4.2: Interval Notation (2024)

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Inequalities slice and dice the real number line into segments of interest or intervals. An interval is a continuous, uninterrupted subset of real numbers. How can we notate intervals with simplicity? The table below introduces Interval Notation.

Inequality	Associated Circle	Associated Endpoint Closures
Either \(<\) or \(>\)		Left Parenthesis: ( or Right Parenthesis: )
Either \(≤\) or \(≥\)		Left Square bracket: [ or Right Square Bracket: ]

Inequalities have \(4\) possible interval closures:

\((A,B)\)	\((A,B]\)	\([A,B)\)	\([A,B]\)

The least number in the interval, \(A\), is always stated first. A comma is placed. The largest number in the interval, \(B\), is stated after the comma. The appropriate closure is considered for each value \(A\) and \(B\).

Four Examples of Interval Notation

\(−2 < x < 3\)	\(−2 < x ≤ 3\)	\(– 2 ≤ x < 3\)	\(– 2 ≤ x ≤ 3\)

\((−2, 3)\)	\((−2, 3]\)	\([−2, 3)\)	\([−2, 3]\)

The Infinities

There are two infinities: positive and negative. Each define a direction on the number line:

Infinity is not a real number. It indicates a direction. Therefore, when using interval notation, always enclose \(∞\) and \(−∞\) with parenthesis. We never enclose infinities with square bracket.

The table below shows four examples of interval notation that require the use of infinity.

\(x < 2\)	\(x ≤ 2\)	\(x > 2\)	\(x ≥ 2\)

\((−∞, 2)\)	\((−∞, 2]\)	\((2, ∞)\)	\([2, ∞)\)

Combinations of Intervals

If two or more intervals are interrupted with a gap in the number line, set notation is used to stitch the intervals together, symbolically. The symbol we use to combine intervals is the union symbol: \(∪\). The table below shows four examples:

Interval Notation	Graph
\((−∞, −2) ∪ [1, ∞)\)
\((−∞, −1) ∪ (−1, ∞)\)
\(\left(−\dfrac{3 \pi}{2} , −\dfrac{\pi}{2} \right) ∪ \left( \dfrac{\pi}{2}, \dfrac{3 \pi}{2} \right)\)
\(\left[−2 \pi, − \dfrac{\pi}{2} \right) ∪ \left[ \dfrac{3 \pi}{2} , ∞ \right)\)

Compound Inequalities

Intervals that have gaps, like the ones shown above, translate to compound inequalities. Real solutions belong in one interval or another. The word “or” plays a key role when translating. For example: the interval \((−∞, −2) ∪ [1, ∞)\) translates to its associated compound inequality:

\(x < -2\) or \(x ≥ 1\)

The word “and” cannot be used between the inequalities because a number cannot belong to both intervals at once. For example, \(x = 5\) is a solution because \(5\) belongs in the interval \(x ≥ 1\), but \(5\) does not belong in the interval \(x < −2\). Nevertheless, because of the word “or,” \(x = 5\) is a solution to the interval \((−∞, −2) ∪ [1, ∞)\).

Try It! (Exercises)

For exercises #1-6, state the inequality and the interval notation associated with the graph.

Graph			Inequality	Interval Notation

For exercises #7-10, state the interval notation and sketch the graph associated with the inequality.

Graph			Inequality	Interval Notation
			\(−3 ≤ x ≤ 1\)
			\(x < 4\)
			\(x ≥ −2\)
			\(0 ≤ x < 3\)

For exercises #11-17, sketch the graph associated with the given interval notation.

Graph			Interval Notation
			\((−∞, 4)\)
			\((−∞, −3) ∪ [0, ∞)\)
			\([−1, 1) ∪ [2, ∞)\)
			\((−∞, −5] ∪ (−1, 5)\)
			\(\left[−\dfrac{\pi}{2} , \dfrac{\pi}{2} \right]\)
			\((−∞, −\pi] ∪ [\pi, ∞)\)
			\(\left(−\dfrac{3\pi}{2} , −\dfrac{\pi}{2} \right) ∪ \left(-\dfrac{\pi}{2} , 0\right)\)

For #18-21

a. Sketch a graph of the compound inequality.

b. State the interval using interval notation.

