Problem 18 For all real numbers $a$ and \... [FREE SOLUTION]

Chapter 3: Problem 18

For all real numbers $a$ and $b$, if \(a

Short Answer

Expert verified

Given that $a<b$, we have shown that $a<\frac{a+b}{2}<b$ by using the properties of inequalities. First, we demonstrated $a<\frac{a+b}{2}$ by adding $a$ to both sides of the inequality and dividing by 2. Then, we demonstrated $\frac{a+b}{2}<b$ by adding $b$ to both sides of the inequality and dividing by 2. Thus, our result holds for any real numbers $a$ and $b$.

Step by step solution

Show that \(a

: We are given that $a<b$. We want to show that $a<\frac{a+b}{2}$. To do so, we're going to use the properties of inequalities to transform our given inequality into the desired one. First, let's add $a$ to both sides of the inequality, so we have: $$a+a<b+a \Rightarrow 2a<b+a$$ Now, we can divide both sides by 2: $$\frac{2a}{2}<\frac{b+a}{2} \Rightarrow a<\frac{a+b}{2}$$ which is what we wanted to show.

Show that \(\frac{a+b}{2}

: We have already shown that $a<\frac{a+b}{2}$. Now, we need to show that $\frac{a+b}{2}<b$. We will again use the properties of inequalities to transform our given inequality into the desired one. We will start with the given inequality, $a<b$. This time, let's add $b$ to both sides of the inequality: $$a+b<b+b \Rightarrow a+b<2b$$ Now, we will divide both sides by 2: $$\frac{a+b}{2}<\frac{2b}{2} \Rightarrow \frac{a+b}{2}<b$$ which is what we wanted to show. Putting both steps together, we have shown that if $a<b$, then $a<\frac{a+b}{2}<b$.

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!

Key Concepts

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

Properties of Inequalities

Inequalities are a fundamental part of mathematics. They help us compare two values and determine the order of numbers.

Understanding the properties of inequalities is crucial when solving mathematical expressions. For example, consider the following properties:

Transitive Property: If $a < b$ and $b < c$, then $a < c$. This property helps in combining inequalities.
Addition and Subtraction: You can add or subtract the same number from both sides of an inequality without changing the inequality's direction. If $a < b$, then adding $c$ gives $a + c < b + c$.
Multiplication and Division: When you multiply or divide both sides of an inequality by a positive number, the direction remains the same. However, if you multiply or divide by a negative number, the inequality's direction reverses.

These properties were used in the exercise to break down and transform the given inequality $a < b$ into smaller, manageable inequalities, showing the validity of $a < \frac{a+b}{2} < b$. Thus, they are invaluable in proving or simplifying complex inequalities.

Mathematical Proofs

Proofs are an essential aspect of mathematics. They ensure that a statement is universally true and not just based on specific instances. In this exercise, a proof was constructed to demonstrate that $a < \frac{a+b}{2} < b$ for any real numbers $a$ and $b$ such that $a < b$.

Mathematical proofs typically involve logical reasoning, clear definitions, and properties of mathematical elements. Here's how the exercise proof was structured:

Initial Assumption: Start with a known fact, in this case, $a < b$.
Logical Steps: Transform the given information step-by-step using valid mathematical operations and properties, like adding or dividing both sides of the inequality.
Conclude with the Desired Result: Each intermediate step brings you closer to the goal, concluding that $a < \frac{a+b}{2} < b$.

In essence, a proof is like a map. It guides you from the start (assumptions) to the end (conclusions), ensuring each step is valid and logical.

Real Numbers

Real numbers are everywhere in mathematics. They include all the numbers on the number line. From simple integers like 1, 2, and -3 to fractions like 1/2, to irrational numbers like $\sqrt{2}$ and $\pi$.

In the exercise, $a$ and $b$ are defined as real numbers. Why is this important?

Continuum Property: The real numbers form a complete set of numbers. This means between any two real numbers, there is another real number, which allows statements like $a < \frac{a+b}{2} < b$ to hold true.
Versatility: Real numbers can describe a vast range of quantities, providing a common ground across various branches of mathematics.
Density: Any real number interval is infinitely dense with real numbers. That is, between any two real numbers $a$ and $b$, there are infinitely many other real numbers.

By understanding that $a$ and $b$ are from the set of real numbers, we recognize the continuity and specific properties of these numbers, which play a critical role in making inequalities, like the one in the exercise, valid.

91影视

For all real numbers \(a\) and \(b\), if \(a

Short Answer

Step by step solution

Show that \(a

Show that \(\frac{a+b}{2}

Key Concepts

Properties of Inequalities

Mathematical Proofs

Real Numbers

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Probability and Statistics

Statistics

Theoretical and Mathematical Physics

Discrete Mathematics

Pure Maths

Study anywhere. Anytime. Across all devices.