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Chapter 3: Problem 25

Which of these two is the greater, \(x\) or \(y\) ? (1) \(x / 3>y\) (2) \(x^3>y^3\) 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
D. 1 alone or 2 alone

Step by step solution

01

Analyze Statement 1

Given the inequality \( x / 3 > y \), multiply both sides by 3 to isolate \( x \). This yields \( x > 3y \). This statement provides a direct comparison between \( x \) and \( y \). Hence, \( x \) is always greater than \( y \).
02

Analyze Statement 2

Given the inequality \( x^3 > y^3 \), take the cube root of both sides to obtain \( x > y \). This statement again provides a direct comparison between \( x \) and \( y \). So, \( x \) is always greater than \( y \) under this statement as well.
03

Conclusion

Both statements independently show that \( x \) is greater than \( y \). Thus, either statement alone is sufficient to determine that \( x \) is greater than \( y \).

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Key Concepts

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

GMAT Preparation
When preparing for the GMAT, especially the quantitative section, you will frequently encounter inequality problems like the one in this exercise. It's crucial to understand how to manipulate and compare inequalities to confidently answer these questions. Here are some tips to improve your preparation:

  • Know the Basics: Make sure you're comfortable with basic algebraic rules and properties of inequalities.
  • Practice Regularly: The more problems you solve, the more familiar you'll become with different types of questions.
  • Tackle Weak Areas: Identify the topics you find difficult and spend extra time on those.
  • Simulate Test Conditions: Practice solving problems under timed conditions to build speed and accuracy.
Using these strategies will help you prepare effectively and approach each inequality problem with confidence.
Inequality Comparison
Inequalities are all about determining the relative sizes of quantities. You are often required to compare two expressions, like in the exercise where you need to find whether x or y is greater.

Here are some key points to remember when comparing inequalities:

  • Isolate the Variable: Try to get the variable on one side of the inequality. For example, in the exercise, we started with \( \frac{x}{3} > y \) and multiplied both sides by 3 to isolate x.
  • Maintain Inequality Direction: Be cautious when multiplying or dividing inequalities by negative numbers, as this will reverse the inequality direction.
  • Understand Transitive Property: If \( a > b \) and \( b > c \), then \( a > c \). Use this to combine inequalities logically.
  • Cube Roots and Powers: When dealing with polynomials or exponential inequalities, remember that raising to an odd power or taking an odd root maintains the inequality's direction. For instance, from \( x^3 > y^3 \), taking cube roots yields \( x > y \) directly.
Comparing inequalities becomes straightforward once you remember these foundational tips.
Algebraic Manipulation
Algebraic manipulation is a valuable skill that helps you transform equations or inequalities to a more solvable form. In this exercise, you saw how algebraic steps can simplify the problem.

Let's break down some critical steps in algebraic manipulation:

  • Simplify Expressions: Combine like terms, cancel out terms, or factor expressions to simplify them. This helps you understand the problem better.
  • Multiplication and Division: Use these operations to isolate variables. In the exercise, multiplying both sides of \(x / 3 > y\) by 3 gave us \(x > 3y\).
  • Exponentiation and Roots: Apply exponents or roots carefully. As shown, taking the cube root of \(x^3 > y^3\) simplified it to \(x > y\).
  • Logical Steps: Follow a logical flow of steps to maintain the integrity of the inequality or equation throughout.
Mastering these techniques requires practice and understanding. Focus on these principles, and you'll find algebraic manipulation much easier. This will aid you significantly in solving GMAT problems efficiently.

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Most popular questions from this chapter

Which of these two is the greater, \(x\) or \(y\) ? (1) \(x=2 a\) (2) \(y=5 \mathrm{a}\) 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

If a ball bearing is dropped into a beaker of water 3 cm deep, by how much does the water rise? (1) The radius of the ball bearing is \(7 \mathrm{~mm}\) (2) The radius of the beaker is \(5 \mathrm{~cm}\) 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

Is the quadrilateral a square? (1) All angles equal \(90^{\circ}\) (2) The diagonals divide the shape into 4 identical triangles 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

The cake was only three-quarters as big as the last one, and to make it the children used an extra one and half eggs and half the sugar. A. three-quarters as big as the last one, and to make it the children used an extra one and half B. a three-quarters as big as the last one, and to make it the children used an extra one and a half C. three-quarters as big as the last one, and to make it the children used an extra one and a half D. a three-quarters as big as the last one, and to make it the children used an extra one and half E. three-quarters of as big as the last one, and to make it the children used an extra one and half

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