/*! 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 1 Classify each of the following s... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 1

Classify each of the following statements as either true or false. \(|x|\) is never negative.

Short Answer

Expert verified
True

Step by step solution

01

Understand Absolute Value

Absolute value, denoted by \(|x|\), refers to the non-negative value of \(|x|\) without regard to its sign. Essentially, it measures the distance of \(|x|\) from zero on the number line.
02

Evaluate the Properties

The absolute value of any real number \(|x|\) is either zero or positive. This means \(|x| \geq 0\).
03

Determine Statement's Truth Value

Since \(|x| \geq 0\) for any real number \(|x|\), \(|x|\) is never negative. Therefore, the statement ' \(|x|\) is never negative ' is true.

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.

Properties of Absolute Value
The absolute value, represented as \(|x|\), has some essential properties that make it unique. One of the most important characteristics is that it indicates the magnitude of a number, irrespective of its sign. This means that whether the number is positive or negative, the absolute value will always be non-negative. For example, \(|3| = 3\) and \(|-3| = 3\). Notice that both yield the same result because absolute value only cares about the distance from zero, not the direction.
Some key properties include:
  • \(|x| \geq 0 \) for all real numbers x.
  • \(|x| = 0 \) if and only if \(x = 0\).
  • \(|x| = |-x| \), which emphasizes that absolute value makes the sign of a number irrelevant.
  • \(|xy| = |x| |y| \) and \(|x/y| = |x| / |y| \) when \(y \eq 0\).
Understanding these properties helps to not only solve mathematical problems but also to grasp the core concept of what absolute value represents.
Non-negative Values
One of the distinctive aspects of absolute value is that it is always non-negative. This stems from the definition of absolute value as the distance a number is from zero on the number line. Distances can never be negative, so absolute values can't be either. This can be summarized as follows:
  • Absolute value \(|x| \) is always \( \geq 0 \).
To illustrate further, consider a few examples:
  • \(|5| = 5 \)
  • \(|-5| = 5 \)
  • \(|0| = 0 \)
No matter what the original number is, the outcome of the absolute value function will always be zero or a positive number. Thus, the statement '\
Number Line Distance
A fantastic way to visualize absolute value is by thinking of it as a measure of distance on a number line. On the number line, any point represents a real number, and the absolute value of that number tells us how far it is from zero鈥攔egardless of direction.
Let's break down some key points:
  • The distance from zero to a number \(x\) is \(|x| \).
  • This calculation ignores direction, meaning \(\|-3\| \) and \(|3| \) are the same because both are 3 units away from zero.
Think of absolute value like walking a certain distance from a starting point: no matter if you walk forwards or backwards, you've still covered a positive amount of ground. This visualization can make problems involving absolute values easier to understand and solve.

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

Write an equivalent inequality using absolute value. $$ x \leq-6 \text { or } 6 \leq x $$

For \(f(x)\) as given, use interval notation to write the domain of \(f\). $$ f(x)=\sqrt{2 x+7} $$

Elevators. Many elevators have a capacity of 1 metric ton \((1000 \mathrm{kg}) .\) Suppose that \(c\) children, each weighing \(35 \mathrm{kg},\) and \(a\) adults, each \(75 \mathrm{kg}\), are on an elevator. Graph a system of inequalities that indicates when the elevator is overloaded.

College Faculty. The number of part-time instructional faculty in U.S. postsecondary institutions is growing at a greater rate than the number of full-time faculty. The number of parttime faculty, in thousands, is approximated by $$ p(t)=27 t+325 $$ and the number of full-time faculty, in thousands, is approximated by $$ f(t)=16 t+500 $$ For both functions, \(t\) represents the number of years after \(1995 .\) Using an inequality, determine those years for which there were more part-time faculty than full-time faculty.

Waterfalls. In order for a waterfall to be classified as a classical waterfall, its height must be less than twice its crest width, and its crest width cannot exceed one-and-one-half times its height. The tallest waterfall in the world is about 3200 ft high. Let \(h\) represent a waterfall's height, in feet, and w the crest width, in feet. Write and graph a system of inequalities that represents all possible combinations of heights and crest widths of classical waterfalls.
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Logic and Functions

Read Explanation

Mechanics Maths

Read Explanation

Pure Maths

Read Explanation

Discrete Mathematics

Read Explanation

Applied Mathematics

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.