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Chapter 1: Problem 51

Students often say "Absolute value is always positive." Is this true? Explain.

Short Answer

Expert verified
Absolute value is always non-negative, meaning it is either positive or zero.

Step by step solution

01

- Define Absolute Value

Absolute value refers to the distance of a number from zero on the number line, regardless of direction. It is denoted by two vertical bars, for example, \(|x|\).
02

- Absolute Value of Positive and Negative Numbers

For any positive number or zero, the absolute value is the number itself. For any negative number, the absolute value is its positive counterpart. For instance, \(|3| = 3\) and \(|-3| = 3\).
03

- Address the Misconception

The statement 'Absolute value is always positive' is a bit misleading. It should be understood that the absolute value is always non-negative. This means it can be positive or zero.
04

- Provide Examples

Consider the numbers \(-5\), \(-2\), 0, and 4. Their absolute values are as follows: \(|-5| = 5\), \(|-2| = 2\), \(|0| = 0\), and \(|4| = 4\). Note that while all are non-negative, zero is not positive.
05

- Conclusion

In conclusion, the absolute value of a number is always non-negative, meaning it is either positive or zero, but not necessarily always positive.

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Key Concepts

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

Absolute Value and Distance on a Number Line
Imagine a number line where each number is a point on this line, with zero at the center. The absolute value of a number simply describes how far the number is from zero, regardless of the direction. For instance, both 3 and -3 are three units away from zero. We write this distance as \(|-3| = 3\) and \(|3| = 3\).
This concept is crucial because it allows us to measure distance without worrying about direction (left or right).

In summary, the absolute value \( |x| \) tells us the distance of \( x \) from zero on a number line.
Understanding Non-Negative Values
Non-negative value means a number that is either positive or zero, but never negative.
Absolute values fall in this category. They can be zero, as with \(|0| = 0\), or positive, like \(|5| = 5\) and \(|-5| = 5\).

This understanding helps to correct the misconception that absolute value is always positive. Instead, remember it is always non-negative, covering both zero and positive numbers but not negative ones.
Positive and Negative Numbers with Absolute Values
Every number has an absolute value that represents its non-negative distance from zero.
Positive numbers and their absolute values are straightforward. For example, \( |7| = 7 \).

For negative numbers, things are a bit different. The negative sign is ignored, and they are converted to their positive counterparts. For instance, \(|-8| = 8\). This makes every absolute value non-negative, as we previously discussed.
By understanding this, we grasp that the concept of absolute value serves to represent distances on the number line, simplifying interactions with both positive and negative numbers.

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Most popular questions from this chapter

Fill in the blanks: The commutative property allows us to change the _____ of the terms in a sum or the factors in a product. The associative property allows us to change the _____ of the terms in a sum or the factors in a product.

Identify each number as prime, composite, or neither. If the number is composite, write it as the product of prime factors. See Example 1 . $$19$$

Write a numerical expression for each phrase and simplify. The sum of \(-7\) and \(-13,\) increased by 14

In each term, give the numerical coefficient of the variable(s). See Objective 2. $$ 3 m^{2} $$

Fill in each blank with one of the following: greater than 0, less than 0, equal to \(0 .\) The product or the quotient of two numbers with different signs is _____.
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