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Chapter 3: Problem 38

Is \(x^3>y^3\) ? (1) \(x>y\) (2) \(x / y>0\) 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

Expert verified
C. 1 and 2 together (need both)

Step by step solution

01

Identify the Given Statements

Start by understanding the information given:(1) \(x > y\)(2) \(\frac{x}{y} > 0\)
02

Analyze Statement (1) Alone

Statement (1) tells us that \(x\) is greater than \(y\). If \(x\) and \(y\) are both positive or both negative, then \(x^3 > y^3\). However, if \(x\) and \(y\) have opposite signs, we cannot conclude \(x^3 > y^3\). Thus, statement (1) alone is insufficient.
03

Analyze Statement (2) Alone

Statement (2) tells us that \(\frac{x}{y} > 0\), which means that \(x\) and \(y\) are either both positive or both negative. However, knowing only this does not tell us anything about the magnitude comparison of \(x\) and \(y\). Therefore, statement (2) alone is insufficient.
04

Combine Statements (1) and (2)

Combine the information from both statements. From (1), \(x > y\), and from (2), \(x\) and \(y\) must be either both positive or both negative. This ensures that if \(x > y\), it follows that \(x^3 > y^3\) since cubing preserves the inequality when both numbers are of the same sign. Hence, combined statements are sufficient to conclude \(x^3 > y^3\).
05

Conclusion

Based on the analysis, both statements (1) and (2) together are necessary and sufficient to determine that \(x^3 > y^3\).

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Key Concepts

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

inequalities in algebra
Inequalities in algebra deal with comparing expressions using inequality signs like >, <, 鈮�, and 鈮�. In this exercise, we are asked if the inequality \(x^3 > y^3\) holds. Given that algebraic inequalities involve various rules and properties, it's crucial to break them down.

When analyzing inequalities:
  • If both sides of the inequality are positive or negative, we can raise both to the same power without changing the direction of the inequality unless the power is even.
  • If the signs are different, comparing powers can become tricky and might require more information.
Understanding these properties helps simplify and solve these problems. Practice and familiarity with these rules are key to mastering inequalities.
statement analysis in GMAT
Statement analysis in GMAT problems is about evaluating given information to determine whether the statements alone or in combination sufficiently answer the question. Here, we analyze two statements:

  • Statement (1): \( x > y \)
  • Statement (2): \( \frac{x}{y} > 0 \)
Each statement needs careful examination:

  • Statement (1) alone tells us one number is larger than the other, but not their signs.
  • Statement (2) alone implies both numbers have the same sign (positive or negative), but it's inadequate to determine which number is larger.
  • Combining both statements (1) and (2) allows us to conclude that cubic inequalities hold when both numbers have the same sign and one is greater in magnitude than the other.
GMAT problems often require combining information from multiple sources to derive a solution, underscoring the importance of critical thinking and detailed analysis.
quantitative reasoning GMAT
Quantitative reasoning on the GMAT involves solving problems that test arithmetic, algebra, and geometry skills. This problem asks us to compare cubic inequalities using given conditions.

To do well in quantitative reasoning, understand these strategies:

  • Break down the problem: Decompose complex problems into simpler parts, as seen here by separately analyzing each statement.
  • Identify key information: Determine what each statement tells you and what it doesn't. Recognizing the necessity for both positive and negative conditions was pivotal here.
  • Combine and conclude: After analyzing parts individually, synthesize the information to draw a holistic conclusion. Here, the combination of inequalities and sign information led to our answer.
Regular practice with GMAT-style questions improves problem-solving speed and accuracy, preparing you to tackle a variety of quantitative challenges.

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