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Chapter 2: Problem 12

Let \(a, b, c, d\) be numbers satisfying \(0

Short Answer

Expert verified
For the condition \(a c<b d\), we can take \(a = 1, b = 2, c = -3, d = -1\), and for the condition \(b d<a c\), we can take \(a = 2, b = 3, c = -4, d = -1\).

Step by step solution

01

Example for \(a c

The first step is to find numbers \(a, b, c, d\) that meet the condition \(a c<b d\). One example of this could be let \(a = 1, b = 2, c = -3, d = -1\). Substituting these numbers into the inequality, we get \(1*(-3)<2*(-1)\), which simplifies to \(-3<-2\), and it's true.
02

Example for \(b d

The second step is to find numbers \(a, b, c, d\) that meet the condition \(b d<a c\). An instance of this would be let \(a = 2, b = 3, c = -4, d = -1\). Substituting these numbers into the inequality, we get \(3*(-1)<2*(-4)\), that simplifies to \(-3<-8\), and it's true.

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Key Concepts

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

Number Properties
Understanding the properties of numbers is critical when solving inequalities. In real analysis, a key focus is on the properties which define the sets of numbers we interact with, such as the real number system. This covers positive numbers, negative numbers, and zero, each possessing unique characteristics particularly relevant to inequalities.

For instance, the multiplication of two positive numbers results in a positive product, while multiplying two negative numbers also gives a positive product because the 'negative' signs cancel each other out. In contrast, a positive and a negative number yield a negative product. Keeping these properties in mind is essential when trying to determine the truth of inequalities involving real numbers, as seen in the textbook exercise where we compare products of different pairs of numbers with positive and negative signs.
Inequality Solutions
When we look at inequality solutions, the aim is to find values that satisfy the inequality's condition. In the given exercise, the solution involves selecting appropriate values for variables to satisfy two distinct inequalities involving the multiplication of positive and negative numbers.

To demonstrate an inequality like \(a c < b d\), we must choose numbers that adhere to the conditions given (\(0
Real Number Examples
Real number examples in the context of inequalities are pivotal for grasping the concept that there is an infinite spectrum between the whole numbers that we are familiar with. Taking our cues from the earlier examples, let's look closely at how real numbers can be used to satisfy inequalities.

In the first scenario, by choosing \(a = 1, b = 2, c = -3, d = -1\), we have two sets of positive and negative numbers that follow the given restrictions and showcase a true inequality. Here, the values of a and c, when multiplied, result in a product that is less than the product of b and d. But it's not the specific numbers that are essential; it's the relationship between them. Applying these concepts to different sets of reals, we could also choose fractions, decimals, or irrational numbers so long as the relationships between them maintain the inequality. Understanding how real numbers interact through inequality provides a fundamental tool for working on more complex real analysis problems.

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Most popular questions from this chapter

Let \(S \subseteq \mathbb{R}\) be nonempty. Show that if \(u=\sup S\), then for every number \(n \in \mathbb{N}\) the number \(u-1 / n\) is not an upper bound of \(S\), but the number \(u+1 / n\) is an upper bound of \(S\). (The converse is also true; see Exercise 2.4.3.)

Let \(S\) be a nonempty subset of \(\mathbb{R}\) that is bounded below. Prove that inf \(S=-\sup \\{-s: s \in S\\}\).

Show that if \(a, b \in \mathbb{R}\), and \(a \neq b\), then there exist \(\varepsilon\) -neighborhoods \(U\) of \(a\) and \(V\) of \(b\) such that \(U \cap V=\emptyset\)

Modify the argument in Theorem \(2.4 .7\) to show that there exists a positive real number \(y\) such that \(y^{2}=3\)

Determine and sketch the set of pairs \((x, y)\) in \(\mathbb{R} \times \mathbb{R}\) that satisfy: (a) \(|x|=|y|\), (b) \(|x|+|y|=1\), (c) \(|x y|=2\) (d) \(|x|-|y|=2\).
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