Problem 71 - Algebra 1

Question

If \(a\) and \(b\) are positive and \(a

Step-by-Step Solution

Answer

Yes, if \(a\) and \(b\) are positive and \(a

Step 1: Understanding what happens when a number is divided by a larger number

When a number is divided by a larger number, the result is a smaller quantity. For example, for any positive number \(p\), \( \frac{p}{2} > \frac{p}{3}\) because 2 is less than 3. Here, although \(p\) is the same, the result depends on whether \(p\) is divided by 2 or 3.

Step 2: Applying this understanding to our problem

In our problem, \(a

Step 3: Providing an example

Let's use an example to illustrate this. Let's say \(a=2\) and \(b=3\). \(a

Key Concepts

Rational Numbers

Rational numbers are numbers that can be expressed as the quotient of two integers, where the denominator is not zero. For example, the numbers \( \frac{1}{2}, \frac{3}{4}, \) and \( -\frac{5}{3} \) are all rational numbers. They are crucial in mathematics because they allow us to perform calculations with precision.

When dealing with rational numbers, it's important to remember that they can be positive, negative, or zero. However, in this context, we focus on positive rational numbers. These can represent quantities like fractions of a whole or probabilities in statistical analyses. Understanding how to manipulate and compare these numbers is essential in solving inequalities involving fractions.

• Rational numbers include positive and negative fractions.
• Both numerator and denominator are integers.
• Common examples: \( \frac{1}{2}, \frac{3}{7}, \frac{-4}{5} \).

Comparison of Fractions

Comparing fractions involves determining which fraction represents a larger or smaller value. There are simpler methods than cross-multiplying for comparing these fractions, especially when comparing fractions with different denominators.

In the given exercise, we compare the fractions \( \frac{1}{a} \) and \( \frac{1}{b} \) where \( a \) and \( b \) are positive numbers, and \( a < b \). When you have fractions of the form \( \frac{1}{x} \), the larger the denominator, the smaller the fraction's value. Therefore, having a larger denominator means the fraction is smaller when the numerators are the same. This understanding helps to confirm that \( \frac{1}{b} < \frac{1}{a} \) if \( a < b \).

• Smaller denominators yield larger fractions.
• Larger denominators yield smaller fractions.
• Use cross-multiplication or common denominators for comparison.

Inverse Relationship

The notion of inverse relationships in mathematics often describes how two values change in opposite directions. For instance, consider how the size of a denominator influences the result of a fraction \( \frac{1}{x} \).

In the example provided, we observe an inverse relationship: as the denominator \(b\) increases (assuming \(a < b\)), the value of the fraction \( \frac{1}{b} \) decreases. This underscores the concept that inversely related values react oppositely to each other's changes, which is crucial in understanding how dividing by larger numbers results in smaller quantities.

• Inversely related variables move in opposite directions.
• Larger denominators lead to smaller fractions in \( \frac{1}{x} \).
• This principle applies across many mathematical contexts.

Previous

Problem 71

Next

Problem 72

Other exercises in this chapter

Problem 71

Evaluate the expression. $$79-v^{3} \text { when } v=4$$

View solution

Problem 71

As parts (a) and (b) of Example 3 show, it is sometimes easier to evaluate an expression by simplifying it before substituting, and sometimes easier if you subs

View solution

Problem 72

FREIGHT TRAINS A train with 150 freight cars is used to haul two types of grain. Each freight car can haul 97.3 tons of barley or 114 tons of corn. Let \(n\) re

View solution

Problem 72

Evaluate the expression. Use estimation to check your answer. GASOLINE PRICES Last month the price of gasoline was \(\$ 1.19\) per gallon. This month the price

View solution

Key Concepts

Rational Numbers

Rational numbers are numbers that can be expressed as the quotient of two integers, where the denominator is not zero. For example, the numbers \( \frac{1}{2}, \frac{3}{4}, \) and \( -\frac{5}{3} \) are all rational numbers. They are crucial in mathematics because they allow us to perform calculations with precision.

When dealing with rational numbers, it's important to remember that they can be positive, negative, or zero. However, in this context, we focus on positive rational numbers. These can represent quantities like fractions of a whole or probabilities in statistical analyses. Understanding how to manipulate and compare these numbers is essential in solving inequalities involving fractions.

• Rational numbers include positive and negative fractions.
• Both numerator and denominator are integers.
• Common examples: \( \frac{1}{2}, \frac{3}{7}, \frac{-4}{5} \).

Comparison of Fractions

Comparing fractions involves determining which fraction represents a larger or smaller value. There are simpler methods than cross-multiplying for comparing these fractions, especially when comparing fractions with different denominators.

In the given exercise, we compare the fractions \( \frac{1}{a} \) and \( \frac{1}{b} \) where \( a \) and \( b \) are positive numbers, and \( a < b \). When you have fractions of the form \( \frac{1}{x} \), the larger the denominator, the smaller the fraction's value. Therefore, having a larger denominator means the fraction is smaller when the numerators are the same. This understanding helps to confirm that \( \frac{1}{b} < \frac{1}{a} \) if \( a < b \).

• Smaller denominators yield larger fractions.
• Larger denominators yield smaller fractions.
• Use cross-multiplication or common denominators for comparison.

Inverse Relationship

The notion of inverse relationships in mathematics often describes how two values change in opposite directions. For instance, consider how the size of a denominator influences the result of a fraction \( \frac{1}{x} \).

In the example provided, we observe an inverse relationship: as the denominator \(b\) increases (assuming \(a < b\)), the value of the fraction \( \frac{1}{b} \) decreases. This underscores the concept that inversely related values react oppositely to each other's changes, which is crucial in understanding how dividing by larger numbers results in smaller quantities.

• Inversely related variables move in opposite directions.
• Larger denominators lead to smaller fractions in \( \frac{1}{x} \).
• This principle applies across many mathematical contexts.