Question

Is \(\mathrm{x}\) negative? (1) \(x y<1\) (2) \(y\) is positive 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

Understand Statement (1)

Statement (1) tells us that the product of x and y is less than 1. This inequality, \(xy < 1\), needs to be analyzed to understand if x is definitely negative.

Analyze Statement (1)

Given \(xy < 1\), x can be either positive or negative depending on the value of y. For example, if y is a small positive number (such as 0.1), x could be a larger positive number less than 10 (like 5) and still satisfy \(xy < 1\). However, it’s also possible for y to be a larger negative number (such as -0.1), in which case x could be a larger negative number (like -5) and still satisfy \(xy < 1\). Thus, statement (1) does not alone determine if x is negative.

Understand Statement (2)

Statement (2) tells us that y is positive. This gives us a specific property about y but does not provide any information directly about x.

Analyze the Combination of Statements (1) and (2)

Now, combining both statements: From statement (1) \(xy < 1\) and from statement (2) \(y > 0\). Since y is positive, for the product \(xy\) to be less than 1, x must necessarily be less than 1/y. Since y is positive, we evaluate \(x < 1/y\). Depending on the value of y, x can be either positive or negative. For example, if y = 2 (positive), x can be less than 0.5 (positive). Thus, even with both statements together, we can't be certain if x is negative.

Conclusion

Since neither statement alone nor combined can definitively determine if x is negative, the answer is that the information provided is not sufficient.

Key concepts

inequality analysis

To solve the problem, understanding inequalities is crucial. An inequality compares two values using symbols like <, >, ≤, or ≥. In our exercise, we use the inequality \( xy < 1 \). This tells us that the product of x and y is less than 1. Analyzing such inequalities requires exploring different scenarios:

For positive values of y, x must be less than \( \frac{1}{y} \) to keep their product below 1.
For negative values of y, x must be greater than \( \frac{1}{y} \), since two negative numbers multiply to a positive product.
This analysis helps evaluate if x can be both positive and negative, based on the value of y.

data sufficiency

Data sufficiency questions test your ability to determine if given information is enough to answer a question without necessarily solving it. The key steps are:

Evaluate each statement individually.
Combine statements if necessary.
In our example, we have two statements: \( xy < 1 \) and \( y > 0 \). Analyzing each:
Statement (1) alone: Given \( xy < 1 \), x can be either positive or negative.
Statement (2) alone: Asserts that y is positive but gives no information about x.
Combining both: Knowing \( xy < 1 \) and \( y > 0 \) still leaves x uncertain since x can be either positive or negative to satisfy \( xy < 1 \). Thus, neither alone nor combined are sufficient to determine if x is negative.

mathematical reasoning

Mathematical reasoning plays a vital role in solving GMAT problems. For this question, we need to reason through:

What does \( xy < 1 \) imply about x if y is positive?
If y is positive, then x must be less than \( \frac{1}{y} \). For example, if y = 0.1, x can be any value less than 10.
However, x could still be either positive or negative (e.g., 0.5 or -0.5 if y is 2) to satisfy \( xy < 1 \).
Generalizing, if y is positive, \( x < \frac{1}{y} \). Thus, x can be both positive and negative. This reasoning is crucial to understand why the combined statements still do not answer if x is negative.

problem-solving steps

Breaking down GMAT problems into defined steps makes them easier to handle. For this problem:

Step 1: Understand Statement (1) \( xy < 1 \) and realize x could be either positive or negative.
Step 2: Analyze Statement (1) with different y values.
Step 3: Understand Statement (2) tells us y is positive.
Step 4: Combine both statements, but realize x can still be positive or negative.
Step 5: Conclude neither statement alone nor together sufficiently determine if x is negative.
Following these structured steps helps ensure no details are overlooked and the logic remains clear throughout.