Problem 4 - Intermediate Algebra

Question

Fill in the blanks. \((2,8)\) is an example of an open ______ \(,[-4,0]\) is an example of a ______ interval, and (0, 9] is an example of a half-______ interval.

Step-by-Step Solution

Answer

Open; Closed; Open (or closed)

Step 1: Identify Interval Components

First, we need to understand the differences between 'open', 'closed', and 'half-open' (or 'half-closed') intervals. An open interval does not include its endpoints, a closed interval includes both endpoints, and a half-open interval includes one endpoint but not the other.

Step 2: Classify the First Interval

The interval \((2,8)\) does not include the endpoints 2 and 8, as it uses parentheses. Therefore, it is an open interval.

Step 3: Classify the Second Interval

The interval \([-4,0]\) includes the endpoints -4 and 0, as it uses square brackets. Therefore, it is a closed interval.

Step 4: Classify the Third Interval

The interval \((0, 9]\) includes the endpoint 9 (indicated by the square bracket) but not 0 (indicated by the parenthesis). Therefore, it is a half-open or half-closed interval.

Key Concepts

Open interval

An open interval in mathematics is a set of numbers between two endpoints. However, importantly, in an open interval, neither endpoint is included as part of the set. For example, the interval \((2, 8)\) means that we consider numbers that are strictly greater than 2 and strictly less than 8.
The notation uses parentheses, like in \((a, b)\), indicating that the endpoints are not part of the interval.
Open intervals are useful when there is a need to describe values that are bounded on both ends but do not include the specific endpoints.

• They are often used in inequalities, such as describing values greater than or less than a certain number.
• In graphical representations, the endpoints of an open interval are depicted with open circles.

Closed interval

A closed interval is distinctly different from an open interval because both of its endpoints are included in the set of numbers. In mathematical notation, closed intervals are written using square brackets, such as \([-4, 0]\).
This indicates that the numbers -4 and 0 are included in the interval.
Closed intervals are especially useful when endpoints are significant in a certain context, necessitating their inclusion in a set.

• They commonly appear in discussions around continuous ranges, like when defining domains in calculus.
• In visual graphs, closed intervals are depicted with solid circles at the endpoints to show their inclusion.

Half-open interval

A half-open interval is an interesting mix between the open and closed intervals. It includes one endpoint but excludes the other. There are two variations for representing half-open intervals:

• The first one, such as \( (0, 9] \), includes the endpoint 9 but not 0.
• The second type, such as \( [-4, 3) \), would include -4 but not 3.

In terms of notation, the endpoint you include uses a square bracket, while the one you exclude uses a parenthesis.
Half-open intervals are useful in problems where starting or ending values are crucial to include, while others are not.

• These intervals are excellent for representing datasets where one side of the data is restricted while the other needs flexibility.
• For visual representation, they combine solid and open circles, corresponding to the bracket and parenthesis used.