Problem 69 - Beginning and Intermediate Algebra

Question

Insert \(<,>,\) or \(=\) in the appropriate space to make a true statement. See Examples 6 through 8 . $$ |-5|\quad-4 $$

Step-by-Step Solution

Answer

|-5| > -4

Step 1: Understand Absolute Value

The absolute value of a number is its distance from zero on the number line, regardless of direction. This means that the absolute value is always non-negative.

Step 2: Calculate |-5|

The absolute value of -5 is calculated as follows: \[ |-5| = 5 \] because -5 is 5 units away from 0.

Step 3: Compare 5 to -4

Now, we need to compare 5 with -4. Since 5 is a positive number and -4 is a negative number, 5 is greater than -4.

Step 4: Insert the Correct Symbol

According to the comparison established: \[ 5 > -4 \] we can substitute back the absolute operation to get the correct equation: \[ |-5| > -4 \]

Key Concepts

Understanding the Number Line

The number line is a visual tool that helps us understand the value and relationships of numbers, including both positive and negative numbers. It is a straight line with equally spaced marks for each integer, extending infinitely in both directions. The central point of a number line is zero, which divides the line into positive numbers on the right and negative numbers on the left.

Using a number line can help us easily visualize the distance of a number from zero, which is crucial when dealing with absolute values. The distance is always positive because it reflects the magnitude, not the direction. This means that no matter if a number is on the positive or negative side, its absolute value is a positive distance from zero.

• For instance, the number 3, which is three units to the right of zero, has an absolute value of 3.
• Similarly, -3 is three units to the left of zero and also has an absolute value of 3.

Understanding how numbers are arranged on this line allows us to easily compare their sizes, both with and without considering their signs.

Making Comparisons on the Number Line

Comparison is an important concept in mathematics, especially when working with numbers that have different properties. On a number line, comparing numbers involves looking at their positions relative to each other. Numbers further to the right are larger, while those to the left are smaller.

When comparing absolute values, we are interested in the distances from zero, without considering the direction on the number line. This can often change the results we get when simply comparing numbers by themselves.

For example, comparing -5 and -4 without considering absolute values tells us -5 is less than -4 because it's more to the left. However:

• When considering the absolute value of -5, which is 5, and comparing it to -4, we see that 5 (absolute value of -5) is greater than -4.

This understanding helps with problems like inserting comparison symbols, where knowing how far numbers are from zero is more relevant than their position on the line.

Positive and Negative Numbers

Positive and negative numbers are fundamental in mathematics, offering a way to express values in a relative position to zero on the number line. Positive numbers are greater than zero and lie to the right of zero on the number line, whereas negative numbers are less than zero and extend to the left.

The concept of positive and negative numbers becomes particularly significant when dealing with operations like absolute value and comparisons. Absolute value disregards whether a number is positive or negative, focusing solely on its distance from zero. This way, the absolute value of both -5 and 5 is 5, indicating a purely positive measure of their magnitude.

• In our case, the absolute value of -5 simply turns it into its positive counterpart, 5.
• When comparing a positive number to a negative one, the positive number is always greater. Thus, in our comparison, 5 (absolute value of -5) is greater than -4.

Understanding this distinction is essential when solving problems that involve both positive and negative numbers, ensuring that we correctly interpret and use them based on their absolute values and positions on the number line.