Problem 52 - Elementary and Intermediate Algebra

Question

Perform the operations. Simplify, if possible. $$ \frac{4}{b-6}+\frac{b}{6-b} $$

Step-by-Step Solution

Answer

The expression simplifies to \( \frac{4-b}{b-6} \).

Step 1: Identify the Problem

We have the expression \( \frac{4}{b-6} + \frac{b}{6-b} \) to simplify. Notice that the second denominator \( 6-b \) is the negative of \( b-6 \).

Step 2: Simplify Denominators

Since \( 6-b = -(b-6) \), rewrite the second fraction by factoring out the negative sign: \( \frac{b}{6-b} = -\frac{b}{b-6} \). Now the expression becomes \( \frac{4}{b-6} - \frac{b}{b-6} \).

Step 3: Combine Fractions

Now, both fractions have the same denominator, \( b-6 \). Combine them: \( \frac{4 - b}{b-6} \).

Step 4: Simplification Check

Check if further simplification is possible. The numerator is \( 4-b \), which cannot be simplified further with respect to the denominator. Therefore, \( \frac{4 - b}{b-6} \) is the simplest form.

Key Concepts

Algebraic Fractions

Algebraic fractions are similar to regular fractions, but instead of integers, they contain algebraic expressions in the numerator, the denominator, or both. For example, the expression \( \frac{4}{b-6} \) is an algebraic fraction because its denominator includes the variable \( b \). These types of fractions require an understanding of how to manipulate expressions containing variables, similar to working with numbers in arithmetic fractions, but with extra steps.Key points:

• They are fractions that include variables as part of their numerators or denominators.
• Just like numerical fractions, the denominator must not be zero. Hence, in algebraic fractions, it is often necessary to specify restrictions on the variable to avoid division by zero.
• The same arithmetic operations (addition, subtraction, multiplication, and division) that apply to numerical fractions can also be applied to algebraic fractions.

These fractions often appear in algebra problems where terms need to be simplified, expressions are factored, or equations solved. Understanding algebraic fractions is essential because they lay the groundwork for more complex algebraic operations and calculus.

Simplifying Expressions

Simplifying expressions refers to reducing them to their simplest form. This usually involves combining like terms, factoring expressions, and making cancellations possible. In the context of the problem with rational expressions, we're dealing with the simplification of algebraic fractions. In the case of \( \frac{4}{b-6} + \frac{b}{6-b} \), it is important to simplify the denominators if possible. Notice that \( 6-b \) is equivalent to \(-(b - 6)\). This observation allows transforming the second term to have the same base denominator as the first.

• Rewrite the expression so that all terms have the same denominator, which allows for combining them into a single fraction.
• Combine numerators by performing the corresponding operation (addition or subtraction).
• Check if the resultant expression can be factored and further reduced.

A simpler form of an expression makes it easier to analyze or use in further operations. Simplification reduces complexity and uncovers the core elements of an expression.

Common Denominators

Finding a common denominator is a crucial step when adding or subtracting fractions, especially algebraic fractions, to ensure that the fractions can be combined correctly. A common denominator is a shared multiple of the denominators found in each term of the expression.For example, in the problem \( \frac{4}{b-6} + \frac{b}{6-b} \), we convert the denominators to match. Recognizing the relationship between \( b-6 \) and \( 6-b \) as negatives of each other helps transform the expression. This becomes evident when \( \frac{b}{6-b} \) is rewritten as \(-\frac{b}{b-6}\).

• Transform all terms to have a common denominator, facilitating their combination into a single fraction.
• By having a uniform denominator, you can directly add or subtract the numerators.
• This aligns with the principle of keeping the operation on the same base, which is the denominator.

Ultimately, finding a common denominator simplifies the process of fraction operations, allowing for easier manipulation and solution finding in algebraic expressions.