Problem 35 - College Algebra

Question

express the distance between the given numbers using absolute value. Then find the distance by evaluating the absolute value expression. \(-19\) and \(-4\)

Step-by-Step Solution

Answer

The distance between -19 and -4 is \( |15| = 15 \)

Step 1: Write the numerical expression

Firstly, write the numbers as a subtraction expression using absolute value (| |), where the larger number comes before the smaller one. So, the expression would be \( |-19 - (-4)|\).

Step 2: Subtract inside the absolute value

Next, find the result of subtraction inside the absolute value (| |). As the subtraction is \( -19 - (-4)\), this becomes \( |-19 + 4|\), because -- turns into +. This then equals \( |-15|\).

Step 3: Determine the absolute value

Finally, the previous expression gives the absolute value of -15 which is 15. So, the final answer expression is \( |15|\).

Key Concepts

Expressing Distance

When dealing with numbers on a number line, expressing the distance between two points is a fundamental concept. This distance is always a positive quantity, regardless of which number is larger or whether the numbers are positive or negative. To express this distance, we use the mathematical notion of absolute value, denoted by two vertical lines (| |). The absolute value of a number is the non-negative value of that number without regard to its sign.

For example, if we consider the numbers -19 and -4, we can express the distance between them using an absolute value expression: \( |-19 - (-4)| \). This numerical statement captures the distance on the number line between the two points but needs to be simplified to find the exact distance. By expressing distance in this way, we can easily determine how far apart any two numbers are.

Evaluating Absolute Value

Evaluating the absolute value involves two main steps: simplifying the expression inside the absolute value symbol and then finding the non-negative value. When the expression inside the absolute value symbols involves subtraction, as in \( |-19 - (-4)| \), we interpret this as \( |-19 + 4| \) since subtracting a negative number is the same as adding its positive counterpart. Thus, the expression simplifies to \( |-15| \).

The absolute value of -15, denoted by \( |-15| \), is simply 15 because absolute value measures the distance a number is from zero on the number line, and distance is never negative. Evaluating absolute value is a crucial skill that allows us to understand and calculate the magnitude of numbers without considering their direction or sign.

Subtraction with Negative Numbers

Subtracting negative numbers can be counterintuitive at first because it involves a double negative, which actually turns the operation into addition. When subtracting a negative number, the two negatives effectively cancel out, leaving you with a positive. For example, \( -19 - (-4) \) becomes \( -19 + 4 \), because here, the subtraction of a negative is equivalent to adding the positive value of that number.

Tips for Subtracting Negative Numbers:

• Remember that two negatives make a positive.
• Always convert the subtraction of a negative into addition to simplify the problem.
• Think of subtraction as 'owing' and converting to addition as 'receiving' to aid conceptual understanding.

Subtraction with negative numbers is a fundamental skill in algebra, and mastering it will help you simplify and solve a wide range of mathematical problems.