Question

Given that \( 0<x<y<1 \), compare the following quantities: \[ 1 - y \quad \text{and} \quad y - x \]

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

Hint:

In problems with inequalities involving variables, it’s important to test with specific values to check the relationship between quantities.

Answer 1

The Correct Option is D

Step 1: Understand the given condition.
We are given that \( 0<x<y<1 \), and we are asked to compare \( 1 - y \) and \( y - x \).
Step 2: Try two approaches.
Approach 1: Number line. On a number line, \( 1 - y \) is the distance between \( y \) and 1, and \( y - x \) is the distance between \( y \) and \( x \). Depending on the values of \( x \) and \( y \), these two distances may or may not be equal.
Approach 2: Plug values for \( x \) and \( y \). For example, if \( x = 0.4 \) and \( y = 0.5 \), then: \[ 1 - y = 1 - 0.5 = 0.5, \quad y - x = 0.5 - 0.4 = 0.1 \] In this case, \( 1 - y \) is greater than \( y - x \). However, if \( x = 0.1 \) and \( y = 0.9 \), then: \[ 1 - y = 1 - 0.9 = 0.1, \quad y - x = 0.9 - 0.1 = 0.8 \] Here, \( y - x \) is greater.
Step 3: Conclusion.
Since the relationship depends on the specific values of \( x \) and \( y \), the correct answer is that the relationship cannot be determined from the information given.
Final Answer: \[ \boxed{\text{The correct answer is (4) The relationship cannot be determined from the information given.}} \]