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Magazine Chapter 8: Problem 6
If \(x
Short Answer
Expert verified
Option (E) must be true: \(\frac{y}{x} > \frac{x}{y}\).
Step by step solution
01
Analyze the Inequality Setup
We have the inequality \(x < y < -1\). This tells us that both \(x\) and \(y\) are negative numbers, with \(x\) being the smallest among them.
02
Evaluate Option (A)
Option (A) suggests \(\frac{x}{y} > x y\). Since both \(x\) and \(y\) are negative, \(\frac{x}{y}\) is positive and smaller in magnitude compared to \(x y\), which is positive but has a greater magnitude. Therefore, this statement is not always true.
03
Evaluate Option (B)
Option (B) suggests \(\frac{y}{x} > x+y\). \(\frac{y}{x}\) is positive, and \(x+y\) results in a neutral-to-negative value as both are negative. It seems possible, but not necessarily always true, depending on specific \(x\) and \(y\) values.
04
Evaluate Option (C)
Option (C) suggests \(\frac{y}{x} > x y\). While positive, \(\frac{y}{x}\) must be less (due to magnitude differences) than \(x y\) which is positive with \(-x \, y\). Not always true.
05
Evaluate Option (D)
Option (D) suggests \(\frac{y}{x} < x+y\). Similar to Step 3 analysis, \(\frac{y}{x}\) tends to be positive with smaller absolute value than a highly negative \(x+y\). Examination shows inconsistency, as often, \(\frac{y}{x}\) exceeds \(x+y\).
06
Evaluate Option (E)
Option (E) suggests \(\frac{y}{x} > \frac{x}{y}\). In this case, both fractions are positive; since \(x < y\), \(\frac{y^2}{x^2}>1\), making \(\frac{y}{x}\) larger than \(\frac{x}{y}\) consistently. This must be true.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Inequalities
Inequalities are mathematical expressions involving the symbols like less than (<), greater than (>), and their variants. In this exercise, we have an inequality: \(x < y < -1\). Inequalities show the relationship between two expressions, indicating one is smaller or larger than the other.
Understanding how to handle inequalities involves:
Understanding how to handle inequalities involves:
- Solving simple inequalities, such as with addition or subtraction.
- Multiplying or dividing both sides by a positive number keeps the inequality sign the same.
- Multiplying or dividing by a negative number reverses the inequality sign.
Negative Numbers
Negative numbers are numbers that are less than zero and are represented with a minus sign (-). In the context of the inequality \(x < y < -1\), both \(x\) and \(y\) are negative and less than -1.
When dealing with negative numbers:
When dealing with negative numbers:
- Adding two negative numbers results in a more negative number.
- The product of two negative numbers is positive.
- Dividing two negative numbers also gives a positive result.
- Among negative numbers, the one with the higher absolute value is actually smaller when compared (e.g., \(-3 < -1\)).
Algebraic Expressions
An algebraic expression is a combination of numbers, variables, and operations. In this exercise, we interact with expressions like \(\frac{x}{y}\), \(x y\), and \(x+y\). Each has its nuances especially when combined with inequalities and negative numbers.
When dealing with algebraic expressions in inequalities:
When dealing with algebraic expressions in inequalities:
- Remember to follow the hierarchy of operations: parentheses, exponents, multiplication/division, and then addition/subtraction.
- Be careful with expressions involving division by a variable that could be negative, as it may impact the inequality direction.
- Handling variable-based expressions involves careful substitution and simplification.
Fractions
Fractions are a part of algebraic expressions where one quantity is divided by another. In our problem, we encounter fractions such as \(\frac{x}{y}\) and \(\frac{y}{x}\). Working with fractions typically involves:
- Finding a common denominator when adding or subtracting fractions.
- Multiplication and division are straightforward, involving multiplying numerators and denominators.
- A fraction represents a division, where the sign changes only if the numerator or denominator is negative.
- Fractions like \(\frac{y}{x}\) can provide insights into the relationships between variables. It helps to know if fractions stay positive or negative based on the signs of numerator and denominator.
