Question

Is \(x\) positive? (1) \(x^3<0\) (2) \(x^5<0\) 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 statement (1)

Statement (1) says that \( x^3 < 0 \. \). A cube of a number is negative only if the number itself is negative. Therefore, if \( x^3 < 0 \, \) then \, x < 0 \. \ Therefore, x is not positive.

Analyze statement (2)

Statement (2) says that \( x^5 < 0 \.\) A fifth power of a number is negative only if the number itself is negative. Therefore, if \( x^5 < 0 \), \( x \) must also be negative, and hence x is not positive.

Evaluate the sufficiency of statements

Since each statement alone can determine that \( x \) is not positive, either statement alone is sufficient to answer the question. Therefore, the answer is D: 1 alone or 2 alone.

Key concepts

Analytic Reasoning

Analytic reasoning involves breaking down complex problems into simpler parts and understanding the relationships between them. In the given GMAT problem, we analyze the statements to determine whether they are sufficient to answer the question, 'Is x positive?' By understanding that real numbers follow certain rules when raised to different powers, we can deduce the answer. For example, a negative number raised to an odd power will always result in a negative number. This logical approach helps us solve the problem effectively.

Positive and Negative Numbers

Understanding positive and negative numbers is crucial for solving algebraic problems. In this exercise, both statements given are inequalities involving powers of x. For statement (1), if we know that x cubed (\( x^3 \)) is less than zero, this means x itself must be negative. Similarly, in statement (2), if the fifth power of x (\( x^5 \)) is less than zero, x must also be negative. Remember:

- Any negative number raised to an odd power remains negative.

- Therefore, the initial condition, whether it is in the form of cubes or fifth powers, will always point to the fact that x is negative, implying x is not positive.

Algebraic Inequalities

Algebraic inequalities involve expressions where two sides are compared using inequality symbols like < and >. In this problem, both inequalities are in the form of powers of x. We evaluate these by:

- Considering the base value: If the result (\( x^3 \)) is negative, then x itself must be negative.

- Applying the property to higher powers: The same reasoning applies to (\( x^5 \)). This helps in establishing whether the statements are sufficient to answer the question. Understanding these fundamentals makes it easier to tackle similar problems.

Sufficiency Analysis

In GMAT problems, determining whether the information provided is sufficient to answer the question is crucial. Sufficiency analysis requires evaluating each statement independently and together to see if they provide enough information. In this example:

- Analyzing statement (1) tells us that x is not positive because (\( x^3 < 0 \)) implies x is negative.

- Analyzing statement (2) leads to the same conclusion since (\( x^5 < 0 \)) also implies x is negative.

- Hence, both statements independently are sufficient to determine that x is not positive. The sufficiency analysis ensures that we methodically verify each piece of information to arrive at a logical conclusion. Because each statement alone is sufficient, the answer is D: 1 alone or 2 alone.