequality is excluded but the right endpoint is included.
- [ ): One round parenthesis on the right side means that the left endpoint of the inequality is included but the right endpoint is excluded.
Representation of Intervals on Number Line
Solved Examples
Example `1`: Draw a number line for the inequality for `-2 ≤ x ≤ 5`.
Solution:
The inequality is based on the closed interval. So, the number line will include both the endpoints which are `-2` and `5`. The number line will be drawn as shown below:
Example `2`: Write the inequality for the number line shown below:
Solution:
The interval shown on the number line has a filled circle on `20` and a hollow circle on `29` which means that `20` should be included in the interval and `29` should not be included in the interval.
So, the inequality can be represented as `20 ≤ x < 29`.
Practice Questions
Q`1`. Express the inequality \(-2 < x \leq 4\) using interval notation.
- \( (-2, 4) \)
- \( (-2, 4] \)
- \( [-2, 4] \)
- \( [-2, 4) \)
Answer: b
Q`2`. Express the line graph in the interval notation.
- \( (-5, 2] \)
- \( (-5, 2) \)
- \( [-5, 2] \)
- \( [-5, 2) \)
Answer: d
Q`3`. Represent the following statement in interval notation:
Rocky went to the market to buy a dozen bananas. His mother suggested that the price for the same should be not more than `$6` and not less than `$4`.
- \( (4, 6] \)
- \( (4, 6) \)
- \( [4, 6] \)
- \( [4, 6) \)
Answer: c
Q`4`. What is the inequality for the interval notation \([-1, 6)\)?
- \(-1 < x \leq 6\)
- \(-1 \leq x \leq 6\)
- \(-1 < x < 6\)
- \(-1 \leq x < 6\)
Answer: d
Q`5`. Express the line graph in interval notation.
- \( [-2, \infty) \)
- \( (-2, \infty) \)
- \( (-2, \infty] \)
- \( (-2, \infty] \)
Answer: a
Frequently Asked Questions
Q`1`. How do you convert a compound inequality into interval notation?
Answer: To convert a compound inequality into interval notation, find the common solution range. For example, \(1 < x \leq 5\) can be expressed as the interval \((1, 5]\).
Q`2`. Why is the use of parentheses and brackets important in interval notation?
Answer: Parentheses \((\,)\) and brackets \([\,]\) in interval notation convey whether the endpoints are included or excluded. Parentheses indicate exclusion, and brackets indicate inclusion. For instance, \((a, b]\) includes \(a\) and excludes \(b\), while \([a, b]\) includes both \(a\) and \(b\).
Q`3`. Is it appropriate to use square brackets to denote infinity in interval notation?
Answer: No, square brackets are not used to represent infinity in interval notation. Infinity is considered an "open" endpoint, indicating that the interval continues indefinitely. Instead, a round parenthesis \((\infty)\) or \((-\infty)\) is used to denote infinity in interval notation.
For example:
- \((a, \infty)\) represents all real numbers greater than \(a\) but not including \(a\).
- \((-\infty, b]\) represents all real numbers less than or equal to \(b\), including \(b\).
Q`4`. What is the difference between open and close intervals?
Answer: Open intervals: (\(a, b)\) exclude both endpoints \(a\) and \(b\)
Closed intervals: \([a, b]\) include both endpoints \(a\) and \(b\).