Problem 55 - College Algebra

Question

Exercises \(55-57\) will help you prepare for the material covered in the next section. $$ \text { Subtract: } \frac{3}{x-4}-\frac{2}{x+2} $$

Step-by-Step Solution

Answer

The solution to the subtraction of the given fractions is \(\frac{x + 14}{x^{2} - 2x - 8}\)

Step 1 - Identify Common Denominator

The most important part of performing the subtraction of fractions is to get a common denominator. So for the two groups \(x - 4\) and \(x + 2\), there is no initial common denominator that exists. Hence, you gain one by multiplying both denominators. So, the common denominator will be \((x - 4) * (x + 2)\)

Step 2 - Apply Common Denominator to Each Fraction

To reach common denominator, multiply top and bottom of first fraction by \((x + 2)\) and top and bottom of second fraction by \((x - 4)\). You get \(\frac{3 * (x + 2)}{(x - 4) * (x + 2)} - \frac{2 * (x - 4)}{(x - 4) * (x + 2)}\)

Step 3 - Simplify the Fractions and Perform the Subtraction

Expand the multiplication in the numerators of the fraction in the previous step. You get \(\frac{3x + 6}{x^{2} - 2x - 8} - \frac{2x - 8}{x^{2} - 2x - 8}\). Since the denominators are same, you can simply subtract the numerators to obtain: \(\frac{3x + 6 - 2x + 8}{x^{2} - 2x - 8}\) = \(\frac{x + 14}{x^{2} - 2x - 8}\)

Key Concepts

Common Denominator

When subtracting fractions, one of the first steps is to find a common denominator. This approach enables you to combine the fractions by aligning their denominators so that the subtraction can be carried out smoothly. The common denominator must be a multiple of both individual denominators.

Finding the least common denominator (LCD) minimizes the complexity of the following operations. However, when it comes to algebraic fractions where the denominators are polynomials, such as \(x-4\) and \(x+2\), you may not have the luxury of a 'least' common denominator. Instead, your common denominator will be the product of these distinct algebraic expressions, resulting in \(x-4\) times \(x+2\) or \(x^2-2x-8\). After obtaining this, both fractions are transformed to have this shared denominator which sets the stage for simplifying the expressions.

Algebraic Fractions

Algebraic fractions are similar to regular fractions but involve variables. In these types of fractions, the numerator and/or the denominator contain algebraic expressions. For the subtraction \(\frac{3}{x-4}-\frac{2}{x+2}\), each fraction's denominator has a binomial with a variable.

When working with algebraic fractions, we handle them the same as numerical fractions. Aligning the denominators is usually achieved through multiplication or factoring. Once lined up, you can add or subtract the numerators directly. The tricky part with algebraic fractions can be simplifying after combined. It's vital to factor and reduce wherever possible to achieve the simplest form of the expression.

Simplifying Expressions

Simplifying expressions is an essential skill in algebra. It entails reducing an algebraic expression to its simplest form without changing its value. To simplify \(\frac{3x + 6}{x^2 - 2x - 8} - \frac{2x - 8}{x^2 - 2x - 8}\), we combine the numerators while the denominator remains the same. This process sometimes unveils further opportunities for factorization or cancellation.

In the provided exercise, simplifying the subtracted numerators \(3x + 6 - 2x + 8\) results in \(x + 14\). No further reduction can occur between the numerator and the denominator \(x^2 - 2x - 8\), giving us the final simplified algebraic fraction. It's important to check for any factorable expressions both before and after the subtraction to ensure the expression is fully simplified.