/*! 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 7 Exer. 3-12: Express the inequali... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 7

Exer. 3-12: Express the inequality as an interval, and sketch its graph. $$ -2

Short Answer

Expert verified
The interval is \((-2, 4]\). Open circle at \(-2\), closed circle at \(4\), shade between.

Step by step solution

01

Understand the Inequality

The inequality given is \(-2 < x \leq 4\). This tells us that \(x\) is greater than \(-2\) and at the same time \(x\) is less than or equal to \(4\).
02

Express the Inequality in Interval Notation

To express \(-2 < x \leq 4\) in interval notation, we use a parenthesis to indicate that \(-2\) is not included and a bracket to show that \(4\) is included. The interval is written as \((-2, 4]\).
03

Sketch the Graph of the Interval

To sketch \((-2, 4]\), draw a number line. Place an open circle at \(-2\) to show it is not included and a closed circle at \(4\) to indicate it is included. Shade the region between \(-2\) and \(4\) to represent all values \(x\) can take.

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 method for representing inequalities in algebra. It provides a concise way to express ranges of values that a variable can take. In our exercise, we have the inequality \(-2 < x \leq 4\), which translates into interval notation as \((-2, 4]\). Here鈥檚 how it works:
  • A parenthesis ( indicates that the endpoint is not included in the interval. This is known as an open interval.
  • A bracket [ signifies that the endpoint is included. This is called a closed interval.
In the case of \(-2 < x \leq 4\), we use a parenthesis for \(-2\) because \(-2\) itself is not a possible value for \(x\). Conversely, since \(4\) is part of the acceptable values, we use a bracket for it. Interval notation is both efficient and helpful when you have to write the domains of functions or solutions to inequalities.
Graphing Inequalities
Graphing inequalities involves representing the solution set of an inequality on a number line. This visual approach helps in understanding which values of the variable satisfy the inequality condition. Let鈥檚 break down the process using our inequality example \(-2 < x \leq 4\):
  • First, draw a horizontal line to represent the number line.
  • Identify and mark the critical points, which are \(-2\) and \(4\) in our example.
  • Use an open circle at \(-2\) to show that \(-2\) is not included in the solution set.
  • Place a closed circle at \(4\) to indicate that it is included.
After marking these points, shade the region between \(-2\) and \(4\) to depict all possible values of \(x\) that make the inequality true. Graphing provides a clear visual confirmation of the interval, facilitating a better grasp of the range of solutions.
Number Line Representation
The number line is an essential tool in algebra for visually representing intervals and inequalities. Here鈥檚 a step-by-step way to use this tool effectively:
  • Begin by drawing a horizontal line, labeling it with regular intervals of numbers.
  • Locate your critical points on this line. For \(-2 < x \leq 4\), these points are \(-2\) and \(4\).
  • Use symbols such as circles to indicate whether an endpoint is included in the interval. Open circles mean the number isn't part of the set, while closed circles indicate inclusion.
  • Shade or highlight the section of the number line fitting within your inequality. This shaded area visually represents all values that satisfy the inequality.
By using simple lines and symbols, a number line offers a clear, intuitive way to understand and communicate inequalities. It simplifies complex concepts by allowing visual representation, which aids in comprehension.

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 the equation. $$\frac{1}{x+4}+\frac{3}{x-4}=\frac{3 x+8}{x^{2}-16}$$

A person's intelligence quotient (IQ) is determined by multiplying the quotient of his or her mental age and chronological age by 100 . (a) Find the IQ of a 12-year-old child whose mental age is \(15 .\) (b) Find the mental age of a person 15 years old whose IQ is 140 .

The formula occurs in the indicated application. Solve for the specified variable. \(s=\frac{1}{2} g t^{2}+v_{0} f\) for \(v_{0}\)

Solve the equation. $$\frac{3}{2 x-4}-\frac{5}{3 x-6}=\frac{3}{5}$$

Below the cloud base, the air temperature \(T\) (in \({ }^{\circ} \mathrm{F}\) ) at height \(h\) (in feet) can be approximated by the equation \(T=T_{0}-\left(\frac{55}{1000}\right) h\), where \(T_{0}\) is the temperature at ground level. (a) Determine the air temperature at a height of 1 mile if the ground temperature is \(70^{\circ} \mathrm{F}\). (b) At what altitude is the temperature freezing?
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Discrete Mathematics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Probability and Statistics

Read Explanation

Applied Mathematics

Read Explanation

Geometry

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.