Problem 106 - Prealgebra and Introductory Algebra

Question

Place the correct symbol, \(<,=,\) or \(>,\) between the two numbers. $$|-8| \quad|3|$$

Step-by-Step Solution

Answer

The correct inequality to express the relationship between \(|-8|\) and \(|3|\) is \(|-8| > |3|\).

Step 1: Calculate the absolute values

The absolute value of a number is its distance from zero on the number line, ignoring its sign. Therefore, \(|-8|=8\) and \(|3|=3\).

Step 2: Compare the absolute values

Now, 8 and 3 need to be compared. As 8 is greater than 3, the correct symbol to place between the two numbers would be '>'.

Key Concepts

number line

A number line is a simple visual representation of numbers that helps you understand their order and magnitude. It is drawn as a straight horizontal line with numbers placed at equal intervals. The center or origin of the number line is usually marked as zero.

Numbers to the right of zero are positive, and those to the left are negative. This visualization aids in understanding operations like addition, subtraction, and, importantly, absolute values, as you can see distances directly.

An absolute value represents how far a number is from zero, regardless of direction. Hence, whether you are at -8 or 8 on the number line, the distance from zero is 8 units in both cases. Similarly, for the number 3, its absolute value is simply its distance from zero, which is 3 units.

comparison of numbers

Comparing numbers is about evaluating which number is larger, smaller, or if they are equal. Absolute values simplify this process because they convert all numbers to a positive format, focusing only on magnitude.

In the case of comparing \(|-8|\) and \(|3|\), we first calculate their absolute values which are 8 and 3, respectively. On a number line, 8 is further right than 3, indicating that it is a larger number.

When comparing absolute values, such as 8 and 3, we use symbols like <, >, or = to describe the relationship. Here, because 8 is greater than 3, we use the \(>\) symbol.

prealgebra concepts

Prealgebra is a critical foundation for all higher-level math, dealing with basic arithmetic and the introduction of algebraic thinking. Concepts like absolute value and number comparison are essential.

The absolute value is a key prealgebra concept, as it prepares students to handle both positive and negative numbers seamlessly by focusing on distance from zero. This concept is fundamental to understanding more complex algebraic expressions later on.

Another core aspect of prealgebra is the comparison of numbers, which fosters numeracy and aids in logical reasoning. This lays the groundwork for inequality solving in algebra and real-world problem-solving skills, where determining the larger quantity is crucial.

Prealgebra also includes understanding the number line, which ties together these concepts by providing a tangible framework to visualize numeric relationships and operations.