Problem 51 - Elementary Algebra

Question

For exercises \(25-68\), evaluate or simplify. $$ \frac{5-\frac{1}{x+3}}{2+\frac{4}{x-1}} $$

Step-by-Step Solution

Answer

The expression simplifies to \(\frac{(5x + 14)(x - 1)}{(x + 3)(2x + 2)}\).

Step 1 - Simplify the Inner Fractions

Look at the numerator and the denominator of the main fraction. The numerator is 5 - 1/(x + 3) and the denominator is 2 + 4/(x - 1). Focus on simplifying these parts first.

Step 2 - Combine the Terms in the Numerator

The numerator is 5 - 1/(x + 3). To combine these terms, convert 5 to a fraction with the same denominator as 1/(x + 3). This becomes: \[ \frac{5(x + 3)}{(x + 3)} - \frac{1}{(x + 3)} = \frac{5x + 15 - 1}{(x + 3)} = \frac{5x + 14}{(x + 3)} \]

Step 3 - Combine the Terms in the Denominator

The denominator is 2 + 4/(x-1). To combine these terms, convert 2 to a fraction with the same denominator as 4/(x - 1). This becomes: \[ \frac{2(x - 1)}{(x - 1)} + \frac{4}{(x - 1)} = \frac{2x - 2 + 4}{(x - 1)} = \frac{2x + 2}{(x - 1)} \]

Step 4 - Simplify the Main Fraction

Now the expression is: \[ \frac{\frac{5x + 14}{(x + 3)}}{\frac{2x + 2}{(x - 1)}} \] To simplify, multiply the numerator by the reciprocal of the denominator: \[ \frac{5x + 14}{(x + 3)} \times \frac{(x - 1)}{2x + 2} \]

Step 5 - Simplify the Resulting Expression

Simplify the final expression by performing the multiplication: \[ \frac{(5x + 14)(x - 1)}{(x + 3)(2x + 2)} \] Simplify further, if possible, by canceling out common factors. In this case, there are no common factors to cancel, so this is the simplified form.

Key Concepts

Algebraic Expressions

Algebraic expressions are a combination of numbers, variables, and operations. For instance, in our problem, we have expressions like \(5 - \frac{1}{x + 3}\) and \(2 + \frac{4}{x - 1}\). These expressions can involve addition, subtraction, multiplication, and division involving variables. Learning to handle algebraic expressions is crucial as it allows us to manipulate and simplify complex mathematical problems.

Fractions Simplification

Simplifying fractions involves reducing them to their simplest form. With algebraic fractions, it’s important to first address the smaller fractions within the larger expression. As seen in our exercise, the first step was to convert the terms in the numerator and denominator into common denominators:

• \( \frac{5(x + 3)}{(x + 3)} - \frac{1}{(x + 3)} = \frac{5x + 15 - 1}{(x + 3)} \)
• \( \frac{2(x - 1)}{(x - 1)} + \frac{4}{(x - 1)} = \frac{2x - 2 + 4}{(x - 1)} \)

Understanding how to combine and reduce these parts can help in simplifying the entire fraction.

Rational Expressions

Rational expressions are fractions where the numerator and the denominator are algebraic expressions. The key to solving problems involving rational expressions is to handle the numerator and denominator independently first. In the solution provided, we methodically broke down the problem:

• Simplify the numerator \( \frac{5x + 14}{x + 3} \)
• Simplify the denominator \( \frac{2x + 2}{x - 1} \)
• Then, combine them by multiplying the numerator by the reciprocal of the denominator: \( \frac{5x + 14}{x + 3} \times \frac{x - 1}{2x + 2} \)

This process showcases how rational expressions can be managed through systematic simplification.

Numerator and Denominator Operations

Working with numerators and denominators is a fundamental aspect of fraction handling. In our given problem, converting terms into a common denominator within the numerator and denominator helped us organize them into simpler fractions. Here’s a quick breakdown:

• Combine \( 5 \) and \( \frac{1}{x+3} \) in the numerator by writing them as fractions over a common denominator.
• Do the same for \( 2 \) and \( \frac{4}{x-1} \) in the denominator.

After this, perform multiplication: \( \frac{(5x + 14)(x - 1)}{(x + 3)(2x + 2)} \). This approach simplifies complex expressions by isolating and simplifying each part systematically.