Problem 61 Suppose $a$ and $b$ are numb... [FREE SOLUTION]

91影视

Precalculus A Prelude to Calculus

Sheldon Axler

$Math Studyset 91影视 Explanations$ Math

2 Edition

Chapter 0: Problem 61

Suppose $a$ and $b$ are numbers. Explain why either $ab$.

Short Answer

Expert verified

Considering the properties and orderings of real numbers, we can analyze the possible relative positions of a and b on a number line. There are only three possibilities: a is at the same position as b (a = b), a is to the left of b (a < b), or a is to the right of b (a > b). As there are no other possible positions on the number line, either $a < b$, $a = b$, or $a > b$.

Step by step solution

Understanding the properties and orderings of real numbers

In order to understand the possible relationships between two numbers a and b, it's important to first recognize that we are working within the set of real numbers. Real numbers have some inherent properties and their orderings are well-defined. Real numbers follow the properties of commutativity, associativity, and distributivity, and have inverse and identity elements for addition and multiplication. Ordering of real numbers is determined based on their positions on a number line. A number is considered smaller if it is to the left and larger if it is to the right of another number on the number line.

Consider all possible relative positions of a and b

In order to explain why a and b must be either less than, equal to, or greater than each other, we will consider all possible positions of a and b on the number line. - Case 1: a and b occupy the same position on the number line. - Case 2: a is to the left of b on the number line. - Case 3: a is to the right of b on the number line.

Relate the positions of a and b to their ordering

Based on the cases outlined in Step 2, we can now relate these positions to the possible relationships between a and b: - Case 1: If a and b occupy the same position on the number line, then a = b, as both numbers have the same value. - Case 2: If a is to the left of b on the number line, then a < b, as per the ordering of real numbers. - Case 3: If a is to the right of b on the number line, then a > b, as per the ordering of real numbers. Since there are no other possibilities for the relative positions of a and b on a number line, these three cases cover all possible relationships between a and b. Therefore, either a < b, a = b, or a > 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

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

Suppose \(a\) and \(b\) are numbers. Explain why either \(ab\).

Short Answer

Step by step solution

Understanding the properties and orderings of real numbers

Consider all possible relative positions of a and b

Relate the positions of a and b to their ordering

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Geometry

Theoretical and Mathematical Physics

Discrete Mathematics

Probability and Statistics

Logic and Functions

Study anywhere. Anytime. Across all devices.