/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 3 Write each interval in set notat... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 3

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

Short Answer

Expert verified
Interval: \(\{x \in \mathbb{R} \mid x \leq 2\}\); Graph: Shaded from 2 to left, including 2.

Step by step solution

01

Understanding the Interval

The interval \((-fty, 2]\) denotes a set of real numbers. This particular interval includes all numbers from negative infinity up to and including the number 2. It is important to note that it does not include a specific endpoint for the negative side because infinity is conceptual, and thus open.
02

Writing in Set Notation

In set notation, intervals are expressed in terms of conditions that the elements must satisfy. The interval \((-fty, 2]\) in set notation is expressed as \(\{x \in \mathbb{R} \mid x \leq 2\}\). This reads as the set of all real numbers \(x\) such that \(x\) is less than or equal to 2.
03

Graphing the Interval on a Number Line

To graph this interval, draw a number line. Mark the point 2 on the number line with a filled dot to indicate that 2 is included in the interval. Shade the line extending from 2 towards the left, meaning it includes everything less than 2 but extends indefinitely without stopping on the left. Use an arrow to show this indefinite extension towards negative infinity.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Interval Graphing
Interval graphing helps us visually represent ranges of numbers on a number line. This concept is particularly useful in identifying which numbers belong in a particular set or solution. To effectively graph an interval:
  • Identify the endpoints of the interval. For example, in (- 鈭�, 2], the endpoint is 2.
  • Decide if the endpoints are included or not. An endpoint included in the interval is marked by a filled dot.
  • If an endpoint is not included, often noted with a parenthesis, use an open circle.
  • Shade the appropriate section of the number line to signify all numbers in that direction are included.

For (- 鈭�, 2] on a graph, place a filled dot at 2 and shade everything extending leftward. Use an arrow pointing towards negative infinity to indicate that the interval continues indefinitely. This representation is straightforward once you understand these visual markers.
Real Numbers
Real numbers include all the rational and irrational numbers that can be found on the number line. The concept of real numbers combines both subsets:
  • Rational numbers: These numbers can be expressed as the fraction of two integers, such as 1/2, -4, or 0.75.
  • Irrational numbers: Numbers that cannot be perfectly expressed as fractions. Examples include 蟺 and 鈭�2.

Real numbers encompass all these, ranging from negative to positive infinity. When dealing with intervals, the flexibility of real numbers allows any portion of this continuity to be isolated and focused upon. It's essential to appreciate this diversity, as it helps in understanding the wide scope covered when discussing real numbers in mathematical contexts.
Inequalities in Mathematics
Inequalities are mathematical expressions that show the relative size or order of two values. They can tell us how one number compares to another. Key inequality symbols include:
  • < (less than)
  • > (greater than)
  • 鈮� (less than or equal to)
  • 鈮� (greater than or equal to)

In the interval (- 鈭�, 2], the inequality x 鈮� 2 indicates that any x value is less than or equal to 2. This translates into a set of values which graphically appears as a shaded region on a number line. Inequalities are crucial because they allow us to describe not just singular values, but entire sets of potential solutions in mathematics.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

For the quadratic function \(f(x)=a x^{2}+b x+c\), what condition on one of the coefficients will guarantee that the function has a highest value? A lowest value?

BEHAVIORAL SCIENCES: Smoking and Educatior According to a study, the probability that a smoker will quit smoking increases with the smoker's educational level. The probability (expressed as a percent) that a smoker with \(x\) years of education will quit is approximately \(y=0.831 x^{2}-18.1 x+137.3\) (for \(10 \leq x \leq 16\) ). a. Graph this curve on the window \([10,16]\) by \([0,100]\). b. Find the probability that a high school graduate smoker \((x=12)\) will quit. c. Find the probability that a college graduate smoker \((x=16)\) will quit.

Which of the following is not a polynomial, and why? $$ x^{2}+\sqrt{2} \quad x^{\sqrt{2}}+1 \quad \sqrt{2} x^{2}+1 $$

The following function expresses an income tax that is \(10 \%\) for incomes below \(\$ 5000\), and otherwise is \(\$ 500\) plus \(30 \%\) of income in excess of \(\$ 5000\). \(f(x)=\left\\{\begin{array}{ll}0.10 x & \text { if } 0 \leq x<5000 \\\ 500+0.30(x-5000) & \text { if } x \geq 5000\end{array}\right.\) a. Calculate the tax on an income of \(\$ 3000\). b. Calculate the tax on an income of \(\$ 5000\). c. Calculate the tax on an income of \(\$ 10,000\). d. Graph the function.

SOCIAL SCIENCE: Age at First Marriage Americans are marrying later and later. Based on data for the years 2000 to 2007 , the median age at first marriage for men is \(y_{1}=0.12 x+26.8\), and for women it is \(y_{2}=0.12 x+25\), where \(x\) is the number of years since 2000 . a. Graph these lines on the window \([0,30]\) by \([0,35] .\) b. Use these lines to predict the median marriage ages for men and women in the year 2020 . [Hint: Which \(x\) -value corresponds to 2020 ? Then use TRACE, EVALUATE, or TABLE.] c. Predict the median marriage ages for men and women in the year 2030 .
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Calculus

Read Explanation

Applied Mathematics

Read Explanation

Geometry

Read Explanation

Probability and Statistics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.