Problem 75 - Beginning and Intermediate Algebra

Question

Insert \(<,>,\) or \(=\) in the appropriate space to make a true statement. See Examples 6 through 8 . $$ |0| \quad|-8| $$

Step-by-Step Solution

Answer

|0| < |-8|

Step 1: Understand Absolute Value

The absolute value of a number is its distance from zero on the number line, irrespective of direction. Therefore, it removes any negative sign.

Step 2: Calculate Absolute Values

Determine the absolute values of the numbers involved: - \(|0| = 0\), because zero is already non-negative.- \(|-8| = 8\), because the absolute value of \(-8\) is its positive equivalent.

Step 3: Compare Calculated Values

Compare the absolute values you found: - \(0\) is less than \(8\).

Step 4: Insert the Correct Symbol

Since \(0 < 8\), insert the symbol \(<\) between \(|0|\) and \(|-8|\). Thus, the true statement is \(|0| < |-8|\).

Key Concepts

Understanding the Number Line

The number line is a visual representation of all real numbers arranged in a straight line. It usually has zero in the center, with positive numbers increasing to the right and negative numbers decreasing to the left.

• Each point on the line corresponds to a real number.
• The further to the right a number is, the larger it is compared to those on its left.
• Negative numbers are found on the left side of zero.
• Zero is the neutral point that divides positive and negative numbers.

When dealing with concepts like absolute value, the number line helps us visualize the distance of any number from zero, ignoring the direction (left or right). This understanding of the number line is crucial, as it reflects how numbers relate to each other spatially and helps in comparing them efficiently.

Working with Negative Numbers

Negative numbers are values that are less than zero and are located to the left of zero on the number line. They are used to represent values below a baseline, such as debts in finances, temperatures below freezing, or elevations below sea level.

• The opposite of a positive number. For instance, the negative of 5 is -5.
• Negative numbers get smaller as their absolute magnitude gets larger. For example, -8 is smaller than -3.
• They turn positive when we consider their absolute values, which represent their distance from zero on a number line.

This representation and understanding of negative numbers become particularly important when we apply the concept of absolute value, as it gives us the tool to discuss magnitudes without considering directionality or sign.

Comparison of Numbers Using Absolute Value

Comparing numbers using their absolute values involves evaluating which number is further from zero on the number line, regardless of on which side (left or right) the number lies.

To compare absolute values, follow these steps:

• Find the absolute values of each number involved.
• The number with the smaller absolute value is closer to zero on the number line.
• If one number is zero, its absolute value is the smallest possible.

For example, when comparing \(|0|\) and \(|-8|\), we find:- The absolute value of 0 is 0 because it is neither positive nor negative.- The absolute value of -8 is 8, the positive distance from zero.Since 0 is less than 8, we insert the correct inequality symbol (\(<\)) to show that \(|0|\) is less than \(|-8|\). This method simplifies the task of comparing numbers, especially when negative numbers are involved.