Question

Is \(x\) negative? (1) \(x y=3\) (2) \(x^2=y\) 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

C. 1 and 2 together (need both)

Step-by-step solution

Analyzing Statement (1)

Statement (1) provides the equation: \[ xy = 3 \] To determine if this tells us if \( x \) is negative, we note that \( x \) and \( y \) could be either positive or negative. Possible cases include both \( x \) and \( y \) being positive (making their product positive), and both being negative (which also makes their product positive). Therefore, knowing \( xy = 3 \) is not enough to determine the sign of \( x \).

Analyzing Statement (2)

Statement (2) provides the equation: \[ x^2 = y \] This tells us that \( y \) is always non-negative since \( x^2 \) is non-negative for all real numbers. However, \( x \) could be either positive or negative (e.g., both \( 2^2 \) and \( (-2)^2 \) equal 4). Therefore, this statement alone does not dictate whether \( x \) is negative.

Combining Statements (1) and (2)

Combining statements (1) and (2), we have: 1) \( xy = 3 \) 2) \( x^2 = y \) Substituting \( y \) from statement (2) into (1), we get: \[ x \times x^2 = 3 \] simplifying, we obtain: \[ x^3 = 3 \] Taking the cube root of both sides, we find: \[ x = \root 3 \rates{3} = \root{3}{3} \] Cube root of a positive number is always positive, thus \( x \) is positive, so it is not negative.

Conclusion

From the combined information, we have determined that \( x \) is positive, so it cannot be negative. Therefore, both statements together are sufficient to answer the question.

Key concepts

GMAT Algebra

Understanding algebra is crucial for success in the GMAT exam. Algebra comes into play in various question types, such as problem-solving and data sufficiency. In this specific problem, we deal with algebraic equations to determine if a variable, let's call it x, is negative. To solve such questions, you must know how to manipulate equations and understand properties of numbers. In our example, analyzing the equations given is essential for deducing whether the variable is negative or positive. When combining the equations properly, we see they reveal critical information about the value of x, ultimately leading us to conclude whether x is positive or negative. Being comfortable with algebraic manipulation is key to solving these types of GMAT problems efficiently.

GMAT Data Sufficiency

Data sufficiency questions on the GMAT are designed to test your ability to analyze given information and determine whether it is enough to solve a problem. These questions often present two statements and ask if either statement alone or both statements together can answer the question at hand. In our problem, the statements provide equations involving x and y. First, we examine each statement independently to see if it provides enough information to deduce whether x is negative. Next, we combine both statements to see if together they offer sufficient data to answer the question. The core skill in data sufficiency is not just in solving the equations but knowing when to combine the information from both statements effectively.

Graduate Management Test Preparation

GMAT preparation is critical for aspiring business school students. Understanding how to tackle different types of questions can significantly improve your score. The typical question types include problem-solving and data sufficiency. Data sufficiency questions specifically can be tricky, as you must decide the sufficiency of the provided data rather than just solving a problem. For effective prep, practice is key. Continuously practice GMAT-style questions,especially focusing on algebra and data sufficiency sections. Use these practice sessions to familiarize yourself with common question patterns and develop strategies for how to approach each problem. Remember, the goal is not just to find an answer but to determine the quickest and most efficient path to see if you have enough information to solve the problem at all. Good preparation involves honing these skills to perform confidently and accurately under timed conditions.