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Chapter 2: Problem 75

Fill in the blank with \(<,>,\) or \(=\) $$|-12| \square|12|$$

Short Answer

Expert verified
\(|-12| = |12|\).

Step by step solution

01

Understand Absolute Value

The absolute value of a number is its distance from zero on the number line, regardless of direction. Therefore, the absolute value of both negative and positive numbers is always non-negative.
02

Evaluate Each Number's Absolute Value

Calculate the absolute value of both numbers given. First, find \(|-12|\): \(|-12| = 12\). Next, find \(|12|\): \(|12| = 12\).
03

Compare the Absolute Values

Now, compare the values obtained: \(|-12| = 12\) and \(|12| = 12\). Hence, \(|-12| = |12|\).
04

Fill in the Blank

Given \(|-12| \square |12|\), based on the previous comparison, insert \(=\) into the blank to make the statement true.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Absolute Value
The concept of absolute value is very important in mathematics. The absolute value of a number represents its distance from zero on the number line. It doesn't matter whether the number is positive or negative; its absolute value is always non-negative. For example, the absolute value of both -12 and 12 is 12. This concept is crucial because it helps to simplify equations and inequalities that involve negative numbers. When you see the notation \(|a|\), you can think of it as the 'magnitude' of \(a\), without considering whether \(a\) is to the left or right of zero on the number line.
Number Line
The number line is a visual representation that helps us understand and compare numbers. Imagine a horizontal line with zero at its center. Positive numbers are placed to the right of zero, while negative numbers are placed to the left. The farther you move from zero, the greater the absolute value of the number, regardless of direction. For instance, -12 and 12 are both 12 units away from zero, making their absolute values 12. The number line is useful in many areas of mathematics, especially when solving inequalities and absolute value problems. It visually demonstrates the concept of distance and helps clarify why \(|-12|\) and \(|12|\) are equal.
Inequality
In mathematics, an inequality is a relationship between two values that indicates that one value is larger or smaller than the other. Inequalities use symbols like <, >, and =. When comparing absolute values, you're often asked to determine which distance from zero is greater or whether they are equal. For instance, in the exercise, \(|-12| \square |12|\), you compare the absolute values of -12 and 12. Since both are 12, the correct inequality symbol to use is =, making the complete statement \(|-12| = |12|\). Understanding how to work with inequalities is essential, as they are used in various mathematical contexts, from algebra to calculus.

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