Problem 73 - College Algebra Essentials

Question

Express the distance between the given numbers using absolute value. Then fi nd the distance by evaluating the absolute value expression. -3.6 and -1.4.

Step-by-Step Solution

Answer

The distance between -3.6 and -1.4 is 2.2.

Step 1: Write Down the Absolute Value Expression

We can form the absolute value expression for the distance between the two numbers by subtracting one number from the other within the absolute value notation. In this case, we can find the distance between -3.6 and -1.4. Thus, the expression becomes |(-3.6) - (-1.4)| or |-3.6 + 1.4|

Step 2: Simplify Within the Absolute Value

Add the value inside the absolute value brackets first, before taking the absolute value. This means we will add -3.6 and 1.4. The result is -2.2

Step 3: Apply the Absolute Value

Finally, the absolute value of -2.2 is determined. Since the absolute value of a number is its distance from zero, it is always a non-negative number. The absolute value of -2.2 is therefore 2.2

Key Concepts

Absolute Value Expression

Understanding absolute value is crucial in math because it represents the distance of a number from zero on the number line, regardless of direction. In simpler terms, it's the number's 'numerical value' without considering its sign (positive or negative). An absolute value expression is written with vertical bars (| |), symbolizing this non-negative value. For example, the absolute value of -3 is 3, written as \( |{-3}| = 3 \).

In the given exercise, the distance between two numbers -3.6 and -1.4 is expressed using absolute value. This type of representation allows us to evaluate distance in a straightforward, sign-independent manner. We can think of these vertical bars as a mathematical function that 'strips away' any negative signs, ensuring that we only consider the magnitude of the difference.

Evaluating Absolute Values

When evaluating absolute values, the primary goal is to determine the non-negative difference between a number and zero or between two numbers. Follow these steps to correctly assess the absolute value:

• Identify the numbers involved.
• Form an expression with the numbers inside the absolute value notation. If comparing two numbers, this will often involve subtraction, such as \( |a - b| \).
• Simplify the expression inside the vertical bars.
• Apply the rule that if the expression inside the bars is non-negative, it stays the same; if it's negative, convert it to positive.

This process translates to our problem as first combining the numbers -3.6 and 1.4 (the opposite of -1.4, to subtract it) to get -2.2 and then converting this to a positive number, which reflects the absolute difference.

Algebraic Distance Calculation

The concept of algebraic distance calculation is grounded on the idea that distance is an absolute value--it cannot be negative. To calculate the distance between any two points on the number line, we take the difference of their positions and apply the absolute value. This yields a positive distance.

For instance, to find the distance between two points, a and b, on the number line, the distance is \( |a - b| \) or \( |b - a| \), as the order of subtraction doesn't affect the result after taking the absolute value. In our exercise, the algebraic distance between -3.6 and -1.4 is the absolute value of their difference, which is \( |{-3.6} - {-1.4}| = |{-2.2}| = 2.2 \). This method ensures that distance is always presented as a positive value, consistent with its real-world interpretation.