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Chapter 1: Problem 4

Write each interval in set notation and graph it on the real line. \([7, \infty)\)

Short Answer

Expert verified
The interval \([7, \infty)\) in set notation is \(\{ x \mid x \geq 7 \}\); graph it by marking 7 with a solid dot and extending a line to the right.

Step by step solution

01

Understanding Interval Notation

The given interval is \([7, \infty)\). This represents all real numbers greater than or equal to 7. The square bracket \([\) indicates that 7 is included in the set, while the parenthesis \()\) indicates that infinity is not a number, but instead a concept, and thus cannot be included.
02

Expressing in Set Notation

Set notation expresses the same interval using a different format. In set notation, the interval \([7, \infty)\) can be written as \(\{ x \mid x \geq 7 \}\). This reads as 'the set of all \(x\) such that \(x\) is greater than or equal to 7.'
03

Graphing on the Real Line

To graph \([7, \infty)\), start by marking the number 7 on the real line with a solid dot, indicating that 7 is included in the interval. Then, draw a continuous line or arrow extending to the right to illustrate all numbers greater than 7 without an endpoint, representing infinity.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Set Notation
Set notation is a mathematical way of representing a collection of items or numbers, often relating them to a specific condition or rule. It offers a precise and concise method to describe sets and intervals:
  • Elements in the set are typically described by conditions they satisfy.
  • For the interval \([7, \infty)\), the set notation is written as \(\{ x \mid x \geq 7 \}\).
  • The vertical bar \(\mid\) reads as 'such that', indicating the condition that members of the set fulfill.
In \(\{ x \mid x \geq 7 \}\), it denotes the set of all real numbers \(x\) where \(x\) is greater than or equal to 7.
Set notation is particularly useful for efficiently conveying mathematical concepts in the realm of inequalities.
Graphing on the Real Number Line
The real number line is a visual tool used to represent all numbers, including negative, positive, and zero. It's a straight line with numbers placed sequentially and utilized for illustrating intervals:
  • To graph the interval \([7, \infty)\), begin by locating the number 7 on the line.
  • A solid dot is placed at 7, signifying that 7 is part of the interval.
  • An arrow is drawn to the right from 7, indicating numbers greater than 7 extending towards infinity.
This graph helps visualize the concept of infinity, suggesting that the numbers continue indefinitely.
Graphing on the real number line simplifies the understanding of concepts related to intervals and inequalities.
Exploring Inequalities
Inequalities are mathematical expressions that compare different values, frequently indicating that one quantity is larger or smaller than another. They are fundamental in discussing intervals and set notation:
  • The symbol \(\geq\) used in \(x \geq 7\) shows that \(x\) is greater than or equal to 7.
  • Inequalities are represented graphically on the real number line by using open or closed dots (also known as unfilled or filled circles), depending on whether a boundary value is included.
  • A closed dot as in \([7, \infty)\) is used for \(\geq\) and \(\leq\) to indicate the boundary number is part of the set, whereas an open dot is for excluding the boundary number.
Understanding inequalities involves recognizing how they describe the relationships between numbers and provide a foundation for solving mathematical problems involving limits and ranges.

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