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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
The interval is \((- finity, 2]\) and it includes 2, extending leftwards toward \(-\infty\).

Step by step solution

01

Understanding Set Notation

The interval \((-finity, 2]\) includes all real numbers less than or equal to 2. This is why we use a parenthesis \(()\) on \(-\infty\), indicating \(-\infty\) is not included, and a bracket \([]\) on 2, indicating that 2 is included.
02

Graphing the Interval

To graph \((-finity, 2]\) on a real number line: 1. Draw a number line.2. Locate the point at 2 on the number line and place a closed dot at this position to signify that 2 is included in the set. 3. Shade the region to the left of 2 toward the direction of \(-\infty\) to represent all numbers less than 2 extending infinitely.
03

Finalizing the Graph

Ensure your graph has: 1. A closed dot at 2, representing that 2 is part of the interval. 2. An arrow to the left (or the segment extending indefinitely to the left), illustrating inclusion of all numbers less than 2.

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Key Concepts

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

Intervals on the Real Line
Intervals on the real line are a fundamental concept in mathematics that help us describe a continuous range of numbers. An interval tells us which numbers are included in a set, often using set notation to clearly specify its bounds.
For example, the interval \((-\infty, 2]\) includes all real numbers that are less than or equal to 2, stretching all the way from negative infinity up to and including 2.
  • The parenthesis \((\) on \(-\infty\) indicates that \(-\infty\) is not part of the interval because infinity is not a finite number we can reach.
  • The bracket \([\) on 2 shows that 2 is included in this interval, meaning that 2 is a boundary point that is part of the set.
By using intervals, we can succinctly express sets of numbers without needing to list each one individually.
Graphing Intervals
Graphing intervals on a real number line is a visual way to represent the numbers that are part of an interval. This representation provides a quick and intuitive understanding of which numbers are included.
When graphing the interval \((-\infty, 2]\):
  • Start by drawing a horizontal line to represent the real number line.
  • Locate the number 2 on this line, then place a closed dot over it. A closed dot signifies that 2 is included in the interval.
  • Shade the line extending to the left of 2, illustrating that all numbers less than 2 are part of the interval. Use an arrow pointing to the left to show it extends indefinitely towards \(-\infty\).
This graphing method clarifies which numbers belong to the interval and helps visualize the set's scope.
Inequalities in Mathematics
Inequalities are expressions that indicate that two values are not equal and can be used to describe ranges of values on the real number line. They come in different forms such as less than \(<\), greater than \(>\), less than or equal to \(\leq\), and greater than or equal to \(\geq\).
In our example, \((-\infty, 2]\) can be expressed as an inequality where \(x \leq 2\). This is saying that \(x\) represents any number which can be 2 or any number less than 2.
  • Inequalities are useful in mathematical expressions and equations where finding just one specific number isn't enough, and a range of numbers is involved.
  • They help solve problems requiring constraints and can be further graphed, showing the range of possible solutions visually.
Understanding how to work with inequalities allows students to solve a variety of mathematical problems that require considering multiple conditions or constraints.

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