Problem 120 - Introductory Algebra for College Students

Question

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. If \(a\) and \(b\) are negative numbers, then \(a-b\) is sometimes a negative number.

Step-by-Step Solution

Answer

The statement is false. If \(a\) and \(b\) are negative numbers, then \(a-b\) is never a negative number.

Step 1: Understanding the Problem

The statement is: 'If \(a\) and \(b\) are negative numbers, then \(a-b\) is sometimes a negative number.' We need to ascertain the validity of this statement.

Step 2: Evaluating the Negative Numbers Operation

If \(a\) and \(b\) are negative numbers, that means both \(a\) and \(b\) are less than 0. In subtraction of the form \(a - b\), if \(b\) is negative, subtracting negative \(b\) is equivalent to adding |b| (absolute value of \(b\)). So the operation \(a - b\) becomes \(a + |b|\), and since \(a\) is also negative, adding a positive number |b| to a smaller negative number \(a\) will make the result closer to 0 or become positive, but it will never be negative.

Step 3: Creating a True Statement

To make the statement true, we can say, 'If \(a\) and \(b\) are negative numbers, then \(a-b\) is never a negative number.'

Key Concepts

Negative Numbers

Negative numbers are numbers that are less than zero. They are represented with the minus sign in front of the numeral, like -3 or -7. These numbers are the opposite of positive numbers. They're found to the left of zero on the number line.

Negative numbers have some unique properties. When you multiply or divide two negative numbers, the result is positive. However, when you multiply or divide a negative and a positive number, the result is negative. It's important to be attentive to these properties when solving algebraic problems.

• If you add a negative number to a positive number, imagine you're taking away from the positive number's value.
• Adding two negative numbers will result in a more negative number.

Understanding the nature of negative numbers is crucial. They often appear in various contexts such as in financial debts, temperatures below zero, or when discussing depths below sea level.

Subtraction

Subtraction is a basic mathematical operation. It involves taking one number away from another. In general, subtraction can be seen as the inverse operation of addition. This means instead of combining quantities, you are removing a quantity.

When you subtract a negative number, it is similar to adding its absolute value (the positive version of the number). For instance, subtracting -5 from 3 is the same as adding its absolute value, which is 5. Thus, 3 - (-5) equals 3 + 5, resulting in 8.

• The order of numbers in subtraction matters. If you reverse the order, the result can greatly change.
• Subtracting a larger number from a smaller one results in a negative number.
• Subtracting zero leaves the number unchanged.

Understanding subtraction's properties, especially involving negative numbers, is beneficial in simplifying expressions and solving equations.

Absolute Value

Absolute value refers to the distance of a number from zero on the number line, always expressed as a non-negative value. For any number whether positive or negative, the absolute value turns it into its positive counterpart.

For instance, the absolute value of -4 is 4, and the absolute value of 7 is 7. It is often denoted by two vertical bars surrounding the number, such as |a|.

Absolute value is very useful in situations where only the size of a number matters and not its direction. It can help clarify distance, magnitude, and financial gains or losses, regardless of whether these values were negative or positive in the first place.

• Absolute values are never negative.
• Calculating absolute value helps in solving equations involving sum or difference of numbers.
• When comparing two absolute values, it determines the magnitude without considering the direction.

This concept aids in understanding distance measures and is essential for interpreting real-world data with both positive and negative variables.