Problem 69 - Beginning and Intermediate Algebra

Question

Select the lesser of the two given numbers. |-3.5|,|-4.5|

Step-by-Step Solution

Answer

\(|-3.5| < |-4.5|\). Hence, the lesser number is |-3.5|.

Step 1: Understand Absolute Values

Absolute value refers to the distance of a number from zero, regardless of direction on the number line. For instance, the absolute value of -3.5 is written as \(|-3.5|\) and represents 3.5.

Step 2: Calculate the Absolute Values

Calculate the absolute value for each number: \(|-3.5| = 3.5\) and \(|-4.5| = 4.5\).

Step 3: Compare the Absolute Values

Compare the results from step 2. We have two absolute values: 3.5 and 4.5. The lesser of these two values is 3.5.

Step 4: Conclusion

Since \(|-3.5| = 3.5\) and \(|-4.5| = 4.5\), the correct answer is the value corresponding to 3.5.

Key Concepts

absolute value

The concept of absolute value is crucial in mathematics and represents a number's distance from zero on the number line, regardless of its direction. For instance, whether it is a positive or a negative number does not matter; the absolute value definition ensures we consider only how far the number is from zero. Let's look at an example to make this clearer:

• For \(-3.5\), the absolute value is written as \(|-3.5|\) and is equal to 3.5.
• Similarly, \(|4|\) is just 4.

Absolute value is always non-negative, giving us an easy way to compare distances and magnitudes.

comparing numbers

In mathematics, comparing numbers allows us to determine which number is larger or smaller. This skill helps in various applications, such as ordering or sorting numbers. When comparing absolute values, we're essentially comparing distances.
For example:

• Given the absolute values of -3.5 and -4.5, we have \(|-3.5| = 3.5 \) and \(|-4.5| = 4.5|\).
• To find which is lesser, we compare 3.5 and 4.5 directly. Here, 3.5 is clearly the smaller number.

By understanding absolute values, you can easily determine which of two absolute values is the lesser or greater.

number line

A number line is a visual representation that helps us easily see the relationship between different numbers. The line extends infinitely in both directions from zero, with positive numbers are to the right and negative ones to the left.
Using a number line makes comparing absolute values straightforward because you can see how far each number is from zero:

• -3.5 is three and a half units to the left of zero.
• -4.5 is four and a half units to the left of zero.

Visualizing these numbers on the number line, we observe that -3.5 is closer to zero than -4.5. Hence, \(|-3.5|\) is less than \(|-4.5|\).

distance from zero

Understanding the distance from zero helps to simplify mathematical problems involving absolute values. Distance in this context refers to how far a number is from the origin point, zero, on the number line.
Here's a simple breakdown:

• The distance of -3.5 from zero is 3.5.
• Similarly, the distance of -4.5 from zero is 4.5.

By focusing on distances (absolute values), we remove concerns about the directions (negative or positive) and concentrate on magnitude. This ability to measure distance efficiently is crucial not only in math but also in various fields such as physics and engineering, where understanding magnitudes is essential.