Problem 102 - Elementary Algebra

Question

Simplify. $$ |-20| $$

Step-by-Step Solution

Answer

The simplified expression is 20.

Step 1: Understand Absolute Value

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

Step 2: Apply Absolute Value Definition

To find the absolute value of -20, determine its distance from zero on the number line. Since it is 20 units away, disregarding direction, the absolute value of -20 is 20.

Step 3: Write the Simplified Result

The absolute value of -20 simplifies to 20, written as \(|-20| = 20\).

Key Concepts

Number Line

A number line is a visual representation of numbers laid out in order from smallest to largest. It acts as a tool to visualize and compare numbers. On a number line, zero is typically placed in the center, with positive numbers to the right and negative numbers to the left. This arrangement helps to easily see relationships between numbers, such as greater than or less than.

For example, on a number line:

• -20 is to the left of 0
• 0 is the point of origin dividing the positives from the negatives
• 20 is to the right of 0

The number line is crucial in understanding absolute value. It helps in identifying how far a number is from zero, regardless of whether it's a positive or negative. It's a simple yet effective way to visualize math concepts and gain deeper insights into numbers.

Distance From Zero

Distance from zero is a key aspect of understanding absolute values. In mathematical terms, it refers to how far a number is from zero on the number line, without considering its direction. This distance is always expressed as a non-negative number and is known as the absolute value.

Here's how you determine distance from zero:

• Locate the number on the number line.
• Count how many units it is away from zero.
• Ignore the direction by making the value positive.

For example, for the number -20, we find it is 20 units away from zero on the number line. Thus, the absolute value, or the distance from zero, is 20. Understanding this distance helps in simplifying expressions and solving real-world problems involving measurements and quantities.

Simplifying Expressions

Simplifying expressions in mathematics is about making an expression as concise and clear as possible without changing its value. When we simplify expressions involving absolute values, we convert them into their simplest form.

• Apply the definition of absolute value to simplify the expression.
• Remove negative signs if they are inside the absolute value symbol.
• Ensure the final result is non-negative.

For instance, \(|-20|\) simplifies to 20. The negative sign is ignored because absolute value only cares about the distance, not direction. This simplification process is crucial for solving equations, as it ensures values are clearly understood and correctly used in further calculations.