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Chapter 1: Problem 2

Write each interval in set notation and graph it on the real line. $$ (-3,5] $$

Short Answer

Expert verified
The interval in set notation is \( \{ x \in \mathbb{R} \, | \, -3 < x \leq 5 \} \) and it is graphed with an open circle at \(-3\) and a filled circle at \(5\).

Step by step solution

01

Identify Interval Type

The interval given is (-3, 5]. It is a mixed-type interval which is open on the left at \(-3\) and closed on the right at \(5\). This means \(-3\) is not included in the interval, while \(5\) is included.
02

Write Interval in Set Notation

In set notation, we express the interval as \( \{ x \in \mathbb{R} \, | \, -3 < x \leq 5 \} \). This states that \(x\) is a real number such that \(x\) is greater than \(-3\) and less than or equal to \(5\).
03

Graph the Interval on the Real Number Line

To graph the interval, draw a number line. Place open parentheses (an open circle) at \(-3\) to denote that it is not included, and place a bracket (a filled circle) at \(5\) to denote that it is included. Shade the number line between these two points to represent all numbers \(x\) that satisfy \(-3 < x \leq 5\).

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Key Concepts

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

Mixed-Type Interval
An interval is considered mixed-type when it is neither fully open nor fully closed. In the interval \((-3, 5]\), there are two distinct ends:
  • The open end at \(-3\)
  • The closed end at \(5\)
This means that \(-3\) is not part of the interval, and any number greater than \(-3\) may be included. When we talk about mixed-type intervals, we are focusing on the different ways numbers can be part of or excluded from a set.
This concept is crucial because it showcases flexibility in defining sets, allowing for a wide range of solutions to mathematical problems.
Real Number Line
The real number line is a visual representation of all possible real numbers. Real numbers encompass
  • Integers
  • Fractions
  • Decimals
  • Irrationals
It extends infinitely in both directions, forming a straight line.
In mathematics, the real number line is used frequently because it helps to visualize intervals and solutions to equations. When graphing intervals like \((-3, 5]\), positions on this line illustrate which numbers are included or excluded as part of the interval by visually showing spaces or points along the line.
Understanding how the real number line works helps to comprehend more complicated mathematical concepts that build upon these foundational ideas.
Open and Closed Intervals
Intervals in math are ranges of numbers that have clear beginning and ending points. These intervals can be open, closed, or a mix of both:
  • Open interval: Neither endpoint is included. Notated with parentheses: \((a, b)\)
  • Closed interval: Both endpoints are included, wrapped with brackets: \([a, b]\)
  • Mixed interval: Combines one open and one closed end, using both a parenthesis and a bracket: \((a, b]\) or \([a, b)\)
For our specific case, \((-3, 5]\), describes all numbers greater than \(-3\) and up to, including the number \(5\). Understanding these intervals helps in solving and plotting solutions on the real number line efficiently.

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Most popular questions from this chapter

Find the slope (if it is defined) of the line determined by each pair of points. \((3,-1)\) and \((5,7)\)

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