Question

If \(x>1\) and \(y>2\), then: (A) \(x-y>-1\) (B) \(x / y>1 / 2\) (C) \(x / y>0\) (D) \(x-y>0\) (E) \(y-x>1\)

Short answer

A and C

Step-by-step solution

Analyze Condition x > 1

Given that x is greater than 1, we have the inequality: \[ x > 1 \].

Analyze Condition y > 2

Given that y is greater than 2, we have the inequality: \[ y > 2 \].

Evaluate Option (A)

For option (A) \( x - y > -1 \), let's consider if this is true. Since \( x > 1 \) and \( y > 2 \), \[ x - y > 1 - 2 \]\[ x - y > -1 \].This condition is true. Therefore, option (A) is a valid choice.

Evaluate Option (B)

For option (B) \( \frac{x}{y} > \frac{1}{2} \), consider that \( x > 1 \) and \( y > 2 \), \[ \frac{x}{y} > \frac{1}{2} \]While this might seem plausible, there is no concrete guarantee that \( \frac{x}{y} > \frac{1}{2} \), as it will heavily depend on the values of both x and y. Therefore, option (B) is not necessarily true.

Evaluate Option (C)

For option (C) \( \frac{x}{y} > 0 \), considering \( x > 1 \) and \( y > 2 \), \[ \frac{x}{y} > 0 \] Since x and y are both positive and \( y > 0 \), \( \frac{x}{y} \) must be greater than 0. Hence, option (C) is true.

Evaluate Option (D)

For option (D) \( x - y > 0 \), consider \( x > 1 \) and \( y > 2 \). Thus,\[ x - y > 0 \]\[ x - y > 2 - 2 \]\[ x - y > 0 \] is not necessarily true as x could be very slightly greater than y but not satisfying \( x - y > 0 \). Therefore option (D) is false.

Evaluate Option (E)

For option (E) \( y - x > 1 \), consider \( x > 1 \) and \( y > 2 \). Since \( y \) can be much larger than x, \[ y > x + 1 \] could be true. However, it is not guaranteed for any values, thus option (E) is not necessarily true.

Conclusion

Based on the analysis, the valid options are: (A) and (C).

Key concepts

Inequalities

Understanding inequalities is crucial to solving algebraic problems. Inequalities tell us about the relative size of two values. A statement like \( x > 1 \) indicates that \( x \) is greater than 1. Similarly, \( y > 2 \) tells us that \( y \) is greater than 2. Such inequalities often appear in GMAT problem-solving questions.
To solve inequalities, you often need to combine them or analyze them separately. For example, if you know \( x > 1 \) and \( y > 2 \), you could look at the difference \( x - y \). Substituting the minimum values gives \( 1 - 2 = -1 \), so \( x - y > -1 \). This kind of reasoning helps eliminate wrong answer choices and guides you to the correct one.

Fraction Comparison

Comparing fractions involves determining which fraction is larger, smaller, or if they are identical. In GMAT problems like the one given, you might need to compare fractions like \( \frac{x}{y} \).
When comparing fractions, it's helpful to understand a few principles. For instance, if \( x > 0 \) and \( y > 0 \), then \( \frac{x}{y} \) is positive. Given \( x > 1 \) and \( y > 2 \), \( \frac{x}{y} \) is certainly greater than \(0\), since both the numerator and the denominator are positive values.
However, ensuring that \( \frac{x}{y} > \frac{1}{2} \) requires more specific information about the relationship between \( x \) and \( y \). Without concrete values or further constraints, you cannot confirm this inequality. This highlights the importance of context and exact numerical relationships when comparing fractions.

Step by Step Solution

Approaching GMAT problems with a step-by-step solution strategy can simplify complex problems. Let's walk through the given exercise.

• **Step 1:** Determine what each inequality tells you. Given \( x > 1 \) and \( y > 2 \), you get two separate inequalities to work with.
• **Step 2:** Analyze each answer choice. For example, to prove \( x - y > -1 \), look at the minimum values \( x \) and \( y \) can take based on the given conditions.
• **Step 3:** Validate options. For \( \frac{x}{y} > 0 \), since both \( x \) and \( y \) are positive, it follows that their fraction is positive.

Breaking down the problem into smaller pieces makes it manageable.
Additionally, evaluating each option systematically confirms which are true or false, helping to find the correct answers efficiently.