Chapter 3: Problem 16
If \(x>1\) and \(y>-1\), then:
(A) \(x y>-1\)
(B) \(x y<-1\)
(C) \(-x>y\)
(D) \(-x
Short Answer
Expert verified
Option (D)
Step by step solution
01
- Analyze the given inequalities
The problem statement provides the inequalities: 1. \(x > 1\)2. \(y > -1\)
02
- Option (A) Analysis
To check if \(xy > -1\), we need to analyze the product of \(x\) and \(y\). Since \(x > 1\) and \(y > -1\), multiplying these can give us various values. However, as \(x\) is positive and greater than 1,\(xy\) can still be negative if \(y\) is close to but greater than \(-1\). Therefore, \(xy > -1\) cannot be guaranteed.
03
- Option (B) Analysis
Next, check if \(xy < -1\). From the given conditions, \(y\) can be a very small negative number such that \(xy\) becomes less than \(-1\). So, this option can be possible.
04
- Option (C) and (D) Analysis
For option (C), check if \(-x > y\): Since \(x > 1\), \(-x < -1\) and the inequality is not possible because \(y > -1\). For option (D), check if \(-x < y\): Since \(-x < -1\) and \(y > -1\), which satisfies the inequality.
05
- Option (E) Analysis
Finally, check if \(x < y\): Since \(x > 1\) and \(y > -1\) doesn't necessarily give a direct relationship to \(x\) being less than \(y\), this inequality cannot be guaranteed.
06
Conclusion
From the analyses, option (D) \(-x < y\) is the only one that is guaranteed to be true given the conditions.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Understanding Inequalities
Inequalities are mathematical expressions that show the relative size or order of two values. In this problem, we have two inequalities: \(x > 1\) and \(y > -1\). These inequalities tell us that \(x\) is any number greater than 1 and \(y\) is any number greater than -1. This means \(x\) could be any positive number greater than 1, such as 2, 3, or even 100, while \(y\) could be a small positive number or a slight negative number like -0.1. Understanding how these ranges interact helps us solve the given problem effectively. Remember, solving inequalities often involves looking at the boundaries and considering various possible values to test each option.
Product of Variables
When dealing with inequalities involving the product of variables, we multiply the given expressions. Here, we are asked to analyze \(xy\) given that \(x > 1\) and \(y > -1\). This analysis involves understanding how different values of \(x\) and \(y\) affect the product. For example, if \(x\) is 2 (since 2 > 1) and \(y\) is -0.5 (since -0.5 > -1), then \(xy \) would be 2 * -0.5 = -1, showing that \(xy\) can be exactly \(-1\). If we take even larger values for \(x\), like 3, and smaller values for \(y\), like -0.9, then \(xy\) becomes 3 * -0.9 = -2.7, showing the product can indeed be less than \(-1\). Thus, examining the product helps infer the possible outcomes for given inequalities.
Logical Reasoning
Logical reasoning is crucial when interpreting inequalities and arriving at conclusions about their implications. This requires systematically evaluating each option against the given conditions. For example, let's look at Option (D) from the given problem: \(-x < y\). Given \(x > 1\), we can rewrite this inequality as \(-x\) being less than \(-1\). Since \(y > -1\), the inequality \(-x < y\) holds true because if \(-x < -1\) and \(-1 < y\), it follows that \(-x < y\). Applying logical steps like these helps ensure comprehensive evaluation of all possible outcomes and avoids errors.
Mathematical Analysis
Mathematical analysis involves detailed breakdown and examination of the problem to detect underlying patterns or contradictions. Initiating the analysis with a set of given inequalities, \(x > 1\) and \(y > -1\), various values of \(x\) and \(y\) should be tested. For example, checking whether \(xy > -1\) requires pinpointing scenarios where the product \(xy\) could be around the boundary (e.g., \(x = 2\), \(y = -0.6\)). Through such experimentation and rearrangement of variables, mathematical analysis helps ascertain statements' truthfulness. This methodical approach underscores the importance of thoroughly examining each scenario for robust mathematical reasoning.