An interval is an unbroken range of numbers. An interval may have zero, one or two endpoints. The interval may or may not include the endpoints. An interval that includes its endpoints is called a closed interval. The interval -1 ≤ x ≤ 3 is a closed interval. An interval that does not include its endpoints, or that does not have endpoints, is called an open interval. The interval 1 < r < 3 is an open interval. If one endpoint of an interval is included, and the other endpoint is not or the interval has only one endpoint, the interval is called a half-open interval. The intervals -10 < t ≤ -5 and 2 ≤ g < 6.2 are half-open intervals.
If an interval includes an endpoint, the interval is said to be inclusive of the endpoint. If an interval does not include an endpoint, the interval is said to be exclusive of the endpoint. The interval 1 < a ≤ 9 is exclusive of 1 and inclusive of 9.
Intervals can be represented in several different ways:
Number Line | ||||
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Inequality | Interval Notation | Set Notation | ISO 31-11 Notation | Description |
3 < x < 7 | (3, 7) | {x ∈ ℝ | 3<x<7 } | ]3, 7[ | This is an open interval between 3 and 7. The interval is exclusive of 3 and 7. |
-3 ≤ x < 4 | [-3, 4) | {x ∈ ℝ | -3 ≤ x < 4} | [-3, 4[ | This is a half-open interval between -3 and 4. The interval includes -3 and excludes 4. |
-5 < x ≤ -1 | (-5, -1] | {x ∈ ℝ | -5 < x ≤ -1 } | ]-5, -1] | This is a half-open interval between -5 and -1. The interval is exclusive of -5 and inclusive of -1. |
2 ≤ x ≤ 8 | [2, 8] | {x ∈ ℝ | 2 ≤ x ≤ 8 } | [2, 8] | This is a closed interval between 2 and 8. The interval includes 2 and 8. |
-1 < x | (-1, ∞) | {x ∈ ℝ | -1 < x } | ]-1, ∞[ | This is an open interval from -1 to positive infinity. The interval does not include -1. |
x < 3 | (-∞, 3] | {x ∈ ℝ | x < 3 } | ]-∞, 3] | This is a half-open interval from negative infinity to 3. The interval includes 3. |
not applicable | (-∞, ∞) | {x | x ∈ ℝ } | ]-∞, ∞[ | This is the open interval of all real numbers. |
Table 1: Representing intervals |
Click on the blue points and drag them to change the figure. Click on the check boxes to change the figure. Which interval notation do you prefer? Why? |
Manipulative 2 - Interval Notation Created with GeoGebra. |
# | A | B | C | D |
E | F | G | H | I |
J | K | L | M | N |
O | P | Q | R | S |
T | U | V | W | X |
Y | Z |
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