Problem 70 - Algebra A Combined Function

Question

Insert \(<,>,\) or \(=\) in the appropriate space to make each statement true. $$ |-12| \quad|0| $$

Step-by-Step Solution

Answer

The statement is |-12| > |0|.

Step 1: Understanding Absolute Values

Before comparing, understand that the absolute value of a number is its distance from zero on the number line, without considering direction. Thus, absolute values are always non-negative.

Step 2: Calculate Absolute Values

First, find the absolute values of the numbers involved: \(|-12| = 12\) and \(|0| = 0\).

Step 3: Compare the Absolute Values

Now, compare the values obtained: Since \(12 > 0\), therefore \(|-12| > |0|\).

Step 4: Insert the Correct Symbol

Based on the comparison, insert '>' between \(|-12|\) and \(|0|\) to make the statement \(|-12| > |0|\) true.

Key Concepts

Understanding the Number Line

A number line is a visual representation of numbers ordered along a straight line. It helps us understand how numbers relate to each other in terms of size and distance.

• Position of numbers: Numbers grow larger as you move to the right and smaller as you move to the left on the number line.
• Zero's importance: Zero is the central point on a number line, dividing positive numbers from negative numbers.
• Absolute value: This involves the concept of distance from zero. For any number, its absolute value is the number of units away from zero on the number line, regardless of direction.

Visualizing the number line can help you better grasp absolute values and comparisons. The further right a number is, the larger the value.

Making Comparisons

Comparisons between numbers involve understanding which is greater, less, or whether they are equal. Symbols are used to make these comparisons clear:

• "<" symbol: This symbol means "less than." When one number is smaller than the other, this symbol is used.
• "=" symbol: Used when two numbers are exactly the same or equal.
• ">" symbol: This indicates "greater than." It is used when one number is larger than the other.

For instance, in the comparison \(|-12| > |0|\), the ">" sign tells us that \(|-12|\) which equals 12 is greater than \(|0|\) which equals 0. Skillfully using these symbols is crucial in accurately describing relationships between numbers.

Significance of Non-Negative Numbers

Non-negative numbers include all positive numbers and zero. These are vital in the context of absolute values:

• Absolute values always result in non-negative outcomes. For instance, the absolute value of \(-12\) is 12.
• Non-negative numbers are intuitive since they express magnitude without direction, and thus making it easier when comparing sizes on the number line.
• Being aware of non-negative numbers can help eliminate confusion, especially when dealing with distances or measuring values in real-world scenarios.

Every absolute value is considered non-negative, emphasizing your focus on size rather than direction.

Using Mathematical Symbols Correctly

Mathematical symbols are shortcuts that convey meaning succinctly in equations and comparisons:

• Using these symbols properly is essential for mathematical clarity.
• They establish relationships between numbers, such as identifying which numbers are larger or smaller.
• Symbols like ">", "<", and "=" are fundamental in representing numerical relationships on a number line, especially when denoting absolute values.

For example, in the exercise where you had \(|-12| > |0|\), the ">" symbol indicates that the absolute value of \(-12\) is greater than zero, highlighting a key relationship facilitated through proper symbol use.