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Chapter 1: Problem 3

Write each interval in set notation and graph it on the real line. $$ (-\infty, 2] $$

Short Answer

Expert verified
Set notation: \(\{ x \mid x \leq 2 \}\). Graph: Line extending left from 2 with closed circle on 2.

Step by step solution

01

Understand Interval Notation

The interval \((-\infty, 2]\) represents all real numbers that are less than or equal to 2. The parenthesis \((-\infty\) means that the interval does not include negative infinity, and the bracket \([2]\) indicates that the interval includes the number 2.
02

Convert to Set Notation

To express the interval in set notation, we identify the set of real numbers satisfying the condition: \(\{ x \mid x \leq 2 \}\). This denotes the set of all numbers \(x\) such that \(x\) is less than or equal to 2.
03

Graph the Interval on the Real Line

To graph \((-\infty, 2]\) on the real line, draw a line extending to the left from 2 towards negative infinity. Use an open dot on the side of negative infinity (since it is not included) and a closed dot on 2 (since it is included).

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Key Concepts

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

Set Notation
Set notation is a way to describe a collection of elements, in this case, numbers, within certain constraints or conditions. It's often used to express all potential solutions or members that satisfy a specific property. For example, in our exercise, the set notation \( \{ x \mid x \leq 2 \} \) includes all numbers \( x \) where \( x \) is less than or equal to 2. The bar \( \mid \) stands for "such that," and it separates the variable from its condition.

When using set notation, keep in mind:
  • It provides a concise and precise way to specify sets of numbers.
  • The expression inside braces defines the rule or property that members of the set must satisfy.
  • It can handle complex mathematical expressions and inequalities.
Understanding set notation is crucial as it forms the foundation for more advanced mathematical concepts you'll encounter later.
Real Line Graph
A real line graph is a visual representation of an interval on a number line, which helps us see all the numbers included in the interval. In our exercise, we need to graph the interval \((-\infty, 2]\) on the real line. To do this:
  • Draw a number line: The number line should include the key number mentioned in the interval (in this case, 2).
  • Use dots: Place a closed dot at 2 to indicate that 2 is included in the set. Any included endpoint uses a closed dot.
  • Extend the line: Draw an arrow extending from 2 towards negative infinity, using an open line to show numbers less than 2 are included.
  • No dot for infinity: Infinity is never reached or included, so it never has a corresponding dot.
This graph helps visualize which numbers are part of the set defined by the interval.
Real Numbers
Real numbers are all the numbers on the continuous number line, making up all rational and irrational numbers. They include integers, whole numbers, fractions, and numbers that can have decimal representations. It's essential to understand this entire range to grasp how intervals include these values.

In our exercise, the interval \((-\infty, 2]\) involves real numbers, specifically those less than or equal to 2. These numbers can be both positive and negative, as well as fractions or decimals. Some key points about real numbers include:
  • They form the basis for intervals, set notation, and number line graphs.
  • They help us describe a complete, unbroken line of possible values.
  • Real numbers can be found everywhere on the number line, even between whole numbers.
Having a solid understanding of real numbers will enhance your capability to work with mathematical concepts that involve continuous values.

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