\(x ≥ 4\) or \(x ≤ 0\)
\(x ≤ – 2\pi\) or \(x > \pi\)
\(−1 > x\) or \(2 ≤ x\)
\(x > 3\pi\) or \(x < – \pi\)

FAQs

How do I write in interval notation? ›

To write in interval notation, use parentheses for opened ends and square brackets for closed ends. The first and last values within the interval are then written within and separated by a comma.

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What is an interval of 4? ›

Summary

Number of half steps	Common Spelling	Example, from C
3	Minor Third (m3)	D sharp
4	Major Third (M3)	F flat
5	Perfect Fourth (P4)	E sharp
6	Tritone (TT)	F sharp or G flat

9 more rows

What is the interval notation for negative infinity? ›

The interval of all real numbers in interval notation is (-∞, ∞). All real numbers is the set of every single real number from negative infinity, denoted -∞, to positive infinity, denoted ∞. Therefore, the endpoints of this interval are -∞ and ∞.

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How to write range in interval notation? ›

Interval Notation

Use brackets, [], when the endpoints are included and parentheses, (), when the endpoints are excluded. Using the graph above, the range would be: R:(−1,2] Because −1 is not included we use a parentheses. 2 is included so we use a bracket.

What do () and [] mean in interval notation? ›

The notation may be a little confusing, but just remember that square brackets mean the end point is included, and round parentheses mean it's excluded. If both end points are included the interval is said to be closed, if they are both excluded it's said to be open.

How do you write the set in interval form? ›

Generally, an interval contains infinitely many points. Also, the given set of numbers can be written in the form of intervals and vice versa. Let's have a look at the examples given below. The set {x : x ∈ R, –4 < x ≤ 9}, written in set-builder form, can be written in the form of the interval as (–4, 9].

What is a perfect 4 interval? ›

A fourth is a musical interval encompassing four staff positions in the music notation of Western culture, and a perfect fourth () is the fourth spanning five semitones (half steps, or half tones).

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What is a 4 3 interval? ›

4/3 is the frequency ratio of the just perfect fourth, which is easily one of the more heavily discussed intervals outside of xenharmony- in fact, some of these usages have gone on to inspire other music theories within xenharmonic contexts, such as certain ideas about tetrachords.

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How do I calculate the interval? ›

To determine the class interval, the lower limit of the class is subtracted from the upper limit. The class interval formula is given as follows: Class interval = Upper Limit - Lower Limit.

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What is an example of a set of interval notation? ›

Examples of Interval Notation

Suppose we want to express the set of real numbers {x |-2 < x < 5} using an interval. This can be expressed as interval notation (-2, 5). The set of real numbers can be expressed as (-∞, ∞).

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What comes first in interval notation? ›

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.

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How to write in interval notation? ›

Intervals are written with rectangular brackets or parentheses, and two numbers delimited with a comma. The two numbers are called the endpoints of the interval. The number on the left denotes the least element or lower bound. The number on the right denotes the greatest element or upper bound.

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What do curly brackets mean in math? ›

In mathematics, curly brackets are used to denote a set. For example, a set containing the numbers 2, 4, 6, 8, and 10 would be written like this: {2, 4, 6, 8, 10}. The curly brackets show that the numbers inside belong together, just as a set of dishes belong together.

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What is an example of an interval set notation? ›

Interval notation is a way of writing subsets of the real number line . A closed interval is one that includes its endpoints: for example, the set { x | − 3 ≤ x ≤ 1 } . An open interval is one that does not include its endpoints, for example, { x | − 3 < x < 1 } .

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What is the symbol for interval notation? ›

Interval notation symbols

[] - brackets denote a closed interval. () - parenthesis denote an open interval. ∪ - union represents the joining together of two sets.

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How to write constant in interval notation? ›

A function f is constant on any interval if, for every (x₁) and (x₂), we have: f ( x 1 ) = f ( x 2 ) This tells us as we move right from x₁ to x₂, the y-values are constant.

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