Problem 71 - Algebra and Trigonometry

Question

Express the distance between the given numbers using absolute value. Then find the distance by evaluating the absolute value expression. $$-19\quad and\quad -4$$

Step-by-Step Solution

Answer

The distance between -19 and -4 is 15.

Step 1: Write the absolute value expression

First, represent the distance between -4 and -19 using the absolute value. To do this, subtract one number from the other and write it as an absolute value expression: \(|-19 - (-4)|\). Note that subtracting negative four translates into adding four in this context.

Step 2: Simplify the absolute value expression

Next, simplify the absolute value expression. It will look like this: \(|-19 + 4|\). Further simplifying: \(|-15|\).

Step 3: Evaluate the absolute value expression

To find the actual distance, evaluate the absolute value expression. The absolute value of -15 is just the positive version of 15. So, the distance between -4 and -19 on the number line is 15.

Key Concepts

Understanding Absolute Value Expressions

An absolute value expression reflects the magnitude of a number regardless of its sign. It is a measure of how far a number is from zero on the number line, which makes it always non-negative. For example, the absolute value of both -3 and 3 is 3, represented as \( |{-3}| = |3| = 3 \).

Whenever students encounter problems involving absolute values, they are often asked to represent some quantity without considering the direction, such as the distance between two points. This is why the exercise prompts expressing the distance between two numbers using an absolute value. In the given exercise, the absolute value expression is used to determine the distance between -19 and -4.

Evaluating Absolute Value

To evaluate the absolute value of a number or an expression, the main rule is to convert whatever value is inside the absolute value notation to a positive value, if it isn't already. Suppose a number 'a' is given within the absolute value bars, such as \( |a| \). If 'a' is positive or zero, the absolute value remains as 'a'. If 'a' is negative, then the absolute value becomes '-a', essentially stripping off the negative sign.

Let's relate this to the exercise: when we have \( |-15| \), we disregard the negative sign, making it 15, which is the distance between the given numbers on the number line. This step is crucial because many students might forget to remove the negative sign and incorrectly interpret the distance as negative.

Calculating Number Line Distance

When envisioning the number line distance, one can imagine a horizontal line with zero at its center, positive numbers extending to the right, and negative numbers to the left. The distance between two points is the number of units you would need to move from one point to reach the other.

Assessing the distance on the number line for the numbers in the exercise, -19 and -4, involves subtracting one number from the other and taking the absolute value of the result. This operation nullifies the direction of distance, focusing solely on how far apart the numbers are. In our example, after simplifying the expression \( |-19 - (-4)| \) as explained in the textbook steps, we're left with \( |-15| \) or simply 15 units. This distance represents the objective space between the two numbers without regard to direction.