/*! 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 55 Explain why $$ |a b|=|a||b| ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 0: Problem 55

Explain why $$ |a b|=|a||b| $$ for all real numbers \(a\) and \(b\).

Short Answer

Expert verified
We analyzed all possible scenarios involving the signs of real numbers \(a\) and \(b\) and found that \(\boldsymbol{|ab| = |a||b|}\) holds true for all of them. This is because the absolute value of a product is equal to the product of the absolute values in each case. Therefore, the given expression is true for all real numbers \(a\) and \(b\).

Step by step solution

01

Understand the properties of absolute values

First, let's recall the properties of absolute values. The absolute value of a real number represents the distance of the number from zero on the number line. In essence, the absolute value of a number is always non-negative. By definition, we have: For any real number x, 1. If \(x \ge 0\), then \(|x| = x\). 2. If \(x < 0\), then \(|x| = -x\).
02

Analyze different scenarios

Now, let's analyze the possible scenarios involving the signs of real numbers \(a\) and \(b\). We have four cases to consider: 1. Both \(a\) and \(b\) are non-negative. 2. \(a\) is non-negative, and \(b\) is negative. 3. \(a\) is negative, and \(b\) is non-negative. 4. Both \(a\) and \(b\) are negative.
03

Case 1 - Both a and b are non-negative

When both \(a\) and \(b\) are non-negative, we have: a. \(|a| = a\) b. \(|b| = b\) Now let's compute the absolute value of their product: \(|ab| = ab\) Now, let's compute the product of their absolute values: \(|a||b| = ab\) So we have \(\boldsymbol{|ab| = |a||b|}\) for this case.
04

Case 2 - a is non-negative and b is negative

When \(a\) is non-negative, and \(b\) is negative, we have: a. \(|a| = a\) b. \(|b| = -b\) Again, let's compute the absolute value of their product: \(|ab| = -ab\) And let's compute the product of their absolute values: \(|a||b| = a(-b) = -ab\) So we have \(\boldsymbol{|ab| = |a||b|}\) for this case too.
05

Case 3 - a is negative and b is non-negative

When \(a\) is negative and \(b\) is non-negative, following the same process, we have: a. \(|a| = -a\) b. \(|b| = b\) Compute the absolute value of their product: \(|ab| = -ab\) Compute the product of their absolute values: \(|a||b| = (-a)b = -ab\) So we have \(\boldsymbol{|ab| = |a||b|}\) for this case as well.
06

Case 4 - Both a and b are negative

When both \(a\) and \(b\) are negative, we have: a. \(|a| = -a\) b. \(|b| = -b\) Compute the absolute value of their product: \(|ab| = ab\) Compute the product of their absolute values: \(|a||b| = (-a)(-b) = ab\) So we have \(\boldsymbol{|ab| = |a||b|}\) for this case too.
07

Conclusion

From our analysis, we have covered all possible scenarios involving the signs of real numbers \(a\) and \(b\) and found that \(\boldsymbol{|ab| = |a||b|}\) holds true for all of them. Therefore, we can conclude that the given expression is true for all real numbers \(a\) and \(b\).

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!

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

Expand the given expression $$ (x+1)(x-2)(x+3) $$

Show that if \(a\) and \(b\) are real numbers such that $$ |a+b|<|a|+|b| $$ then \(a b<0\)

The intersection of two sets of numbers consists of all numbers that are in both sets. If \(A\) and \(B\) are sets, then their intersection is denoted by \(A \cap B .\) In Exercises \(31-40,\) write each intersection as a single interval. $$ (-\infty,-3) \cap[-5, \infty) $$

In Exercises \(7-16,\) write each union as a single interval. $$ (-3, \infty) \cup[-5, \infty) $$

Suppose \(a\) and \(b\) are numbers. Explain why ei. ther \(ab\)
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Probability and Statistics

Read Explanation

Applied Mathematics

Read Explanation

Logic and Functions

Read Explanation

Discrete 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.