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Chapter 1: Problem 12

Find each absolute value. $$ |-3| $$

Short Answer

Expert verified
The absolute value of -3 is 3.

Step by step solution

01

Understand Absolute Value

The absolute value of a number is its distance from zero on the number line, without considering direction. Essentially, it turns negative values into positive ones and leaves positive values unchanged.
02

Apply Absolute Value to -3

To find \(|-3|\), consider the distance of -3 from zero. Since absolute value negates any negative sign, \(|-3| = 3|\).

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Key Concepts

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

number line
Imagine a straight line, like a ruler, that extends infinitely in both directions. This is a number line. The center point is zero, and all other numbers are placed to the left or right of zero. Positive numbers go to the right, while negative numbers go to the left.

Why use a number line?
  • Helps visually see the relationship between numbers.
  • Makes understanding addition and subtraction easier.
  • Aids in understanding concepts like absolute value.
To grasp absolute value, simply look at how far a number is from zero, disregarding the direction.
distance from zero
Distance from zero is at the heart of the concept of absolute value. When we talk about 'distance,' we mean how many steps it takes to reach zero from any given number.

For example, the distance from -3 to 0 is 3 steps. This is why \(|-3| = 3 \). Notice that we don鈥檛 care if we started from a negative or a positive number, we only look at the number of steps.

Key Points:
  • Distance is always a positive value.
  • Absolute value simply reflects this distance.
By thinking about distance, we grasp that negative and positive numbers both have absolute versions that are non-negative.
positive and negative numbers
Numbers can be both positive and negative. It's essential to distinguish between the two:

Positive Numbers:
  • Located to the right of zero on the number line.
  • Examples include 1, 2, 3, etc.


Negative Numbers:
  • Located to the left of zero on the number line.
  • Examples include -1, -2, -3, etc.
When working with absolute values, the sign doesn鈥檛 matter. For example:
  • The absolute value of -3 and 3 is the same, which is 3.
  • This is because both are 3 steps away from zero.
Understanding the nature of positive and negative numbers, and consequently their absolute values, is crucial in various areas of math and real-world applications.

Always remember, the absolute value is like saying, 'I only care how far away you are from zero, not which direction!'

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Most popular questions from this chapter

To the student and the instructor: Synthesis exercises are designed to challenge students to extend the concepts or skills studied in each section. Many synthesis exercises require the assimilation of skills and concepts from several sections. 5\. Is the following true or false, and why? $$ \\{2,4,6\\} \subseteq\\{2,4,6\\} $$

To the student and the instructor: Throughout this text selected exercises are marked with the icon Ahal. Students who pause to inspect an Ahal exercise should find the answer more readily than those who proceed mechanically. This may involve looking at an earlier exercise or example, or performing calculations in a more efficient manner. Some Ahal exercises are left unmarked to encourage students to always pause before working a problem. Evaluate each expression using the values provided. $$ x^{2}-[3(x-y)]^{2}, \text { for } x=6 \text { and } y=4 $$

Which numbers in the list provided are (a) whole numbers? (b) integers? (c) rational numbers? (d) irrational numbers? (e) real numbers?. $$ -6.08,-5,0,1, \sqrt{17}, \frac{99}{2} $$

Calculate using the rules for order of operations. If an expression is undefined, state this. $$ \frac{4^{3}-(7-4)^{2}}{3^{2}-7} $$

Use set-builder notation to write each set. \(\\{24,26,28,30,32\\}\)
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