Question

$$\begin{array}{c} x

Short answer

D. The relationship cannot be determined from the information given.

Step-by-step solution

- Understand the inequalities

The given inequalities are: 1. \(x < y < z\) 2. \(0 < z\). From inequality (2), we know that \(z\) is a positive number. Since \(z\) is the largest in the chain of inequalities \(x < y < z\), we know both \(x\) and \(y\) are less than \(z\).

- Compare \(x\) and 0

Since \(x < z\) and \(0 < z\), we can compare \(x\) directly with 0. Knowing \(0 < z\) does not provide enough information about whether \(x\) is greater or less than 0 because \(x\) could be negative, zero, or a positive number smaller than \(z\).

- Conclude the relationship

Given \(x\) could be a negative number, zero, or a positive number less than \(z\), there is insufficient information to compare \(x\) and 0 definitively. Thus, the relationship between Quantity A (\(x\)) and Quantity B (0) cannot be determined.

Key concepts

Inequalities

Inequalities are a fundamental part of mathematics and are essential in understanding relationships between values. In this exercise, we have two inequalities: \(x < y < z\) and \(0 < z\). An inequality tells us how one number compares to another.

Here, \(x < y < z\) means that 'x' is the smallest value, and 'z' is the largest.

The second inequality, \(0 < z\), identifies that 'z' is positive.

Knowing how to interpret inequalities is vital in problems where you need to compare different quantities. Whether you are dealing with one-variable inequalities like \(x > 0\) or multi-variable like the one above, understanding their real meaning helps in approaching the problem correctly.

In this exercise, while we know 'z' is positive, we have no information relating 'x' directly to 0, other than that it is less than 'z'. This can leave us wondering how 'x' compares to 0, which leads us to other core concepts.

Quantitative Comparison

Quantitative comparison questions are a key feature of standardized tests like the GRE. They require you to compare two quantities and determine the relationship between them.

The common answer choices are always similar to:

• Quantity A is greater.
• Quantity B is greater.
• The two quantities are equal.
• The relationship cannot be determined from the information given.

For instance, in our exercise, Quantity A is 'x', and Quantity B is '0'.

We need to determine whether 'x' is greater than 0, less than 0, or equal to 0. However, given that \(0 < z\) and \(x < z\), we note that 'x' could vary: it can be a positive number, less than 'z', or even a negative number. This uncertainty implies that we cannot draw a definitive conclusion.

In such problems, focus on breaking down the given information and seeing how you can relate one quantity to another. Always think about each possibility and whether there could be multiple scenarios that change the outcome.

Number Properties

Understanding properties of numbers is crucial as it helps to solve problems efficiently. These properties include recognizing whether numbers are positive, negative, or zero, and understanding how they interact under different operations.

Here are some crucial points to remember:

• Positive Numbers: Greater than zero.
• Negative Numbers: Less than zero.
• Zero: Neither positive nor negative.

In this problem, 'z' is clearly positive (since \(0 < z\)). However, 'x' and 'y' could be positive, negative, or even zero. Since all we know is \(x < y < z\), and nothing specific about 'x' or 'y' being positive or negative, it presents multiple possibilities for 'x'.

Consider these points while analyzing the problem:

• If 'x' were negative, it would mean \(x < 0 < z\).
• If 'x' were zero, we would have \(0 < y < z\).
• If 'x' were positive, then \(x\) could be any value lesser than 'z' but greater than zero.

This lack of determination makes the answer clear: there is insufficient data to conclude the exact relationship between 'x' and 0.