Question

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

E. 1 and 2 together are not sufficient

Step-by-step solution

Analyze Statement (1)

Given that statement (1) is: \(y > x^3\). If \(y > x^3\), it means that \(y\) is greater than the value of \(x^3\). To decide whether this helps us determine \(y > x^2\), compare \(x^3\) with \(x^2\) under different conditions.

Compare \(x^3\) and \(x^2\)

For \(x > 1\), notice that \(x^3 > x^2\). For \(0 < x < 1\), \(x^3 < x^2\). Therefore: - When \(x > 1\), \(y > x^3 > x^2\), so \(y > x^2\). - When \(0 < x < 1\), \(y > x^2\) is not necessarily true.- When \(x = 1\), \(x^3 = x^2 = 1\), and since \(y > x^3 = 1\), it follows that \(y > x^2 = 1\). - For \(x \leq 0\), \(x^3\) and \(x^2\) relationship might make outcome different.

Determine Sufficiency of Statement (1)

Based on the analysis, statement (1) alone is not sufficient to definitively conclude that \(y > x^2\) holds for all \(x\) because in the range \(0 < x < 1\), \(y > x^2\) is not guaranteed.

Analyze Statement (2)

Given statement (2): \(x > 0\). This only restricts \(x\) to positive values but provides no direct relationship between \(y\) and \(x^2\).

Determine Sufficiency of Statement (2)

Statement (2) alone does not give any information about \(y\), so it is not sufficient to conclude \(y > x^2\).

Combine Statements (1) and (2)

Using both statements together: - From statement (1), \(y > x^3\). - From statement (2), \(x > 0\). Here, combine the conditions where \(x > 0\) and review the results:- For any \(x > 1\), \(x^3 > x^2\), leading to \(y > x^3 > x^2\), so \(y > x^2\).- For \(0 < x < 1\), \(x^3 < x^2\) and hence, \(y > x^3\) does not guarantee \(y > x^2\). Results still inconclusive to prove \(y > x^2\) for all positive \(x\).

Conclusion

Both statements (1) and (2) together are not sufficient to conclude \(y > x^2\) for all \(x\). Neither are alone sufficient. Therefore, based on (1) and (2), it is not possible to definitively assert \(y > x^2\).

Key concepts

Inequalities

Understanding inequalities is crucial for GMAT problem solving. An inequality compares two values, showing if one is greater, less, or not equal to the other. In algebraic terms, expressions like \(y > x^2\) or \(y < x^3\) indicate that the value of \(y\) is respectively greater or less than the algebraic expressions on the other side.
Inequalities are not just about finding which value is larger or smaller, but also understanding the nature and range of possible values. For instance, if \(x > 0\), it sets a range of possible values for \(x\) that impacts the outcomes of any inequality involving \(x\).
Although they may seem simple, inequalities require careful analysis, especially when dealing with algebraic expressions. Comparing \(x^3\) and \(x^2\) can be tricky because their relationship changes based on the value of \(x\). For example, when \(x > 1\), \(x^3 > x^2\), while for \(0 < x < 1\), \(x^3 < x^2\). Always consider multiple scenarios to fully understand inequalities in an equation.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations like addition and multiplication. They represent mathematical relationships without an equality sign. In the problem above, expressions like \(x^2\) and \(x^3\) are crucial.
Understanding algebraic expressions helps in comparing and manipulating these relationships. For instance, knowing how \(x^2\) and \(x^3\) differ depending on whether \(x\) is positive or negative, or greater or less than one, helps you evaluate statements involving these terms. You must break down and manipulate these expressions to see how they influence each other.
Let’s take another look at the exercise: To determine if \(y > x^2\) with the information that \(y > x^3\), consider the different behavior of \(x^3\) and \(x^2\) over different intervals of \(x\). This requires not only solving for \(y\) and comparing values but also understanding the underlying properties of the expressions.

Sufficiency Analysis

Sufficiency analysis is key in GMAT questions. It involves determining whether the given information is enough to solve a problem. In our exercise, we need to check if the statements alone or combined are enough to definitively answer the question \(y > x^2\).
First, look at each statement individually:

• Statement (1): Given \(y > x^3\), we observe that for some values; \(0 < x < 1\), \(x^3 < x^2\), thus, \(y > x^2\) is not guaranteed.
• Statement (2): Given \(x > 0\), this does not directly link \(y\) and \(x^2\). It simply tells us \(x\) is positive.

When statements are combined: they still don’t conclusively determine if \(y > x^2\) for all enforced conditions. The combined statement also leaves out scenarios where the comparison might not hold. Therefore, both statements together are not sufficient.
This type of logical breakdown is what sufficiency analysis in GMAT entails – dissecting given statements to see if they provide enough information to answer the central question without assumptions.

GMAT Preparation

Preparing for the GMAT involves mastering various concepts and problem-solving techniques. Here's how to tackle topics like inequalities, algebraic expressions, and sufficiency analysis:

• **Practice Regularly**: Consistent practice with diverse problems helps understand underlying concepts and improves accuracy and speed.
• **Understand Core Concepts**: Make sure you fully grasp foundational ideas. For example, recognizing how algebraic expressions like \(x^2\) and \(x^3\) behave under different conditions is crucial.
• **Apply Logical Reasoning**: Learn to logically deduce information from given statements. This is particularly important for sufficiency analysis questions in data sufficiency sections.
• **Review Solutions Carefully**: After solving an exercise, always review each step of the solution. Understand why each step is taken and how it leads to the final answer.
• **Focus on Weak Areas**: Identify and spend extra time on areas where you're struggling, whether it’s inequalities, algebra, or another topic.

The GMAT is not just about knowing the right formulas but also about applying logical reasoning and analytical skills to solve problems effectively.