Problem 74 - Algebra A Combined Function

Question

Insert \(<,>,\) or \(=\) in the appropriate space to make each statement true. $$ |-5.01|-|-5| $$

Step-by-Step Solution

Answer

|-5.01| > |-5|.

Step 1: Understand Absolute Value

Begin by understanding that the absolute value of a number is its distance from zero on the number line without considering the direction. For any number \(x\), its absolute value is denoted as \(|x|\).

Step 2: Calculate the Absolute Values

Calculate the absolute values of the numbers given: \(|-5.01|\) which equals \(5.01\) since absolute value makes numbers positive; and \(|-5|\) which equals \(5\).

Step 3: Compare the Absolute Values

With \(|-5.01| = 5.01\) and \(|-5| = 5\), compare the two: clearly, \(5.01 > 5\).

Step 4: Insert the Correct Symbol

Since \(5.01 > 5\), insert the greater than symbol (\(>\)) between \(|-5.01|\) and \(|-5|\). The final expression is \(|-5.01| > |-5|\).

Key Concepts

Understanding the Number Line

The number line is a visual representation of numbers where each point on the line corresponds to a real number.
This line typically includes all real numbers, extending infinitely in both the positive and negative directions. On a number line, zero is the central point, dividing positive numbers from negative numbers.

• Positive numbers are located to the right of zero.
• Negative numbers are found to the left of zero.
• The absolute value of a number refers to its distance from zero on this line.

Using the number line is beneficial in visualizing operations like addition and subtraction. More importantly, it helps in understanding the absolute value, as it simply measures how far a number is from zero without considering direction.

Comparing Numbers with Absolute Values

Comparing numbers using their absolute values involves looking at their size in terms of distance from zero, ignoring their signs. This technique is incredibly useful in solving many mathematical problems, especially when dealing with negative numbers. When given the numbers \(|-5.01|\) and \(|-5|\):

• The absolute value of \(-5.01\) is \(|-5.01| = 5.01\).
• The absolute value of \(-5\) is \(|-5| = 5\).

To compare these numbers:

• We ignore the negative signs, while focusing on the distance values.
• Here, \(|-5.01| = 5.01\) is greater than \(|-5| = 5\).

Thus, \(|-5.01| > |-5|\). This method simplifies comparing numbers, by translating them into non-negative distances.

Using Mathematical Symbols: Greater Than, Less Than, or Equal To

Mathematical symbols are powerful tools for expressing relationships between numbers. The symbols \(>\), \(<\), and \(=\) are commonly used to compare numbers:

• \(>\): Indicates that the number on the left is greater than the number on the right.
• \(<\): Indicates that the number on the left is less than the number on the right.
• \(=\): Shows that the numbers on both sides are equal.

In the context of absolute values, let's determine which symbol fits between the numbers \(|-5.01|\) and \(|-5|\): 1. We calculate their absolute values: \(5.01\) and \(5\).2. Since \(5.01 > 5\), the correct symbol to use is the greater than sign, making the relationship \(|-5.01| > |-5|\).Understanding these symbols and how to use them is crucial in mathematics, as they allow us to convey precise relationships between different values clearly and concisely.