Problem 73 - Beginning and Intermediate Algebra

Question

Insert \(<,>,\) or \(=\) in the appropriate space to make a true statement. See Examples 6 through 8 . $$ |-2| \quad|-3| $$

Step-by-Step Solution

Answer

|-2| < |-3|.

Step 1: Understanding Absolute Value

The absolute value of a number is its distance from zero on the number line, regardless of direction. It is always non-negative. So, for any number or expression inside absolute value signs, evaluate it to get the non-negative version of that number.

Step 2: Evaluate Absolute Values

First, find the absolute values of both numbers: \(|-2|\) is equal to 2 because the distance from -2 to 0 is 2. Similarly, \(|-3|\) is equal to 3 because the distance from -3 to 0 is 3.

Step 3: Compare the Absolute Values

Now, compare the two absolute values: 2 and 3. Since 2 is less than 3, \(|-2| < |-3|\).

Step 4: Insert the Appropriate Symbol

Insert the comparison symbol based on the evaluation: \(|-2| < |-3|\) is the true statement because 2 is less than 3.

Key Concepts

Number Line

A number line is a visual representation of numbers on a straight horizontal line. It is an essential tool for understanding both positive and negative numbers. Every point on this line corresponds to a number, and the position of each point relative to zero determines its value. Numbers to the right of zero are positive, while numbers to the left are negative.

• The importance of the number line is its ability to visually demonstrate the concept of distance, direction, and order of numbers.
• When dealing with absolute values, the number line helps by showing how far a number is from zero, disregarding direction.

For example, when we look at \(-2\) on a number line, we can see it is two units from zero in the negative direction. The same goes for \(-3\), which is three units from zero. Using the number line can make understanding abstract concepts such as absolute value more tangible.

Comparison

Comparison involves determining the relative size or order of two or more quantities. In mathematics, it often uses comparison symbols like \(<\), \(>\), and \(=\). These symbols help to establish if numbers are smaller, larger, or equal.

• The symbol \(<\) means "is less than."
• The symbol \(>\) means "is greater than."
• The symbol \(=\) means "is equal to."

When comparing absolute values, we evaluate the size of the values without considering their sign. For instance, comparing \(|-2|\) with \(|-3|\), we determine the values as 2 and 3 respectively. Since 2 is less than 3, the appropriate comparison symbol is \(<\). Utilizing comparison helps in developing essential mathematical reasoning skills and understanding the relationship between numbers.

Inequality

Inequality is a mathematical explanation that one value is not equal to another and indicates whether one number is greater or lesser than another. It uses signs such as \(<\), \(>\), \(\leq\), and \(\geq\) to show these relationships. Inequalities can appear straightforward like \(x > 3\), which means that \(x\) could be any number greater than 3, or more complex like \(-5 < x \leq 2\), which describes a range of values for \(x\). In our context, inequalities help express the relation of absolute values, showing one is smaller or larger than the other without implying actual numeric equality.When dealing with absolute value scenarios like in this exercise, inequalities show \(|-2| < |-3|\), which solidifies understanding that the absolute value of \(-2\) is less than that of \(-3\). Recognizing and interpreting inequalities is fundamental in both simple arithmetic and more complex algebraic applications.