Problem 64 - Algebra for College Students

Question

Add or subtract the rational expressions as indicated. Be sure to express your answers in simplest form. $$ \frac{7 x}{x+4}-2 $$

Step-by-Step Solution

Answer

\(\frac{5x - 8}{x+4}\)

Step 1: Identify the Rational Expression

The expression given is \( \frac{7x}{x+4} - 2 \). Notice there is a rational expression (a fraction: \( \frac{7x}{x+4} \)) and a whole number (or integer: \( -2 \)). The goal is to combine these into a single rational expression.

Step 2: Express Integers as Fractions

Convert the integer \(-2\) into a fraction that has the same denominator as the first rational expression, which is \(x+4\). Therefore, \(-2\) can be written as \(\frac{-2(x+4)}{x+4}\).

Step 3: Combine the Rational Expressions

Now that both expressions have the same denominator, combine them into a single rational expression: \(\frac{7x}{x+4} - \frac{2(x+4)}{x+4}\). Since they share a common denominator, you can combine them as follows: \(\frac{7x - 2(x+4)}{x+4}\).

Step 4: Distribute and Simplify the Numerator

Distribute the \(-2\) in the numerator: \(7x - 2(x+4) = 7x - 2x - 8\). This simplifies to \(5x - 8\).

Step 5: Write the Final Simplified Expression

Combine these results into the simplified single expression: \(\frac{5x - 8}{x+4}\). This is now the simplest form of the initial expression.

Key Concepts

Simplifying Rational Expressions

Simplifying rational expressions is like cleaning up an equation to make it shorter and easier to work with. Imagine trying to pack a suitcase and needing to make everything fit neatly. We do this in math, too. When we simplify a rational expression, we are taking a fraction and reducing it to its simplest form. This means we eliminate any unnecessary complexity in the numerators or denominators. In the exercise you saw, the expression is \( \frac{5x - 8}{x+4} \), which came from simplifying \( \frac{7x}{x+4} - 2 \). The process involves:

• Matching denominators
• Distributing any constants in the numerators
• Combining like terms

By the end, we have something that's a lot easier to understand and work with.

Algebraic Fractions

Algebraic fractions are a type of rational expression that includes variables in the numerator, the denominator, or both. When dealing with algebraic fractions, it's just like dealing with regular fractions, but with a touch of algebra.For example, the term \( \frac{7x}{x+4} \) is an algebraic fraction because it includes the variable \( x \). Anything you can do with regular fractions— such as finding a common denominator or simplifying—you can do with algebraic fractions. The key steps involve:

• Understanding the role of the variable
• Ensuring operations keep the fraction equivalent
• Recognizing how algebraic expressions affect simplification

Working with these can be a bit more intricate than basic fractions, but the same foundational rules apply.

Combining Fractions

Combining fractions involves adding or subtracting them into a single fraction. In arithmetic, you need a common denominator first. The same rule applies to algebraic fractions.To combine fractions like \( \frac{7x}{x+4} - 2 \), you first transform the integer \( -2 \) into a fraction: \( \frac{-2(x+4)}{x+4} \).Next, because both fractions now share \( x+4 \) as a denominator, we combine them:

• Maintain the shared denominator
• Handle the numerators using distributive property

The combined expression becomes \( \frac{5x - 8}{x+4} \), showcasing how different terms are neatly merged into one.

College Algebra

College algebra often introduces students to more complex algebraic expressions and requires a robust understanding of how to manipulate them. By this stage, students encounter various types of expressions, including those involving rational expressions like the one we discussed.College algebra builds on fundamental skills:

• Simplifying expressions
• Performing operations with algebraic fractions
• Understanding the intricacies of polynomial expressions
• Developing problem-solving strategies for intricate algebraic equations

The expression \( \frac{7x}{x+4} - 2 \) is typical for this level, requiring students to apply an array of algebraic tactics, including knowledge of fractions and distribution, to condense it to \( \frac{5x - 8}{x+4} \). Mastering these skills is critical and lays the groundwork for even more advanced mathematical concepts.