Problem 45 - Prealgebra

Question

Find each of the following absolute values. $$|-200|$$

Step-by-Step Solution

Answer

The absolute value of \\(-200\\) is \\(200\\).

Step 1: Understand Absolute Value

The absolute value of a number is its distance from zero on the number line, regardless of its direction. This means it is always non-negative.

Step 2: Apply the Absolute Value Definition

The absolute value of a negative number is its positive counterpart. Hence, \(|-200|\) becomes \(200\), because \(-200\) is \(200\) units away from zero on the number line.

Key Concepts

Negative Numbers

Negative numbers are numbers that are less than zero. They are found to the left of zero on the number line. Unlike positive numbers, negative numbers have a minus sign (-) in front of them. For example, -1, -2, and -200 are all negative numbers.

Negative numbers are essential when we want to describe a value that is below zero in situations like temperatures, debts, or elevators moving down. Understanding how negative numbers relate to positive numbers helps us grasp concepts such as temperature changes or financial loss and gain.

Number Line

A number line is a straight, horizontal line that visually represents numbers in sequence. Each point on the line corresponds to a number. Numbers are placed at equal intervals along the line.

The center of the number line is zero, which divides positive numbers (on the right) from negative numbers (on the left). The number line helps us easily see the relationships between numbers, including which are larger or smaller. For instance:

• Positive numbers increase as you move right.
• Negative numbers increase as you move left, even though they become numerically smaller.

This tool is very useful in understanding concepts like addition and subtraction, as well as understanding absolute value by measuring distance from zero.

Distance From Zero

The distance from zero on a number line is known as the absolute value. It's a way to assess how far a number is from zero, regardless of direction. This means:

• The absolute value of -200 is 200 because it is 200 units away from zero on the number line.
• For positive 200, the distance is the same, 200.
• The absolute value of any number is always positive or zero.

By focusing on distance, absolute value simplifies calculations involving both positive and negative numbers. For any number \( x \), the absolute value can be represented as \(|x|\). When processing negative numbers, just convert them to their positive counterparts to find their absolute value.