Intervals
Main Concept
An interval is a set of real numbers that includes all real numbers between one endpoint, a, and another endpoint, b. If both a and b are included in the interval, it is known as a closed interval, and if neither is included it is an open interval. If an endpoint is ±∞, then the interval is unbounded, otherwise, it is bounded.
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Standard Interval Notation
The standard interval notation is to write the endpoints of the interval separated by a comma, using round brackets to signify that the endpoint is not included, and square brackets to signify that it is. If an endpoint is ±∞, round brackets are used, since ∞ is not a real number, and cannot be included in an interval.
Note: Since this is interval notation, and not an ordered pair, the brackets do not need to match: one endpoint may be included while the other is excluded. This is represented by closed and open brackets, respectively.
Inequalities
Intervals can also be represented by inequalities. The exclusion of an endpoint is represented by a strict inequality, > or <, and the inclusion of an endpoint is represented by ≥ or ≤. When the interval is unbounded, the variable is restricted by only one or no inequalities. When the interval is bounded, the variable is restricted by two inequalities: one above, and one below.
Graphical Representation
When representing an interval graphically, a closed or solid point represents that the point is included, while an open point means that it is not included. This is demonstrated in the example below.
Examples
The closed interval from 3 to 4
The open interval from -1 to ∞
The interval from -4 to -2, including -2, but excluding -4
3,4
−1,∞
−4,−2
Inequality
3≤x≤4
−1<x
−4<x≤−2
Click twice to place two endpoints or drag to place an interval on the number line. Choose whether or not to include the two endpoints.
Interval:
Inequality:
Left End Points:
Right End Point:
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