/*! 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 16 Write the given interval as an i... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 16

Write the given interval as an inequality. $$ [1,15) $$

Short Answer

Expert verified
The inequality is \(1 \leq x < 15\).

Step by step solution

01

Understand the Interval Notation

The interval \([1, 15)\) is presented in interval notation. The square bracket \([1\) means that the interval includes the number 1, while the parenthesis \(15)\) means that the interval does not include the number 15. It represents all numbers between 1 and 15, including 1 but not including 15.
02

Convert to Inequality

To convert \([1, 15)\) into an inequality, use the inclusive sign \(\geq\) for the lower bound and the exclusive sign \(<\) for the upper bound. The inequality representation is: \(1 \leq x < 15\). This means that \(x\) is greater than or equal to 1, and \(x\) is less than 15.

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.

Understanding Inequalities
Inequalities are mathematical expressions used to compare two values. Instead of stating that they are equal, inequalities express that one value is less than, greater than, less than or equal to, or greater than or equal to another value. This concept captures a range of possible solutions rather than a single answer.
This can be represented as follows:
  • \( x > 3 \): This means \( x \) is greater than 3.
  • \( x < 7 \): This means \( x \) is less than 7.
  • \( x \geq 5 \): This means \( x \) is greater than or equal to 5.
  • \( x \leq 9 \): This means \( x \) is less than or equal to 9.
Through inequalities, we often describe a set of numbers that satisfy certain conditions. When you see equations with these symbols, remember that it's about comparison, not just equal value.
Working with Intervals
Intervals are used in mathematics to represent a range of values between two endpoints. When working with intervals, it's important to know whether endpoints are included or not included in the interval. This determines the type of bracket to use for each endpoint.
There are two main types of intervals:
  • Closed Interval: Denoted with square brackets, such as \([a, b]\), which includes both endpoints \(a\) and \(b\).
  • Open Interval: Denoted with parentheses, such as \((a, b)\), which does not include either endpoint.
It's also possible to have half-open, or half-closed intervals:
  • \([a, b)\): Includes \(a\) but not \(b\).
  • \((a, b]\): Includes \(b\) but not \(a\).
Understanding intervals helps us in interpreting portions of the number line and can be translated into inequalities for solving problems.
Using Mathematical Notation
Mathematical notation is a universal language that allows us to convey mathematical ideas precisely and concisely.
By using symbols and structured expressions, we avoid lengthy verbal explanations and minimize possible misunderstandings.
  • Symbols such as \(<\), \(>\), \(\leq\), and \(\geq\) in inequalities provide a shorthand way of expressing conditions and comparisons.
  • Brackets, like ‘[’ and ‘(’, offer clarity on whether endpoints are part of an interval.
  • Equations, functions, and expressions are all part of the powerful notation toolkit in math.
Through understanding notation, we enhance our ability to interpret and represent mathematical concepts and can communicate them across diverse audiences effectively.

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

Use binomial expansion to simplify the given expression in part (a). Then, if instructed, find the indicated limit in part (b). (a) \(2(h+1)^{3}-5(h+1)^{2}+3\) (b) \(\lim _{h \rightarrow 0} \frac{2(h+1)^{3}-5(h+1)^{2}+3}{h}\)

Discuss what algebra is necessary to evaluate the given limit. Carry out your ideas. $$ \lim _{x \rightarrow 0} \frac{\sqrt[3]{x+27}-3}{x} $$

Sketch the set of points in the \(x y\) plane whose coordinates satisfy the given inequality. \(1 \leq x^{2}+y^{2} \leq 4\)

Find any intercepts of the graph of the given equation. Determine whether the graph of the equation possesses symmetry with respect to the \(x\) -axis, \(y\) -axis, or origin. Do not graph. \(y=\frac{x^{2}-10}{x^{2}+10}\)

Find any intercepts of the graph of the given equation. Determine whether the graph of the equation possesses symmetry with respect to the \(x\) -axis, \(y\) -axis, or origin. Do not graph. \(y=\frac{(x+2)(x-8)}{x+1}\)
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Probability and Statistics

Read Explanation

Geometry

Read Explanation

Applied Mathematics

Read Explanation

Discrete Mathematics

Read Explanation

Mechanics Maths

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.