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# Positive and Negative Intervals

In mathematics, an interval is a set of real numbers between two endpoints. Intervals can be classified as either positive or negative depending on the sign of the numbers in the interval.

• Positive and Negative Intervals
• Formula to determine the sign of an interval
• Solved Examples for Positive Intervals
• Solved Examples for Negative Intervals
• FAQs

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## Positive and Negative Intervals

A positive interval is an interval that contains only positive numbers. For example, the interval (0, 5) is a positive interval because all the numbers between 0 and 5 are positive.

A negative interval is an interval that contains only negative numbers. For example, the interval (-10, -5) is a negative interval because all the numbers between -10 and -5 are negative.

It’s also possible to have intervals that contain both positive and negative numbers. For example, the interval (-3, 3) contains both positive and negative numbers.

To determine the sign of an interval, you can look at the signs of the endpoints. If both endpoints are positive, then the interval is positive. If both endpoints are negative, then the interval is negative. If one endpoint is positive and the other is negative, then the interval contains both positive and negative numbers.

Intervals are commonly used in algebra and calculus to represent ranges of values for functions and to describe the behavior of functions over certain ranges.

## Formula to determine the sign of an interval

The formula to determine the sign of an interval is as follows:

If both endpoints of the interval are positive, then the interval is positive. If both endpoints are negative, then the interval is negative. If one endpoint is positive and the other is negative, then the interval contains both positive and negative numbers.

In other words, if we have an interval of the form (a, b), where a and b are real numbers, we can determine its sign as follows:

If a > 0 and b > 0, then the interval is positive.
If a < 0 and b < 0, then the interval is negative.
If a < 0 and b > 0, then the interval contains both positive and negative numbers.
For example, the interval (-3, 5) contains both negative and positive numbers, so it is neither a positive nor a negative interval. On the other hand, the interval (-1, -0.5) contains only negative numbers, so it is a negative interval. Similarly, the interval (2, 8) contains only positive numbers, so it is a positive interval.

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## Solved examples for Positive Intervals

(0, 10) – This interval contains all positive numbers between 0 and 10, but does not include 0 or 10.

(1, ∞) – This interval contains all positive numbers greater than 1. The symbol ∞ represents infinity, so the interval continues indefinitely in the positive direction.

(2, 5) – This interval contains all positive numbers between 2 and 5, but does not include 2 or 5.

(0.1, 1) – This interval contains all positive numbers between 0.1 and 1, but does not include 0.1 or 1.

To represent these intervals on a number line, we would draw a line with a tick mark at 0, a tick mark at 10 (for example), and tick marks at any other endpoints, and then shade in the portion of the line that corresponds to the interval.

For example, to represent the interval (0, 10) on a number line, we would draw a line with a tick mark at 0 and a tick mark at 10, and then shade in the portion of the line between these two points.

## Solved examples for Negative Intervals

(-10, 0) – This interval contains all negative numbers between -10 and 0, but does not include -10 or 0.

(-∞, -1) – This interval contains all negative numbers less than -1. The symbol ∞ represents infinity, so the interval continues indefinitely in the negative direction.

(-5, -2) – This interval contains all negative numbers between -5 and -2, but does not include -5 or -2.

(-1, -0.1) – This interval contains all negative numbers between -1 and -0.1, but does not include -1 or -0.1.

To represent these intervals on a number line, we would draw a line with a tick mark at 0, a tick mark at -10 (for example), and tick marks at any other endpoints, and then shade in the portion of the line that corresponds to the interval.

For example, to represent the interval (-10, 0) on a number line, we would draw a line with a tick mark at -10 and a tick mark at 0, and then shade in the portion of the line between these two points.

## Positive and Negative Intervals FAQS

##### What does it mean for an interval to be open or closed?

An interval is considered open if it does not include its endpoints, and closed if it does include its endpoints. For example, the interval (0, 1) is open because it includes all numbers between 0 and 1, but does not include 0 or 1. The interval [0, 1] is closed because it includes all numbers between 0 and 1, including 0 and 1.

##### Can an interval be both positive and negative?

No, an interval cannot be both positive and negative. If an interval contains both positive and negative numbers, it is neither positive nor negative.

##### How do I determine if a number is within a given interval?

To determine if a number x is within a given interval, you need to check if x is between the endpoints of the interval. If the interval is (a, b), then x is in the interval if a < x < b.

##### What is an infinite interval?

An infinite interval is an interval that extends indefinitely in one or both directions. For example, the interval (-∞, ∞) contains all real numbers.

##### How do I represent an interval using interval notation?

To represent an interval using interval notation, we use parentheses to indicate that the endpoints are not included, and square brackets to indicate that the endpoints are included. For example, the interval (2, 5) contains all numbers between 2 and 5, but does not include 2 or 5. The interval [2, 5] contains all numbers between 2 and 5, including 2 and 5.

Gloria Mathew writes on math topics for K-12. A trained writer and communicator, she makes math accessible and understandable to students at all levels. Her ability to explain complex math concepts with easy to understand examples helps students master math. LinkedIn

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