Question

Is \((x / y)^2>1\) ? (1) \(y x

Short answer

E. 1 and 2 together are not sufficient

Step-by-step solution

- Understand the Problem

Determine if \((x / y)^2\) is greater than 1 given two conditions: (1) \(yx < x^2\) and (2) both \(x\) and \(y\) have the same sign.

- Analyze Statement 1

Condition (1) states that \(yx < x^2\). Dividing both sides by \(y^2\) (assuming \(y eq 0\)), we get \((x / y)^2 > 1 \) if \(x^2 < yz \), which doesn't always hold true. So, statement 1 alone doesn't guarantee that \((x / y)^2 > 1\).

- Analyze Statement 2

Condition (2) states that both \(x\) and \(y\) have the same sign. If \(x\) and \(y\) are both positive or both negative, then \((x / y)^2\) is the square of a positive number, which is always greater than 1. Statement 2 alone is not sufficient because it doesn't tell us anything about the magnitude of \(x\) and \(y\).

- Combine Statements 1 and 2

When combining both statements, we know that \(yx < x^2\) and both \(x\) and \(y\) have the same sign. Since both variables have the same sign, \((x / y)^2\) is positive. However, statement 1 is still not sufficient to conclude that \((x / y)^2\) is indeed greater than 1 for all values of \(x\) and \(y\) as the magnitude relationships are not fully covered.

- Conclude

Neither statement alone nor together allows us to definitively conclude that \((x / y)^2\) is greater than 1 under all cases. Therefore, both statements together are not sufficient.

Key concepts

Inequalities

Understanding inequalities is crucial for solving algebraic problems involving comparisons of expressions. In this exercise, you need to determine if \((x / y)^2 > 1\). Inequalities like this focus on the relative size of two quantities.

In this problem, to determine whether \((x / y)^2 > 1\), we need to analyze given conditions and manipulate the inequality step by step. Inequalities are not always straightforward, and it's essential to consider both the direction of the inequality and any possible zeroes or negative values involved.

Algebra

Algebra is all about manipulating equations and expressions to solve for unknowns, like variables x and y in our case. The given statements provide conditions involving algebraic expressions.

For instance, statement (1) gives us \( yx < x^2\). To understand this better, let's divide both sides by \( y^2 \) assuming \( y eq 0 \), resulting in \( (x / y)^2 > 1 \) if \( x^2 < y^2 \). However, this requires careful handling since the provided conditions may not simplify directly to \( (x / y)^2 > 1 \).

In statement (2), we know that x and y share the same sign. This implies that \( (x / y)^2 \) is positive but doesn’t specify its value compared to 1. Algebra will often involve such transformations to derive new equations or simplify the inequalities you’re working with. Thus, understanding and manipulating these algebraic conditions are crucial steps in the solution.

Data Sufficiency

Data sufficiency questions are designed to test your ability to determine whether the provided information is enough to solve a problem. Here, we need to check if the given statements independently or together can conclude \( (x / y)^2 > 1\).

In statement (1), \( yx < x^2 \), alone isn’t sufficient because we cannot guarantee \( (x / y)^2 > 1 \) for all possible values of x and y.

Similarly, statement (2), that both x and y have the same sign, doesn’t provide enough information about their magnitudes, making it insufficient by itself.

Combining both statements leads us to positive \( (x / y)^2 \), but even together, they do not cover all scenarios necessary to conclude definitively. In GMAT data sufficiency, understanding when data is complete and when it’s not is key to solving these decisively.

Math Reasoning

Math reasoning involves logical thinking to interpret and solve math problems. Here, we apply reasoning to inequalities and algebra to determine the sufficiency of conditions.

By systematically analyzing each statement, we employ logic to see if the problem’s requirements are met. For instance, determining \( (x / y)^2 > 1 \) from \( yx < x^2 \) logically leads us to weigh both implications and assumptions of the signs and magnitudes.

Math reasoning also means verifying each condition, like combining statements and checking possible exceptions or contradictions to ensure they fit all scenarios. In this exercise, both our algebraic manipulations and data sufficiency checks require thorough math reasoning to conclude that neither statements alone nor together are sufficient.