/*! 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 15 Express interval in set-builder ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 15

Express interval in set-builder notation and graph the interval on a number line. \((-3,0) \cap[-1,2]\)

Short Answer

Expert verified
The set-builder notation for the given interval can be expressed as \(-1 \leq x < 0\). The graphical representation of the interval on the number line will have a filled circle at -1 and an open circle at 0.

Step by step solution

01

Understand the notation

It is important to understand that the parenthesis such as (a,b) denotes an open interval which includes all real numbers from a to b but does not include the numbers a and b themselves, whereas the brackets [a,b] denote a closed interval which includes all real numbers from a to b including the numbers a and b.
02

Intersect the intervals

Since the task is to find the intersection (\(\cap\)) of the intervals (-3,0) and [-1,2], we must find the numbers that are common in both the intervals. In this case, the common numbers are all values from -1 to 0, but 0 is not included in the first interval (-3,0). So we have the intersection as [-1,0).
03

Convert to set-builder notation

To convert this to the set-builder notation, we express it as the set of all real numbers x such that x is greater than or equal to -1 and less than 0. In set-builder notation: \(-1 \leq x < 0\).
04

Draw on Number line

For the graphical representation, represent the interval [-1,0) on the number line. Start by marking -1 and 0 on the number line. Since -1 is included in the interval, represent it as filled circle (鈼�) and as 0 is not included, represent it as open circle (鈼�).

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 Notation
Interval notation is a shorthand used in mathematics to describe a set of numbers that fall within an interval. It uses parentheses and brackets to indicate whether endpoints are included or not.
  • Parentheses "(a, b)" denote an open interval, meaning that the endpoints "a" and "b" are not included.
  • Brackets "[a, b]" denote a closed interval, meaning that the endpoints "a" and "b" are included.

This notation simplifies the expression of complex sets and is widely used in calculus and algebra to convey information regarding solutions to inequalities and domains of functions.
Intersection of Intervals
The intersection of intervals involves finding the set of numbers that are common to two or more intervals. It is denoted by the symbol \(\cap\).
  • To find the intersection, look at the overlapping part of the number lines represented by each interval.
  • Only numbers that lie within both intervals are part of the intersection.

For example, the intersection of the intervals \((-3,0)\) and \([-1,2]\) is \([-1, 0)\), as this is where the two intervals overlap on the number line.
Number Line Representation
Number line representation is a visual method to depict intervals, making it easier to understand and interpret mathematical solutions.
  • Start by marking key points on the number line, such as endpoints of intervals.
  • Visual symbols such as open circles (鈼�) and closed circles (鈼�) are used to indicate whether endpoints are included or not.

For instance, the interval \([-1, 0)\) is illustrated on a number line with a filled circle at "-1" (included) and an open circle at "0" (not included). This approach simplifies the process of identifying overlaps and intersections between intervals.
Open and Closed Intervals
Open and closed intervals help us understand which portions of the interval are included in the set. This distinction is crucial in solutions involving inequalities.
  • Open intervals, indicated by parentheses, do not include their endpoints. For example, \((a, b)\) means that neither "a" nor "b" are part of the interval.
  • Closed intervals, indicated by brackets, include their endpoints. For example, \([a, b]\) means "a" and "b" are included in the interval.

This concept is especially important when solving inequalities, as it guides whether boundary values satisfy the conditions of the solution.

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

Solve compound inequality. \(-3 \leq x-2 \leq 1\)

Use the strategy for solving word problems, modeling the verbal conditions of the problem with a linear inequality. A city commission has proposed two tax bills. The first bill requires that a homeowner pay \(\$ 1800\) plus \(3 \%\) of the assessed home value in taxes. The second bill requires taxes of \(\$ 200\) plus \(8 \%\) of the assessed home value. What price range of home assessment would make the first bill a better deal?

Each group member should research one situation that provides two different pricing options. These can involve areas such as public transportation options (with or without discount passes), cellphone plans, long-distance telephone plans, or anything of interest. Be sure to bring in all the details for each option. At a second group meeting, select the two pricing situations that are most interesting and relevant. Using each situation, write a word problem about selecting the better of the two options. The word problem should be one that can be solved using a linear inequality. The group should turn in the two problems and their solutions.

What is a compound inequality and how is it solved?

Use the strategy for solving word problems, modeling the verbal conditions of the problem with a linear inequality. A truck can be rented from Basic Rental for \(\$ 50\) per day plus \(\$ 0.20\) per mile. Continental charges \(\$ 20\) per day plus \(\$ 0.50\) per mile to rent the same truck. How many miles must be driven in a day to make the rental cost for Basic Rental a better deal than Continental's?
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Decision Maths

Read Explanation

Discrete Mathematics

Read Explanation

Pure Maths

Read Explanation

Statistics

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.