Problem 63 - Algebra A Combined Function

Question

Find each absolute value. $$ |-20| $$

Step-by-Step Solution

Answer

|-20| = 20

Step 1: Understand the concept of absolute value

The absolute value of a number is the distance of the number from 0 on the number line, regardless of direction. It is always a non-negative number. For any number \(x\), the absolute value is denoted as \(|x|\).

Step 2: Apply absolute value definition

To find \(|-20|\), we simply take the positive form of \(-20\) because the absolute value negates any negative sign, indicating distance. Therefore, \(|-20| = 20\).

Key Concepts

Number Line

A number line is a visual way to represent numbers in a straight line. This line is marked with numbers placed at equal intervals. At the center, you have zero, which acts as a neutral starting point. As you move to the right, numbers become positive, and as you move to the left, they become negative.
For example, the numbers to the right of zero might progress as 1, 2, 3, and so on, while those on the left appear as -1, -2, -3, etc.

• The number line helps in visualizing arithmetic operations and understanding concepts like absolute value.
• It is useful in developing an intuition for positive and negative values.
• The number line is an essential tool for grasping the idea of distance, direction, and relative position of numbers.

Non-negative Number

A non-negative number is any number that is either positive or zero.
In contrast to negative numbers, which lie to the left of zero on the number line, non-negative numbers include zero and all positive numbers to the right of zero.

• Non-negative numbers are represented as 0, 1, 2, 3, and so forth.
• These numbers do not include any negatives; hence, you will not see -1, -2, and so on in this category.
• When considering absolute value, the result is always a non-negative number, even if the original number was negative.

Distance from Zero

Distance from zero refers to how far a number is from zero on the number line, regardless of which direction—left or right—the number lies. This concept is essential when understanding absolute value.
For instance, the numbers 5 and -5 are both five units away from zero, making their absolute values equal ( |5| = 5 and |-5| = 5).

• Distance from zero explains why absolute values are non-negative. Negative signs are disregarded as they only indicate direction, not distance.
• By visualizing numbers on the number line, you can easily calculate their distance from zero to find the absolute value.
• Understanding this concept helps in various mathematical problems, offering clarity when dealing with negative and positive numbers.