Question

Which of these two is the greater, \(x\) or \(y\) ? (1) \(1 / x>1 / y\) (2) Both are negative 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 Inequality Condition (1)

The inequality given in condition (1) is \( \frac{1}{x} > \frac{1}{y} \).

Transform the Inequality

Multiplying both sides of \( \frac{1}{x} > \frac{1}{y} \) by \( xy \) (keeping in mind we don't change inequality direction because we don't know the signs yet), we get \( y > x \) assuming \( x \) and \( y \) are of the same sign.

Interpret Condition (2)

Condition (2) states that both \( x \) and \( y \) are negative numbers. This is crucial since it determines the behavior of the inequality.

Combine Conditions (1) and (2)

Given condition (1), \( \frac{1}{x} > \frac{1}{y} \), and condition (2), both \( x \) and \( y \) are negative, we realize that multiplying or dividing by a negative value will reverse the inequality. Therefore, since \( x \) and \( y \) are negative, \( y > x \) from condition (1) becomes \( x > y \) when adjusting for sign conversion.

Conclusion

With both conditions, we find that \( x \) is greater than \( y \). Therefore, both conditions are necessary to determine the relationship between \( x \) and \( y \).

Key concepts

Understanding Inequalities

Inequalities compare two values, showing if one is less than, greater than, or equal to another. For example, the inequality \( \frac{1}{x} > \frac{1}{y} \) tells us that the reciprocal of \( x \) is greater than the reciprocal of \( y \).

It's essential to understand how inequalities change under certain operations, such as multiplication or division. If you multiply or divide both sides of an inequality by a positive number, the direction of the inequality stays the same. However, if you multiply or divide by a negative number, the inequality direction switches.

When tackling GMAT quantitative reasoning problems involving inequalities, it helps to:

• Clear fractions by multiplying both sides by the denominators,
• Be mindful of the sign of the numbers involved,
• Carefully apply the rules for flipping inequalities when dealing with negative numbers.

Working with Negative Numbers

Negative numbers can often complicate inequalities. Remember a negative number is any number less than zero, like \( -1 \), \( -2.5 \), or \( -100 \).

When you multiply or divide both sides of an inequality by a negative number, you must reverse the inequality symbol. For example, if you start with \( x < y \) and both \( x \) and \( y \) are negative, multiplying both sides by \( -1 \) changes the inequality to \( -x > -y \).

In the exercise, since both \( x \) and \( y \) are given as negative numbers, this reversal rule is crucial. If we know that \( \frac{1}{x} > \frac{1}{y} \), then upon considering the signs of \( x \) and \( y \), we determine that \( x > y \). So, understanding how to handle negative values is key to solving inequality problems.

Multiple Conditions in Problem Solving

Sometimes, a problem provides multiple conditions, and you need to consider them together to find a correct solution. In our exercise, we had condition (1) \( \frac{1}{x} > \frac{1}{y} \) and condition (2) stating both \( x \) and \( y \) are negative.

By just looking at condition (1), we might conclude \( y > x \) assuming \( x \) and \( y \) are positive. But, condition (2) tells us they're negative, flipping our understanding to \( x > y \).

It is crucial to:

• Analyze each condition separately first,
• Combine the information from all conditions,
• Apply fundamental rules of arithmetic and algebra consistently.

Using both conditions together provided the correct answer, making the combined conditions necessary to solve the problem accurately. Always remember to contextualize each condition before integrating them into the broader solution.