/*! 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 107 Solve each equation or inequalit... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 107

Solve each equation or inequality. $$ |x+5|>-9 $$

Short Answer

Expert verified
Any real number satisfies the inequality.

Step by step solution

01

Understand the absolute value inequality

Recognize that \(|x+5|>-9\) is an absolute value inequality.
02

Consider the properties of absolute value

The absolute value \(|y|\) for any real number \(y\) is always non-negative, meaning \(|y| \geq 0\).
03

Analyze the inequality

Since \(|x+5|\) represents an absolute value, it will always be greater than or equal to 0. Therefore, it will always be greater than \(-9\), as \(-9\) is a negative number.
04

Conclusion

Since \(|x+5|\) is always non-negative and greater than \(-9\) for any real number \(x\), there are no restrictions on \(x\). Hence, any real number \(x\) will satisfy the inequality.

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.

Inequalities
Inequalities deal with the relationship between two values, determining if one is greater than, less than, equal to, or not equal to another. When solving inequalities, our goal is to find the set of values for which the inequality holds true. In the exercise, we worked with an absolute value inequality, which involves special rules and properties. Inequalities can be represented using various symbols:
  • \textless: less than
  • \textgreater: greater than
  • \textleq: less than or equal
  • \textgeq: greater than or equal
These relationships are crucial for understanding how solutions are derived. For our example with \(\textverbatim|x+5| > -9\), understanding inequalities helps us determine the overall range of possible values for x.
Absolute Value Properties
Absolute value refers to the distance of a number from zero on the number line, always being non-negative. For any real number y, the absolute value is denoted as \(|y|\), and it has the following key properties:
  • \(|y| \geq 0\) - Absolute value is always zero or positive
  • \: for \(|a - b|\): it represents the distance between a and b
  • Equal for positive and negative pairs: \( |-a| = |a|\)
In the given problem, \(\textverbatim|x + 5| > -9\), we use the non-negative nature of absolute value. Since the left side \(|x + 5|\) is always \(\textgeq 0\), it will always be greater than \(-9\), which is a negative number. This insight helps us conclude that the inequality holds for all real numbers x.
Real Numbers
Real numbers include all the numbers along the number line. They can be integers, fractions, or irrational numbers. Real numbers are denoted by the symbol \(\textbb{R}\) and have the following characteristics:
  • Include both positive and negative numbers
  • Include zero
  • Are continuous, meaning there aren't any gaps
In the context of the exercise, we considered that any real number x can satisfy the inequality \(|x + 5| > -9\). This is because absolute value ensures that the expression on the left is always at least zero, and thus it is always greater than any negative number, including \(-9\). Therefore, all real numbers are valid solutions to the inequality.

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 each equation. $$ |3 x+1|=|2 x+4| $$

A real-estate agent earned \(\$ 6300\) commission on a property sale of \(\$ 210,000 .\) What is her rate of commission?

Dr. Mosely has determined that \(99 \%\) of the babies he has delivered have weighed \(x\) pounds, where $$ |x-8.3|<1.5 $$ What range of weights corresponds to this inequality?

The movie Transformers: The Last Knight in DVD was on sale for $16.17. The list price (full price) of this disc was 17.99.To the nearest tenth, what was the percent discount?

Bonnie earned scores of 90 and 82 on her first two tests in English literature. What score must she make on her third test to keep an average of 84 or greater?
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Decision Maths

Read Explanation

Applied Mathematics

Read Explanation

Geometry

Read Explanation

Calculus

Read Explanation

Logic and Functions

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.