Problem 34 - Prealgebra

Question

Place either < or \(>\) between each of the following pairs of numbers so that the resulting statement is true. $$|-3| \quad|-1|$$

Step-by-Step Solution

Answer

|-3| > |-1|

Step 1: Understand the Absolute Value

The absolute value of a number is its distance from zero on the number line, regardless of direction. This means the absolute value is always non-negative. For example, \(|-x| = x\) if \(x\) is a positive number, and \(|-x| = -x\) if \(x\) is a negative number.

Step 2: Calculate Absolute Values

Compute the absolute values of the given numbers. For \(|-3|\), the absolute value of \(-3\) is \(3\). For \(|-1|\), the absolute value of \(-1\) is \(1\).

Step 3: Compare the Absolute Values

Now, compare the absolute values obtained in Step 2. We have \(3\) and \(1\). Since \(3\) is greater than \(1\), the correct inequality between the two numbers is \(3 > 1\).

Step 4: Write the Final Statement

We use the greater than symbol \(>\) to complete the inequality with the absolute values: \(|-3| > |-1|\).

Key Concepts

Inequalities

Inequalities are mathematical expressions used to show the relationship between two values. They help us understand which value is larger or smaller without requiring them to be equal. There are several types of inequalities, including:

• < : Less than
• > : Greater than
• ≤ : Less than or equal to
• ≥ : Greater than or equal to

When solving inequalities, it's essential to isolate the variable or values you want to compare and use the appropriate inequality symbol to indicate their relationship. For instance, if we compare the absolute values of \( |x| \) and \( |y| \), we can write \( |x| > |y| \) if the absolute value of \( x \) is greater than that of \( y \). When interpreting inequalities, always remember that they describe the relative size or order of the numbers involved.

Number Line

A number line is a visual representation of numbers on a straight line. It helps us grasp the concept of absolute values and inequalities better by marking each number with an equal space. Every point on a number line corresponds to a real number, and the position of numbers can help in understanding their relative magnitudes.On a number line:

• Numbers to the right are greater than those to the left.
• The zero point separates positive numbers from negative numbers.
• The absolute value of any number corresponds to its distance from zero.

Visualizing numbers and comparing them using a number line offers a simple yet powerful method to reason about inequalities and absolute values. For example, by plotting \( -3 \) and \( -1 \) on a number line, we can see how far each is from zero. Distances from zero represent absolute values, assisting in comparison.

Comparing Numbers

Comparing numbers is the process of determining the relationship between their sizes. This comparison might involve actual values, such as 5 and 7, or abstract values like absolute values, such as \(|-3|\) and \(|-1|\). We use inequalities like \(<\) and \(>\) to denote these comparisons.When comparing using absolute values:

• First, find the absolute value of each number.
• Then, compare these values.
• Identify and apply the correct inequality sign.

In our example, \(|-3|\) becomes 3 and \(|-1|\) becomes 1, meaning that 3 > 1. This shows that the absolute value of \(-3\) is greater than that of \(-1\), so \(|-3| > |-1|\). Mastering the skill of comparing numbers is vital for solving problems involving inequalities in math.