Question

Is \((x-y)(x+y)>0\) ? (1) Both \(x\) and \(y\) are positive (2) \(x>y\) 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

Understand the Question

Determine the condition under which \((x-y)(x+y)>0\). Break down the problem using given statements.

Analyze Statement 1

Statement 1: Both \(x\) and \(y\) are positive. If both variables are positive, \(x+y>0\) and \(x-y\) also potentially can be positive if \(x>y\). However, this statement alone doesn't guarantee that \(x>y\). So this statement alone is not sufficient.

Analyze Statement 2

Statement 2: \(x>y\). This implies that \(x-y>0\). However, for \( (x-y)(x+y)>0 \), we also need to ensure \( x+y>0 \). Since \( x \) and \( y \) can be of opposite signs (e.g., \( x=1 \), \( y=-2 \)), this statement alone is also insufficient.

Combine Statements 1 and 2

With both statements 1 and 2, we know \( x \) and \( y \) are positive and \( x>y \). Therefore, \( x+y>0 \) and \( x-y>0 \), making \( (x-y)(x+y)>0 \) always true. Both statements together are sufficient.

Key concepts

inequality analysis

Inequality analysis is a crucial concept in GMAT problem solving. In this question, we must determine the values under which (x-y)(x+y)>0 . For this to hold, two conditions must be met: . This means both expressions, (x+y) and x-y, must be either both positive or both negative. Keep in mind the behavior of positive and negative numbers when analyzing inequalities. Analyzing statement 1: Both x and y are positive. This guarantees that (x+y)>0. However, without knowing the relationship between x and y, we cannot say if x-y >0. Therefore, statement 1 alone is not sufficient. Analyzing statement 2: x>y. This tells us x-y>0. But, it doesn't tell us if x+y >0 since x or y could be negative. Therefore, statement 2 alone is not sufficient.

data sufficiency

Data sufficiency is a unique question type in the GMAT. It requires you to analyze given statements to determine if they provide enough information to solve the problem. In our exercise, we are given two statements. Statement 1: Both x and y are positive. Statement 2: x > y. To decide if they are sufficient to answer if (x-y)(x+y)>0, we need to evaluate each statement individually and then together. First, we analyzed each statement alone: * Statement 1 alone: Tells us both numbers are positive but nothing about the relative size. It’s not sufficient. * Statement 2 alone: Tells us the relationship between x and y, but not their signs. It’s not sufficient. When we combine both statements, we know x and y are positive (from statement 1), and x > y (from statement 2). Now we have enough information. Combining the two statements ensures both (x+y) and x-y are positive, implying (x-y)(x+y) > 0 is true. Thus, both statements together are sufficient. Answer C is correct.

mathematical reasoning

Mathematical reasoning involves using logical and methodical steps to solve problems. For this problem, we use algebraic manipulation and logic to determine when (x-y)(x+y) > 0. We start by understanding what the product of two expressions greater than zero means. This means both expressions must be either both positive or both negative. Now, let’s reason through the problem: * If both x and y are positive (statement 1), we have (x+y)>0. However, we also need to know if x>y to ensure (x-y)>0. * If x>y (statement 2), this ensures (x-y)>0. But without knowing if x and y are both positive, (x+y) could be any value. By logically combining both pieces of information, we ascertain that: - From statement 1: x>0 and y>0 - From statement 2: x>y When both statements are combined, we can conclude both are positive and x>y, meeting all conditions for (x-y)(x+y)>0 to be true. Thus, using mathematical reasoning and combining pieces of information leads us to the correct solution. Combining both statements allows us to confidently determine the truth of the inequality. Answer C is correct.