Problem 38 - College Algebra

Question

express the distance between the given numbers using absolute value. Then find the distance by evaluating the absolute value expression. $$ -5.4 \text { and }-1.2 $$

Step-by-Step Solution

Answer

The distance between -5.4 and -1.2 is 4.2 units.

Step 1: Set up the Absolute Value Expression

First we need to set up the absolute value expression. The distance between two numbers \(-5.4\) and \(-1.2\) can be represented as \(| -5.4 - -1.2 |\) or \(|-5.4 + 1.2|\)

Step 2: Simplify Inside Absolute Value Expression

Next, add the numbers inside the absolute value expression. So the expression simplifies to \(|-4.2|\).

Step 3: Evaluate the Absolute Value Expression

Finally, we need to evaluate the expression. The absolute value of \(-4.2\) is \(4.2\), which means the distance is \(4.2\) units

Key Concepts

Distance Between Numbers

The concept of distance between numbers on a number line is fundamental in mathematics. It tells us how far apart two numbers are, regardless of their direction on the line. To find this distance, we use the absolute value, which ensures the distance is a positive number, as distance cannot be negative.
To express the distance between two numbers using absolute value, you subtract one number from the other and take the absolute value of the result. For instance, given the numbers -5.4 and -1.2, the distance can be represented as \(| -5.4 - (-1.2) |\) or simplified to \(| -5.4 + 1.2 |\).
Understanding this helps you visualize the numbers on a number line and measure how many units they are apart.

Evaluating Absolute Value

Evaluating the absolute value of a number involves converting that number to its positive form. The absolute value is the number's distance from zero on the number line.
If you have a positive number or zero, its absolute value is the number itself. For a negative number, its absolute value is its positive counterpart.
Let's take the expression \(|-4.2|\) as an example. By evaluating this, we find that the absolute value is 4.2. This means that even though -4.2 is negative, its distance from zero is 4.2 units.
This step ensures that when finding the distance between numbers, we get a non-negative result, aligning with the idea that distance cannot be negative.

Simplifying Absolute Value Expressions

To simplify absolute value expressions, begin with any calculations inside the absolute value brackets, just like any other mathematical expression.
In our expression \(|-5.4 + 1.2|\), we first add -5.4 and 1.2, which results in \(-4.2\). This gives us the simplified form \(|-4.2|\).
Simplifying involves ensuring that all operations inside the absolute value are complete before we evaluate. Once simplified, we are ready to find the actual distance, which, as in this example of \(|-4.2|\), evaluates to 4.2.
Simplifying expressions before evaluating is crucial as it reduces computation errors and clarifies the outcome for easier comprehension.