/*! 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 12 Write interval notation for each... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 0: Problem 12

Write interval notation for each of the following. Then graph the interval on a number line. The set of all numbers \(x\) such that \(-5

Short Answer

Expert verified
Interval notation: \((-5, 5)\).

Step by step solution

01

- Understand the interval

Identify the given interval from the inequality provided. The inequality \(-5 < x < 5\) indicates that the variable \(x\) lies between \(-5\) and \(5\) but does not include the endpoints.
02

- Write the interval in interval notation

In interval notation, an open interval (where endpoints are not included) is written with parentheses. Thus, \(-5 < x < 5\) is written as \((-5, 5)\).
03

- Graph the interval on a number line

Draw a number line and mark points at \(-5\) and \(5\). Use open circles to indicate that \(-5\) and \(5\) are not included in the interval, then shade the region between \(-5\) and \(5\).

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.

Inequality
An inequality compares two values, indicating if one is greater, smaller, or equal to the other. In this case, the inequality \(-5 < x < 5\) shows that value \(x\) lies between -5 and 5, but does not include -5 and 5 themselves.
Inequalities help us to understand the range within which a variable can exist. Some common symbols used in inequalities include:
  • \( > \): Greater than
  • \( < \): Less than
  • \( \geq \): Greater than or equal to
  • \( \leq \): Less than or equal to
For instance, \(-5 < x < 5\) states that x can be any real number between -5 and 5, excluding -5 and 5.
Open Interval
An open interval is an interval that does not include its endpoints. In interval notation, open intervals are indicated using parentheses. For our example, \(-5 < x < 5\) in interval notation is written as ( -5, 5 ).
Here are a few key points:
  • Parentheses \( \left( \right) \) denote that the endpoints are not included.
  • For example, \(a < x < b\) translates to \( \left( a, b \right) \) in interval notation.
  • If we wanted to include endpoints, we use brackets \( \left[ \right] \), denoting a closed interval.
So, a closed interval like \[ c, d \] implies that both endpoints c and d are included.
Graphing on a Number Line
Graphing intervals on a number line is a visual representation of the solutions to an inequality. To graph \(-5 < x < 5\):
  • Draw a horizontal line, which represents our number line.
  • Mark the points -5 and 5 on this line.
  • Because it is an open interval, put open circles (or hollow dots) at -5 and 5 to indicate that these values are not included.
Next, shade the region between -5 and 5 to show that x can take any value within this range, but not the endpoints themselves.
To graph an interval that includes endpoints, you would use closed circles (or filled-in dots). This helps clearly convey to anyone reading the graph whether the endpoints are part of the interval or not.

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

An anthropologist can use certain linear functions to estimate the height of a male or female, given the length of certain bones. The humerus is the bone from the elbow to the shoulder. Let \(x=\) the length of the humerus, in centimeters. Then the height, in centimeters, of a male with a humerus of length \(x\) is given by \(M(x)=2.89 x+70.64\) The height, in centimeters, of a female with a humerus of length \(x\) is given by \(F(x)=2.75 x+71.48\) A 26 -cm humerus was uncovered in some ruins. a) If we assume it was from a male, how tall was he? b) If we assume it was from a female, how tall was she?

Find the domain of each function given below. \(g(x)=\frac{x-1}{x^{2}-36} \quad\) (Hint: Factor the denominator.)

Solve for \(y\) in terms of \(x\). Decide whether the resulting equation represents a function. $$2 y^{2}+3 x=4 x+5$$

Find the equilibrium point for each pair of demand and supply functions. Demand: \(q=7-x ;\) \(\quad\) Supply: \(q=2 \sqrt{x+1}\)

Boxowitz, Inc., a computer firm, is planning to sell a new graphing calculator. For the first year, the fixed costs for setting up the new production line are 100,000 dollars. The variable costs for producing each calculator are estimated at 20 dollars. The sales department projects that 150,000 calculators can be sold during the first year at a price of 45 dollars each. a) Find and graph \(C(x),\) the total cost of producing \(x\) calculators. b) Using the same axes as in part (a), find and graph \(R(x),\) the total revenue from the sale of \(x\) calculators. c) Using the same axes as in part (a), find and graph \(P(x),\) the total profit from the production and sale of \(x\) calculators. d) What profit or loss will the firm realize if the expected sale of 150,000 calculators occurs? e) How many calculators must the firm sell in order to break even?
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Logic and Functions

Read Explanation

Probability and Statistics

Read Explanation

Pure Maths

Read Explanation

Calculus

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.