Problem 94 - 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 false; \(|b|-|a|\) is not always positive.

Step 1: Understanding Absolute Values

The absolute value of a number is its distance from zero on the number line, regardless of direction. Thus, for any positive number \(a\), \(|a| = a\) and for a negative number \(b\), \(|b| = -b\).

Step 2: Apply Definitions to the Expression

Given the expression \(|b| - |a|\), using the absolute value definitions, we rewrite it as \(-b - a\).

Step 3: Evaluate the Sign of the Expression

Since \(a\) is positive, the term \(-a\) is negative. Similarly, \(-b\) is positive because \(b\) is negative. Therefore, \(-b - a\) involves subtracting a positive \(a\) from a positive \(-b\).

Step 4: Conclusion Based on the Sign

The expression \(-b - a\) could be positive, zero, or negative, depending on whether \(-b\) is greater than, equal to, or less than \(a\). Hence, the expression \(|b| - |a|\) is not always positive.

Key Concepts

Understanding Positive and Negative Numbers

When we talk about numbers, they can be either positive or negative. Positive numbers are any numbers greater than zero. These are numbers you would typically encounter when counting. For example, 1, 2, 3, and so on. Negative numbers are less than zero and are usually associated with values like debt or temperatures below freezing, such as -1, -2, -3, etc.
It's also helpful to understand the number line:

• Positive numbers are to the right of zero.
• Negative numbers are to the left of zero.
• The distance from any number to zero is its absolute value.

In the problem above, we are dealing with one positive number (\(a\)) and one negative number (\(b\)). Knowing the difference between positive and negative numbers helps us comprehend how their absolute values interact above zero.

Exploring Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operations. They form the foundation of algebra. A variable is a symbol, often a letter, representing a number whose value might change. Unlike equations, expressions don't have an equals sign.
For example, if we consider the expression \(|b| - |a|\) from the exercise:

• \(|b|\) denotes the absolute value of the variable \(b\), which is negative.
• \(|a|\) denotes the absolute value of the variable \(a\), which is positive.

The way expressions are evaluated involves using rules established by mathematics. Here, the rules for absolute values dictate how we rewrite and solve the problem. Knowing how to simplify and interpret algebraic expressions is critical for solving these types of mathematical problems.

Introduction to Real Numbers

Real numbers are a vast group that includes both rational and irrational numbers. Rational numbers can be expressed as fractions, whereas irrational numbers cannot. This collection encompasses all kinds of numbers, including zero, positive numbers, and negative numbers, found on the number line.
When tackling the problem \(|b| - |a|\), it's essential to understand that:

• Both \(a\) and \(b\) are real numbers, highlighting the versatile nature of this number set.
• The primary goal is to discern the behavior of these numbers subject to operations, like subtraction, after determining their absolute values.

Real numbers support operations like addition, subtraction, multiplication, and division, making them highly applicable in various mathematical contexts. Thus, grasping the concept of real numbers is fundamental for analyzing the properties and operations on numbers.