Problem 109 - Prealgebra and Introductory Algebra

Question

Place the correct symbol, \(<,=,\) or \(>,\) between the two numbers. $$|x| \quad|-x|$$

Step-by-Step Solution

Answer

The correct symbol to use is =, so |x| = |-x|.

Step 1: Understand the absolute value

The absolute value of a number is its distance from zero, regardless of the direction. So, the absolute value of x, denoted as |x|, is always a non-negative number.

Step 2: Identify the absolute value of -x

The absolute value of -x, denoted as |-x|, is also non-negative and equals to the absolute value of x, because they are the same distance from zero, but in opposite directions.

Step 3: Compare |x| and |-x|

Since |x| and |-x| are equivalent, it's necessary to use the equal symbol (=) to show their relationship.

Key Concepts

Distance from Zero

Absolute value is a fundamental concept in mathematics, focusing on the idea of "distance from zero." Imagine if you were standing on a number line. The absolute value of any number is simply how far away that number is from zero, without worrying about direction. For instance, both +3 and -3 would have an absolute value of 3. This is because they are both three units away from zero, despite being on different sides. Understanding this distance concept helps explain why absolute values are always non-negative, as distance cannot be negative.

Non-negative Number

When we talk about a non-negative number, we're referring to numbers that are either positive or zero. Absolute values fit perfectly into this category. The symbol \(|x|\) stands for the absolute value of \(x\), and it captures the idea of removing any negative sign from a number. So, whether \(x\) is 5 or -5, \(|x|\) will be 5, a non-negative number. By design, absolute values cannot be less than zero because they measure distance, which only moves forward on the number line or stays at the starting point, zero.

Comparison of Absolute Values

When comparing absolute values, you're essentially checking which distance from zero is greater or smaller. In the given problem, you're tasked to compare \(|x|\) and \(|-x|\).

You might expect that the negative sign in \(-x\) would make its absolute value different from \(|x|\), but it doesn't. Why? Because \(|-x|\) is simply the distance from zero, just as \(|x|\) is. This means no matter what \(x\) is, their absolute values are equal. Hence, for this comparison, the correct symbol to use is "=", indicating both values represent the same distance from zero.

Mathematical Symbols

Mathematical symbols like \(<\), \(=\), and \(>\) are essential for expressing relationships between numbers. They help us succinctly represent ideas like equality or inequality.

In the context of absolute value, understanding these symbols is crucial. For the exercise comparing \(|x|\) and \(|-x|\), the equality symbol \(=\) is used. This conveys that the absolute values are indeed identical when looking at them as distances. Such mathematical symbols facilitate clarity and precision in communication, especially in math, by showing how numbers relate to one another. It's a vital tool for expressing how quantities compare or match, anchoring abstract concepts in concrete terms.