Problem 8 - Beginning and Intermediate Algebra

Question

Insert \(<,>,\) or \(=\) in the appropriate space to make the statement true. See Example 1. $$ -4 \quad-6 $$

Step-by-Step Solution

Answer

-4 > -6

Step 1: Compare Negative Numbers

Negative numbers are further from zero on the number line than positive numbers, and among negative numbers, the one with the lesser absolute value is actually larger. Here, compare -4 with -6. Since -4 is closer to zero than -6, it is greater than -6.

Step 2: Insert the Correct Symbol

Based on the comparison of the numbers, we can determine that -4 is greater than -6. Thus, the correct symbol to insert in the space to make the statement true is \(-4 > -6\).

Key Concepts

Understanding the Number Line

A number line is a visual representation of numbers placed in a straight line. It helps us understand the position and order of numbers, including negative numbers.

On a number line:

• Numbers increase from left to right.
• Positive numbers are located to the right of zero.
• Negative numbers are located to the left of zero.

Knowing how to use the number line is crucial because it shows the distance of numbers from zero. For instance, on the number line, -4 is positioned closer to zero compared to -6, thereby indicating that -4 is greater than -6. This visualization aids in grasping concepts involving positive and negative integers, making it easier to compare them.

The Concept of Absolute Value

Absolute value refers to the magnitude or "distance" of a number from zero on the number line, without considering its direction. It is always a non-negative value.

For any number:

• The absolute value of a positive number is the number itself.
• The absolute value of a negative number is its positive counterpart.

For example, the absolute value of -4 is expressed as \(|-4| = 4\). Similarly, the absolute value of -6 is \(|-6| = 6\). Though -4 and -6 are both negative, looking at their absolute values shows how far each number is from zero, without considering the negative sign. This property of absolute values assists in accurately comparing the sizes of negative numbers.

Comparing Integers

Comparing integers involves determining which of two numbers is larger or smaller. This is done using a number line and the concept of absolute value.

When comparing numbers:

• If one number is positive and the other is negative, the positive number is always larger.
• If both numbers are negative, the number closer to zero is considered larger.
• The number further from zero on the number line has a larger absolute value but is the smaller number if both are negative.

Take for instance the comparison of -4 and -6:-4 is closer to zero than -6, therefore making -4 greater than -6.Understanding these comparisons ensures more accurate placement of numbers in order and assists in selecting the right inequality (\(<\), \(>\), or \(=\)) for given mathematical statements.