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Chapter 1: Problem 49

Simplify by finding the absolute value. \(|6-3|\)

Short Answer

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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 its direction. For any real number x, the absolute value is denoted \(|x|\).
02

- Calculate the Expression Inside the Absolute Value

First, calculate the value inside the absolute value symbols. Here, \(6-3\) equals 3.
03

- Apply the Absolute Value

Since the absolute value of a number is always non-negative, \(|3|\) is simply 3.

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Key Concepts

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

distance from zero
The term 鈥榙istance from zero鈥� is a way to describe the absolute value of a number. When you think about the number line, zero is the central point. Any number on the number line has a certain distance to zero.
For example, the number 5 is 5 units away from zero. This is the same case for -5, which is also 5 units away from zero.
It doesn't matter whether the number is to the left or right of zero; its distance is always positive. So, the absolute value can be thought of as how far a number is from zero without considering its direction.
  • Positive or negative numbers both have absolute values indicating their distance from zero.
  • Distances on the number line are always non-negative.
  • For any real number x, its absolute value \(|x|\) gives the distance from zero.
non-negative
Non-negative numbers are a fundamental concept in mathematics, especially when dealing with absolute values. A non-negative number is any number that is zero or greater.
Whenever you calculate the absolute value of a number, the result is always a non-negative number. This is because the absolute value measures only the distance from zero, which cannot be negative.
  • Examples of non-negative numbers include 0, 1, 2, and so on.
  • Negative numbers, when taken their absolute values, are converted to positive numbers.
  • The term ensures that the focus is on magnitude, not direction.
So, when you see \(|x|\)=y, you can be confident that y is a non-negative number.
number line
A number line is a visual representation of numbers arranged in order on a straight line. Zero is typically at the center, with positive numbers extending to the right and negative numbers to the left.
This tool helps in understanding absolute values as well since you can easily see how far a number is from zero. For instance, if you look at the number 3 on the number line, it is 3 units to the right of zero.
Similarly, -3 is 3 units to the left of zero.
  • Number lines help visualize the concept of distance and direction.
  • They are useful in solving problems that involve ordering and comparing numbers.
  • When determining absolute values, the numbers鈥� distance from zero is what you observe on the number line.

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