Problem 75 - Algebra A Combined Function

Question

Insert \(<,>,\) or \(=\) in the appropriate space to make each statement true. $$ |0| \quad|-8| $$

Step-by-Step Solution

Answer

\(|0| < |-8|\).

Step 1: Understand Absolute Value

The absolute value of a number is the distance of that number from zero on the number line, without considering direction. Therefore, the absolute value of a number is always non-negative.

Step 2: Calculate Absolute Values

Calculate the absolute values for each number given in the exercise. For the number 0, the absolute value is \(|0| = 0\). For the number -8, the absolute value is \(|-8| = 8\).

Step 3: Compare Absolute Values

Now compare the absolute values obtained in Step 2. We have the absolute value of 0 as 0 and the absolute value of -8 as 8. Since 0 is less than 8, we place the symbol \(<\) between them.

Key Concepts

Number Line

To understand the concept of absolute value, it's important to first grasp the idea of a number line. A number line is a straight line on which every point corresponds to an actual number.
This line helps to visually represent numbers so you can clearly see their order and magnitude.
Features of a Number Line:

• It extends infinitely in both directions, from negative infinity on the left to positive infinity on the right.
• Zero is placed in the middle, separating negative numbers from positive numbers.
• Each unit spaced on the line corresponds to a unit increase or decrease in numerical value.
• The farther a number is from zero, the greater its absolute value, regardless of being positive or negative.

Thinking of absolute value in terms of a number line can greatly aid your understanding. Since absolute value measures the distance from zero, both -8 and 8 have an absolute value of 8.

Compare Numbers

Comparing numbers is all about figuring out which number is larger or smaller, or if they are equal.
In terms of absolute value, we compare the distance of numbers from zero.
Steps to Compare Numbers Using Absolute Values:

• First identify and calculate the absolute values of each number.
• Once the absolute values are known, line them up to see which is greater or smaller.
• If one absolute value is larger, it means that number is a greater distance from zero.
• Inserting symbols like \(<\), \(>\), or \(=\) clearly shows the relationship between the numbers.

For example, with \(|0|\) and \(|-8|\), you do the math to find \(|0| = 0\) and \(|-8| = 8\). Clearly, 0 < 8, and therefore, \(|0| < |-8|\). This visual and methodical approach simplifies the process of comparing absolute values.

Non-negative Values

When exploring absolute values, it's crucial to remember that they are always non-negative.
This means the result of an absolute value calculation will never be negative, regardless of the original number.
Why Absolute Values are Non-Negative:

• Absolute value measures how far a number is from zero, not the direction in which it lies.
• Distance is naturally a positive concept. You can't have a negative distance.
• If a number is positive or zero, its absolute value is the number itself.
• If a number is negative, its absolute value is the number without the negative sign.

This non-negative nature of absolute values ensures that they are easy to compare and work with, as shown in our example comparing \(|0|\) and \(|-8|\) leading to \(0 < 8\). Emphasizing this can help demystify the operation for many students.