Problem 81 - College Algebra with Modeling and Visualization

Question

Simplify. $$ \frac{x+3}{x-5}+\frac{5}{x-3} $$

Step-by-Step Solution

Answer

\(\frac{x^2 + 5x - 34}{(x-5)(x-3)}\)

Step 1: Identify the common denominator

To combine the two fractions, we need a common denominator. The denominators we have are \(x-5\) and \(x-3\). The least common denominator (LCD) is the product of these two expressions, \((x-5)(x-3)\).

Step 2: Rewrite fractions with the common denominator

Adjust each fraction to have the common denominator \((x-5)(x-3)\). For the first fraction, \(\frac{x+3}{x-5}\), multiply both the numerator and denominator by \(x-3\). It becomes \(\frac{(x+3)(x-3)}{(x-5)(x-3)}\). For the second fraction, \(\frac{5}{x-3}\), multiply both the numerator and denominator by \(x-5\). It becomes \(\frac{5(x-5)}{(x-3)(x-5)}\).

Step 3: Expand the numerators

Expand the numerators of both fractions: \( (x+3)(x-3) = x^2 - 9 \) and \( 5(x-5) = 5x - 25 \). This gives us \(\frac{x^2 - 9}{(x-5)(x-3)}\) for the first fraction and \(\frac{5x - 25}{(x-3)(x-5)}\) for the second fraction.

Step 4: Add the fractions

Since both fractions now have a common denominator, we can add them by combining their numerators over the common denominator: \(\frac{x^2 - 9 + 5x - 25}{(x-5)(x-3)}\).

Step 5: Simplify the combined numerator

Combine like terms in the numerator: \(x^2 + 5x - 9 - 25 = x^2 + 5x - 34\). So the expression is \(\frac{x^2 + 5x - 34}{(x-5)(x-3)}\).

Step 6: Write the final simplified expression

The expression cannot be factored further so the final simplified expression is \(\frac{x^2 + 5x - 34}{(x-5)(x-3)}\).

Key Concepts

Common Denominator

In algebraic fractions, a common denominator is pivotal for combining fractions effectively. It is especially important because fractions can only be added or subtracted when their denominators match. Here's how you find it:

• Identify the unique denominators in the problem. Here, they are \(x-5\) and \(x-3\).
• Calculate the least common denominator (LCD), which is often the product of the denominators when dealing with polynomial expressions. For our example, the LCD is \((x-5)(x-3)\).

Utilizing a common denominator ensures that you convert multiple fractions into a single, easier-to-manage expression. Once achieved, the numerators can be focused on, leading to further simplifications or operations.

Simplifying Expressions

Simplifying expressions often follows the process of expanding and combining terms to make the equation as concise and clear as possible. Here's a breakdown:

• After establishing a common denominator, rewrite each expression such that their denominators match. Multiply the numerator and the denominator by the necessary terms to achieve this.
• For instance, the expressions in the example are rewritten with the common denominator: \(\frac{(x+3)(x-3)}{(x-5)(x-3)}\) and \(\frac{5(x-5)}{(x-3)(x-5)}\).
• Expand the numerators: simplify \((x+3)(x-3)\) to get \(x^2 - 9\) and \(5(x-5)\) to get \(5x - 25\).

Through simplification, complex equations are transformed into simpler forms. This simplification not only makes the solution clearer but also ensures accurate further calculations.

Combining Fractions

Combining fractions involves carrying out addition or subtraction after rewriting them to have a common denominator. Here's how it works:

• With both fractions now sharing the common denominator \((x-5)(x-3)\), adding them becomes straightforward.
• Add their numerators together: \(x^2 - 9\) from the first fraction and \(5x - 25\) from the second. This results in the new numerator: \(x^2 + 5x - 34\).
• The combined fraction then becomes \(\frac{x^2 + 5x - 34}{(x-5)(x-3)}\).

It's vital to remember that after combining, the result should always be checked for further simplification. In this case, no additional simplification is possible, making \(\frac{x^2 + 5x - 34}{(x-5)(x-3)}\) the final simplified form. Combining fractions efficiently can drastically reduce the complexity of solving algebraic fraction equations.