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Chapter 5: Problem 29

If \(x<-1\) and \(y<-2\), then: A. \(x / y>1 / 2\) B. \(x / y>2\) C. \(x / y>1\) D. \(x-y>0\) E. \(x y>2\)

Short Answer

Expert verified
A, D, E

Step by step solution

01

- Understand the Problem

Given the inequalities, we need to determine which of the provided statements is true given that both conditions, \( x < -1 \) and \( y < -2 \), hold true.
02

- Evaluate Each Option

Go through each option one by one and see if it satisfies the given conditions.
03

- Check Option A

For Option A, \( x / y > 1 / 2 \). Given that \( x < -1 \) and \( y < -2 \), we express this condition as a quotient of two negative numbers. The division of two negative numbers is positive, but let's test if it is greater than \( 1/2 \). Assume \( x = -2 \) and \( y = -3 \). Then, \( x / y = (-2) / (-3) = 2 / 3 \), which is greater than \( 1/2 \). Hence, Option A can be true.
04

- Check Option B

For Option B, \( x / y > 2 \). Considering \( x < -1 \) and \( y < -2 \), using \( x = -2 \) and \( y = -3 \), then \( x / y = 2 / 3 \), which is not greater than 2. Hence, Option B is false.
05

- Check Option C

For Option C, \( x / y > 1 \). With \( x = -2 \) and \( y = -3 \), \( x / y = 2 / 3 \), which is not greater than 1. Hence, Option C is false.
06

- Check Option D

For Option D, \( x - y > 0 \). If \( x = -2 \) and \( y = -3 \), then \( x - y = -2 - (-3) = -2 + 3 = 1 \), which is positive, so Option D is true.
07

- Check Option E

For Option E, \( x y > 2 \). With \( x = -2 \) and \( y = -3 \), then \( x y = (-2) \cdot (-3) = 6 \), which is greater than 2. Hence, Option E is true.
08

- Final Conclusion

From evaluating each option, Options A, D, and E are true given the conditions \( x < -1 \) and \( y < -2 \).

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Key Concepts

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

Inequalities
Inequalities are expressions that define the relative size or order of two quantities. In this problem, we deal with two inequalities: \( x < -1 \) and \( y < -2 \). This means that \( x \) is any number less than -1 and \( y \) is any number less than -2. Understanding inequalities helps you evaluate the conditions and relate them to various scenarios that might hold true for the given values. When you see an inequality like \( x < -1 \), think of all possible values \( x \) can take, such as -2, -3, etc. Similarly, for \( y \), values like -3, -4 are considered. This foundational understanding is essential for assessing the given options correctly.
Logical Reasoning
Logical reasoning involves systematically evaluating given information to arrive at a conclusion. In this exercise, you use logical reasoning to check each option against the given inequalities. For example, to evaluate if \( x / y > 1/2 \), you must verify this by selecting values for \( x \) and \( y \) that satisfy the initial conditions. If \( x = -2 \) and \( y = -3 \):
  • Calculate \( x / y = (-2) / (-3) = 2 / 3 \).
  • Check if \( 2/3 > 1/2 \). This is true, making Option A potentially correct.
The logical steps ensure there is no leap in assumptions and each conclusion is well-supported by actual calculations.
Mathematical Evaluation
Mathematical evaluation is the process of working through calculations to verify or disprove statements. Here we do this by plugging in values and performing operations like division and multiplication. Consider checking Option D, \( x - y > 0 \):
  • Using \( x = -2 \) and \( y = -3 \): \( x - y = -2 - (-3) = -2 + 3 = 1 \).
  • Since 1 is greater than 0, Option D is validated.
Performing these calculations meticulously confirms the validity of each option. It is important as minor mistakes in operations can lead to incorrect conclusions.
Option Analysis
Option analysis involves breaking down each given answer to check its validity based on the provided conditions. Here's how to analyze all options in this problem:
  • For Option A \( ( x / y > 1 / 2 ) \): Test with values \( x = -2 \) and \( y = -3 \). Calculation: \( -2 / -3 = 2/3 > 1/2 \). True.
  • For Option B \( ( x / y > 2 ) \): Test same values. Calculation: \( 2/3 eq > 2 \). False.
  • For Option C \( ( x / y > 1 ) \): Test same values. Calculation: \( 2/3 eq > 1 \). False.
  • For Option D \( ( x - y > 0 ) \): Calculation: \( -2 - (-3) = 1 > 0 \). True.
  • For Option E \( ( x y > 2 ) \): Calculation: \( (-2) \times (-3) = 6 > 2 \). True.
Through this method, we determine that the correct options satisfying \( x < -1 \) and \( y < -2 \) are Options A, D, and E.

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