Question

Is \(1 / x>1 / y ?\) (1) \((1 / x)-(1 / y)>0\) (2) \((y-x) /(x y)>0\) A. 1 alone, not 2 alone B. 2 alone, not 1 alone C. 1 and 2 together (need both) D. 1 alone or 2 alone E. 1 and 2 together are not sufficient

Short answer

D. 1 alone or 2 alone

Step-by-step solution

- Analyze Condition 1

Start by analyzing the first condition: \( \frac{1}{x} - \frac{1}{y} > 0 \). This can be rewritten as \( \frac{1}{x} > \frac{1}{y} \), suggesting that \( x < y \).

- Analyze Condition 2

Now, analyze the second condition: \( \frac{y - x}{xy} > 0 \). Since \(xy \) is positive (as division maintains the sign of \(y - x\)), \(y - x \) must also be positive. Hence, \( y > x \).

- Combine Conditions

Both conditions suggest \( y > x \), implying \( \frac{1}{x} > \frac{1}{y} \). Therefore, each condition alone is sufficient to confirm that \( \frac{1}{x} > \frac{1}{y} \).

- Conclusion

From the analysis, either condition 1 or condition 2 alone is sufficient to conclude that \( \frac{1}{x} > \frac{1}{y} \). Thus, the correct answer is D.

Key concepts

inequalities

Inequalities form an essential part of mathematical reasoning and GMAT problem solving. They help determine the relationship between values that are not necessarily equal. In this exercise, we focus on the inequality \(\frac{1}{x} > \frac{1}{y}\). This signifies that \(\frac{1}{x}\) is greater than \(\frac{1}{y}\). When dealing with inequalities, consider whether you're working with positive or negative values, as the implications can differ significantly. For example, if \x\ and \y\ are both positive, then \x < y\ would imply \(\frac{1}{x} > \frac{1}{y}\). This relationship is vital for understanding how to manipulate and solve the inequality conditions given in the exercise.

mathematical conditions

Mathematical conditions are scenarios expressed in equations or inequalities that must hold true. In our exercise, we have two conditions: \(\frac{1}{x} - \frac{1}{y} > 0 \) and \(\frac{y-x}{xy} > 0\).
The first condition, \(\frac{1}{x} - \frac{1}{y} > 0\), can be rearranged to \(\frac{1}{x} > \frac{1}{y}\), indicating \x < y\ when \x\ and \y\ are positive. The second condition, \(\frac{y-x}{xy} > 0\), suggests \y - x\ should be positive and thus \y > x\.
Both conditions essentially express the same relationship but in different forms. Understanding these conditions and their implications is crucial for verifying the provided answer and determining its correctness.

quantitative reasoning

Quantitative reasoning involves logical thinking to solve problems and make decisions involving numbers. The primary goal is to understand and evaluate mathematical scenarios. This exercise requires you to translate and interpret given conditions using algebra.
By analyzing both conditions, you need to deduce whether \(\frac{1}{x} > \frac{1}{y}\) holds true. Condition 1, when simplified, directly points to \x < y\. Condition 2 needs a bit more manipulation but ultimately indicates \y > x\. Understanding how these deductions support each other shows how quantitative reasoning can simplify complex problems into manageable steps.

fraction comparison

Comparing fractions is a crucial skill in many quantitative problems. In this exercise, we compare \(\frac{1}{x}\) with \(\frac{1}{y}\). When x and y are positive, smaller values of x result in larger values of \(\frac{1}{x}\). Conversely, smaller values of y lead to smaller fractions.
The exercise shows that \(\frac{1}{x} > \frac{1}{y}\) if \x < y\. By using algebraic manipulation and logical reasoning, you can confirm that each condition given holds true independently. This highlights the consistency in comparing fractions and reinforces your understanding of how to approach similar problems.