Problem 68 (a) If \(0... [FREE SOLUTION]

Chapter 4: Problem 68

(a) If \(0

Short Answer

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Question: Determine the relationships between the reciprocals of a and b under the following conditions: (a) both a and b are positive and 0 < a 鈮� b, (b) both a and b are negative and a 鈮� b < 0, and (c) a is negative and b is positive. Answer: (a) When both a and b are positive, and 0 < a 鈮� b, the relationship between their reciprocals is 1/a 鈮� 1/b. (b) When both a and b are negative, and a 鈮� b < 0, the relationship between their reciprocals is 1/a 鈮� 1/b. (c) When a is negative and b is positive, the relationship between their reciprocals is 1/a < 1/b.

Step by step solution

Part (a): Proving the inequality with positive numbers a and b

Given: \(0<a\leq b\) We need to prove that \(1/a \geq 1/b\) Step 1: Find the reciprocal of both sides of the inequality \(0<1/a \leq 1/b\) Step 2: Multiply both sides by a \(0<a \leq a/b\) Since a and b are both positive, the inequality sign remains the same. Now, we have: \(a \leq \frac{a}{b}\) Step 3: Multiply both sides by b \(a*b \leq a\) Step 4: Divide both sides by a \(b \leq 1\) And we have the relationship: \(1/a \geq 1/b\)

Part (b): Proving the inequality with negative numbers a and b

Given: \(a\leq b<0\) We need to prove that \(1/a \geq 1/b\) Step 1: Find the reciprocal of both sides of the inequality \(0 > 1/a \geq 1/b\) Step 2: Multiply both sides by a \(0 < a \leq a/b\) Since a and b are both negative, the inequality sign flips. Now, we have: \(a \geq \frac{a}{b}\) Step 3: Multiply both sides by b \(ab \geq a\) Step 4: Divide both sides by a \(b \geq 1\) And we have the relationship: \(1/a \geq 1/b\)

Part (c): Relationship between \(1/a\) and \(1/b\) when \(a0\)

Given: \(a<00\), their reciprocals would also be opposite in sign, i.e., \(1/a\) would be negative and \(1/b\) would be positive. Thus, the relationship between \(1/a\) and \(1/b\) when \(a<0\) and \(b>0\) is: \(1/a < 1/b\) This concludes our analysis of the given exercise on inequalities and reciprocals of numbers.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Proof Writing

Proof writing is an essential skill in mathematics that allows us to establish the truth of propositions using logical reasoning. When dealing with inequalities, the goal is to clearly demonstrate that one quantity is greater than or equal to another. Here's how you can approach proof writing:

Start by thoroughly understanding the given conditions and assumptions. It's crucial to know what is provided as true and the relationships between those known quantities.
Break down the problem into smaller parts and take logical steps to transform the initial statement into the required form.
In the case of inequalities, often involve manipulation of expressions through operations like addition, subtraction, multiplication, division, and taking reciprocals, while carefully considering the direction of the inequality depending upon each operation.
Finally, articulate each step clearly, explaining why it is valid and how it contributes toward proving the final statement.

By practicing proof writing in different contexts, you become more adept at reasoning logically and effectively communicating your mathematical arguments.

Reciprocal Inequality

The reciprocal inequality is a particular type of inequality that involves expressions in the form of reciprocal values, such as \(1/a\) and \(1/b\). Understanding how these values compare requires knowledge of both the original numbers and their signs. Here are some important facts about reciprocal inequalities:

When dealing with positive numbers \(a\) and \(b\) such that \(0 < a \leq b\), the reciprocals will satisfy \(1/a \geq 1/b\). This occurs because dividing by a larger number results in a smaller reciprocal value.
In contrast, for negative numbers \(a\) and \(b\) where \(a \leq b < 0\), the order of the inequality flips due to the sign flip property of reciprocals of negative numbers, still resulting in \(1/a \geq 1/b\).
When comparing a negative number's reciprocal to a positive one's, like in the case \(a < 0 < b\), regardless of their magnitudes, \(1/a\) is always less than \(1/b\) because any positive number is inherently larger than any negative number.

Reciprocal inequalities are a tool that can help us understand relationships that aren鈥檛 immediately obvious by leveraging the interesting properties of division.

Real Numbers

Real numbers form the backbone of many mathematical concepts being used in various fields of study. These numbers include all the values that can be found on the number line, encompassing both rational and irrational numbers. To grasp the behavior of inequalities in the context of real numbers, consider the following:

Real numbers can be classified into positive, negative, and zero, each playing a unique role in equations and inequalities.
Operations performed on real numbers, such as addition, subtraction, or reciprocation, must adhere to certain rules and properties鈥攆or instance, multiplying or reciprocating a negative number will affect the direction of an inequality.
Knowing whether a real number is positive or negative is critical when you're working with reciprocals, to properly understand the inequality conditions, especially when handling expressions like those involving \(1/a\) and \(1/b\).

As fundamental as real numbers are, their comprehensive understanding contributes significantly to mastering more complex mathematical principles and solving intricate problems.

91影视

(a) If \(0

Short Answer

Step by step solution

Part (a): Proving the inequality with positive numbers a and b

Part (b): Proving the inequality with negative numbers a and b

Part (c): Relationship between \(1/a\) and \(1/b\) when \(a0\)

Key Concepts

Proof Writing

Reciprocal Inequality

Real Numbers

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