Question

x < 0
Column A: |x|
Column B: x

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

Hint:

A simple way to think about absolute value is as the "distance from zero" on a number line, which can never be negative. For any negative number, its absolute value will always be its positive counterpart and therefore greater than the number itself.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
This question tests the definition and properties of the absolute value of a number, specifically when the number is negative.
Step 2: Key Formula or Approach:
The definition of absolute value is: \[ |x| = \begin{cases} x, & \text{if } x \geq 0
-x, & \text{if } x<0 \end{cases} \] Step 3: Detailed Explanation:
We are given the condition that \( x<0 \), which means \(x\) is a negative number.
Column A: The quantity is \( |x| \). Since \( x<0 \), we use the second part of the definition: \( |x| = -x \). If \(x\) is a negative number, then \(-x\) is a positive number. For example, if \(x=-5\), then \(|x| = -(-5) = 5\). Thus, Column A is always positive.
Column B: The quantity is \( x \). We are given that \(x\) is a negative number.
Comparison: Column A represents a positive number, while Column B represents a negative number. Any positive number is greater than any negative number.
Therefore, \( |x|>x \).
Step 4: Final Answer:
The quantity in Column A is greater than the quantity in Column B.