Problem 95 - Beginning and Intermediate Algebra

Question

If \(a\) is a positive number and \(b\) is a negative number, determine whether each statement is true or false. \(|b-a|\) is always a positive number.

Step-by-Step Solution

Answer

The statement is true; \(|b-a|\) is always positive because absolute values are non-negative.

Step 1: Understanding Absolute Value

The absolute value of a number, denoted by \(|x|\), is the distance of that number from zero on the number line. Since distance cannot be negative, the absolute value is always non-negative. For any real numbers \(x\) and \(y\), \(|x - y|\) is the non-negative distance between \(x\) and \(y\).

Step 2: Evaluate the Expression

Consider the expression \(|b-a|\). Since \(a\) is positive and \(b\) is negative, the expression inside the absolute value, \(b-a\), could be negative or zero, but the absolute value will make it non-negative.

Step 3: Consider the Simplest Case

If \(b = -1\) and \(a = 1\), then \(b-a = -1 - 1 = -2\), hence \(|b-a| = |-2| = 2\), which is positive.

Step 4: General Analysis

For the general case, since \(a\) is positive and \(b\) is negative, \(b-a < 0\). The absolute value \(|b-a|\) will convert this negative number (or zero) to a positive number since \(|b-a| = -(b-a) = a-b\), which is positive because \(a > b\).

Step 5: Conclusion

Since absolute values create non-negative results and in this scenario, specifically positive due to \(a > b\), the statement \(|b-a|\) is always a positive number, is true.

Key Concepts

positive numbers

Positive numbers are a fundamental concept in mathematics. They are numbers that are greater than zero and are found to the right of zero on the number line. Some simple examples of positive numbers include 1, 2, 3, and so on.
One of the identifiers of a positive number is that it does not have a negative sign in front of it. When working with mathematical operations, positive numbers can be added, subtracted, multiplied, or divided like any other real numbers. However, when added to negative numbers, positive numbers will influence the result in a way that brings the outcome closer to zero or beyond zero on the positive side.

• Example: The result of adding a positive number to a negative number is determined by the difference in their magnitudes. If the positive number is larger, the result is positive.
• Notation: Typically represented without any sign, e.g., 5 instead of +5.

negative numbers

On the number line, negative numbers are located to the left of zero. They carry a minus sign (-) that differentiates them from positive numbers. Common negative numbers include -1, -2, and -3.
Negative numbers are essential for representing values less than zero, such as temperatures below freezing or debts.

• Negative Times Positive: A negative number multiplied by a positive number results in a negative product.
• Negative Plus Positive: The sum of a negative and a positive number depends on their magnitudes.

In the context of absolute values, such as \(b-a\) where \(b\) is a negative number, the expression inside the absolute value can be negative itself. However, when the absolute value is applied, the output becomes non-negative.

distance on the number line

The concept of distance on a number line is central to understanding absolute value. Distance, in this context, refers to how far apart two numbers are from each other on the number line. Since distance cannot be negative, this naturally leads to the understanding of absolute value as a non-negative measure.
Absolute value is used to determine this distance mathematically, and most simply calculated by taking the difference between two numbers and applying the absolute value function. For example, \|x-y\| finds the distance between \(x\) and \(y\) on the number line.

• Mathematical Representation: \|x-y\| gives the distance between two numbers \(x\) and \(y\).
• Non-negative Result: The result is either zero or a positive number, reflecting that distance is always non-negative.

Understanding this concept is useful for various applications, such as absolute value equations and inequalities.