Problem 70 - Beginning and Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

0 = |0|

Step 1: Understand Absolute Value

Absolute value refers to the distance a number is from zero on the number line, without considering direction. Therefore, the absolute value of zero is zero itself, because zero is exactly zero units away from zero.

Step 2: Calculate Absolute Value

Calculate the absolute value of 0. We have \(|0| = 0\).

Step 3: Compare Values

You are given to compare 0 and \(|0|\), which we calculated as 0 in Step 2. So, compare 0 with 0.

Step 4: Insert Appropriate Symbol

Since 0 is equal to 0, we apply the equality symbol \(=\), so the statement becomes \(0 = |0|\).

Key Concepts

Understanding Inequality

Inequality is a fundamental concept in mathematics that involves the comparison of different values to determine their relationship with one another. When studying inequalities, we use various symbols to indicate whether one number is greater than, less than, or equal to another number. The specific symbols used are:

• Less than \(<\): When one value is smaller than another, such as \(3 < 5\).
• Greater than \(>\): Used to show one value is larger, like \(7 > 2\).
• Equal to \(=\): Indicates two values are identical, as in \((5 = 5)\).

Underneath the surface, inequalities inform many aspects of problem-solving in mathematics, including when determining the boundaries or limits of a set of numbers. Understanding and using inequalities correctly allows students to interpret relationships between quantities, leading to better comprehension in algebra and other mathematical areas.

Representing Values on a Number Line

A number line is a visual tool that helps us understand numbers and their relative positions. It consists of a straight, horizontal line with numbers placed at equal intervals, extending infinitely in both directions.

• Each point on the number line corresponds to a real number, and the position of numbers is crucial for visualizing inequalities.
• Zero is typically at the center, with positive numbers extending to the right and negative numbers to the left.

This tool is particularly useful when discussing absolute values, as it allows us to see how far a number is from zero, regardless of direction. The number line also helps in visually comparing numbers, identifying which is larger or smaller, and facilitating a deeper understanding of mathematical concepts.

Basics of Comparison

Comparison is the act of evaluating two or more quantities to ascertain their mutual relationships. In mathematics, comparing numbers involves determining whether they are equal, greater, or lesser than each other. The key steps involved in making a comparison are:

• Identify the numbers or expressions to compare. For instance, comparing 0 with its absolute, \( |0| \), which equals 0.
• Use the appropriate symbol (\(<\), \(>\), or \(=\)) to denote their relationship based on the values being equal, less, or greater.

Confidence in comparison stems from clear calculations and the understanding of concepts like absolute value, which play into these assessments. Through practice, the process becomes intuitive, allowing students to make quick and accurate judgments about numbers and their relationships.