Problem 39 - Precalculus: Functions and Graphs

Question

Exer. 33-40: Replace the symbol \(\square\) with elther = or \(\neq\) to make the resulting statement true for all real numbers \(a, b\) \(c,\) and \(d,\) whenever the expressions are defined. $$\frac{a-b}{b-a} \square-1$$

Step-by-Step Solution

Answer

Replace \( \square \) with \( = \) since \( \frac{a-b}{b-a} = -1 \).

Step 1: Examine the expression

The given expression is \( \frac{a-b}{b-a} \) and we need to compare it to \(-1\) using either \( = \) or \( eq \). Let's explore the expression more closely.

Step 2: Rewrite the fraction

Notice that \( \frac{a-b}{b-a} \) can be simplified. Observe that \( b-a = -(a-b) \), therefore \( \frac{a-b}{b-a} = \frac{a-b}{-(a-b)} = -1 \).

Step 3: Conclude the relationship

Since the expression \( \frac{a-b}{b-a} = -1 \), we can replace the box with \( = \). The statement is true for all real numbers \( a \) and \( b \) where the expression is defined (i.e., \( a eq b \)).

Key Concepts

Fraction Simplification

Fraction simplification is a fundamental concept in mathematics. It involves changing a fraction to its simplest form without changing its value. In this exercise, we encounter the fraction \( \frac{a-b}{b-a} \). At first glance, this might seem complex, but we can simplify it by understanding a pivotal property of fractions: multiplying or dividing both the numerator and the denominator by the same non-zero number does not change the value of the fraction.

In the expression \( b-a = -(a-b) \), we use this property to simplify the fraction. We observe:

• \( b-a \) is actually the negative of \( a-b \).
• This means that \( \frac{a-b}{b-a} = \frac{a-b}{-(a-b)} \).
• If we divide the whole fraction by \( a-b \), we're left with \( -1 \).

Understanding and visualizing this transformation is critical to grasping fraction simplification. It's about recognizing patterns and relationships within the expression and using basic algebraic rules, such as multiplying both the numerator and the denominator by \(-1\).

Expression Equality

Expression equality occurs when two mathematical expressions represent the same quantity. In this exercise, we're determining whether the fraction \( \frac{a-b}{b-a} \) is equal to \(-1\). This involves determining if the two sides of the expression hold the same value for all allowed real numbers.

• To establish expression equality, the values on either side of the equation need to be identical for all values of the variables involved.
• Once we've simplified \( \frac{a-b}{b-a} \) to \(-1\), we're saying these two expressions are equivalent for all real numbers \( a \) and \( b \), as long as \( a eq b \), due to division by zero restrictions.

The concept of equality is central in algebra, as it provides a foundation for solving equations and understanding the relationships between different expressions. Determining equality requires careful simplification and analysis to ensure you're not mistakenly altering the meaning of the original mathematical statement.

Mathematical Expressions

Mathematical expressions are combinations of numbers, variables, and arithmetic operations. They represent quantities and relationships between different mathematical entities.

In our task, \( \frac{a-b}{b-a} \) is a well-defined mathematical expression. Examining this expression involves several steps:

• Identifying variables: Here, \( a \) and \( b \) are variables that can take on any real number, provided the expression remains defined.
• Performing operations: Simplifying \( \frac{a-b}{b-a} \) through algebraic operations allows us to express it in different forms.
• Evaluating expressions: Calculating or comparing the expression's value under specific conditions helps us determine equality or inequality.

Mathematical expressions function as the building blocks of algebra and calculus, pivotal for constructing equations and inequalities. Understanding how to interpret, manipulate, and simplify these expressions is crucial for problem-solving and theoretical mathematics.