Problem 41 - Precalculus Mathematics for Calculus

Question

Perform the addition or subtraction and simplify. $$\frac{1}{x+5}+\frac{2}{x-3}$$

Step-by-Step Solution

Answer

\( \frac{3x+7}{(x+5)(x-3)} \)

Step 1: Identify the Problem Type

The given problem is an addition of two rational expressions: \( \frac{1}{x+5} + \frac{2}{x-3} \). To simplify this expression, we need a common denominator.

Step 2: Determine the Least Common Denominator (LCD)

The denominators of the expressions are \( x+5 \) and \( x-3 \). The least common denominator is the product of these two terms, \((x+5)(x-3)\).

Step 3: Rewrite Each Fraction with the LCD

To rewrite each fraction with the common denominator \((x+5)(x-3)\), multiply the numerator and the denominator of each fraction by the missing factor. For \( \frac{1}{x+5} \), multiply by \( \frac{x-3}{x-3} \) and for \( \frac{2}{x-3} \), multiply by \( \frac{x+5}{x+5} \).

Step 4: Perform the Multiplications

Calculate each term: \( \frac{1(x-3)}{(x+5)(x-3)} = \frac{x-3}{(x+5)(x-3)} \) and \( \frac{2(x+5)}{(x+5)(x-3)} = \frac{2x+10}{(x+5)(x-3)} \).

Step 5: Add the Numerators

Add the numerators from the previous step: \( (x-3) + (2x+10) = 3x + 7 \).

Step 6: Express the Combined Fraction

Combine the expression into a single fraction: \( \frac{3x + 7}{(x+5)(x-3)} \).

Step 7: Simplify if Possible

Check if the fraction can be simplified by factoring numerators or denominators. In this case, \( 3x + 7 \) does not factor further, and there are no common factors with \( (x+5)(x-3) \). Therefore, the expression is simplified as is.

Key Concepts

Least Common Denominator

When working with rational expressions, especially in addition or subtraction, the least common denominator (LCD) plays a crucial role. The LCD is the smallest expression that can evenly divide both denominators of the fractions you are working with. In our example, we have the rational expressions \( \frac{1}{x+5} \) and \( \frac{2}{x-3} \).
To find the LCD, look at each denominator separately. Since \( x+5 \) and \( x-3 \) are different, you multiply them to get the LCD: \( (x+5)(x-3) \). This product becomes the common denominator for both fractions involved in the operation.

This process ensures that both fractions are expressed with a uniform denominator, allowing you to perform the addition or subtraction effectively. Understanding the concept of the least common denominator is essential for dealing with any fraction operations where the denominators differ.

Simplifying Fractions

Simplifying fractions means reducing them to their simplest form. This involves ensuring that the numerator and the denominator have no common factors besides 1.
In the final expression derived from our example, which after addition becomes \( \frac{3x + 7}{(x+5)(x-3)} \), simplification is critical to present the answer as cleanly as possible. However, before we attempt to simplify, we must check if the numerator or the denominator can be factored further.

• First, examine the numerator, \( 3x + 7 \), to see if it can be factored. In this case, it cannot be factored further.
• Next, look at the denominator, which is already in its factored form, \( (x+5)(x-3) \), as part of our LCD.

Once you've determined that no further simplification is possible, the expression remains unchanged. Simplifying expressions whenever possible is vital, as it makes them easier to understand and work with.

Addition and Subtraction of Fractions

Adding or subtracting fractions involves combining them into a single fraction, but it requires that they have the same denominators. This rule applies equally to regular fractions and rational expressions.
In the exercise given, we have to add \( \frac{1}{x+5} \) and \( \frac{2}{x-3} \).

• First, express each fraction with the least common denominator, which we previously determined to be \( (x+5)(x-3) \).
• Rewrite each fraction so that their denominators match: \( \frac{x-3}{(x+5)(x-3)} \) and \( \frac{2x+10}{(x+5)(x-3)} \).

Now that both fractions share the same denominator, you can directly add their numerators: \( (x-3) + (2x+10) = 3x + 7 \). Combine them into a single fraction: \( \frac{3x + 7}{(x+5)(x-3)} \).

By following this approach, you can easily manage fraction operations, ensuring the results are accurate and simplified where possible.