Problem 41 - College Algebra

Question

Add or subtract as indicated. $$ \frac{3}{x+4}+\frac{6}{x+5} $$

Step-by-Step Solution

Answer

The sum of these fractions is \(\frac{9x+39}{(x+4)(x+5)}\).

Step 1: Identify the Different Denominators.

The different denominators are \(x+4\) and \(x+5\).

Step 2: Get the Common Denominator.

The common denominator is the product of the different denominators, which is \((x+4) \cdot (x+5)\)

Step 3: Transform the Fractions to have the Common Denominator.

The first fraction becomes \(\frac{3 \cdot (x + 5)}{(x + 4)\cdot (x + 5)}\) and the second becomes \(\frac{6 \cdot (x + 4)}{(x + 4)\cdot (x + 5)}\)

Step 4: Add the Fractions.

Adding the fractions results in: \(\frac{3 \cdot (x + 5) + 6 \cdot (x + 4)}{(x + 4)\cdot (x + 5)} = \frac{3x+15+6x+24}{(x+4)(x+5)} = \frac{9x+39}{(x+4)(x+5)}\)

Step 5: Simplify if Possible.

Since there are no common factors between the numerator and the denominator, the fraction cannot be simplified further.

Key Concepts

Common Denominator

When you come across adding or subtracting fractions with different denominators, you need to find a common denominator so that each fraction expresses a part of the same whole. In our case, the fractions have denominators of (x+4) and (x+5). The easiest way to find a common denominator is to multiply these unique denominators together, yielding (x+4)(x+5).

Utilizing the common denominator is essential because it allows us to combine fractions by aligning their parts to a uniform division of the whole, simplifying our calculations. It's the foundational step in solving addition and subtraction problems involving fractions with dissimilar denominators. Without this step, we would be trying to combine pieces of different sizes, which wouldn't make much sense.

Simplify Fractions

After adding or subtracting fractions, we often end up with an answer that can be further reduced to its simplest form, known as simplifying the fraction. This often involves finding factors common to both the numerator and the denominator and eliminating them. However, in our exercise, the fraction \(\frac{9x+39}{(x+4)(x+5)}\) cannot be simplified because there are no common factors.

When Can We Simplify?

Essentially, we simplify a fraction when both the top (numerator) and bottom (denominator) can be divided evenly by the same number. Always check for factors common to the numerator and denominator, including variable terms in algebraic fractions. If we find them, we divide both parts of the fraction to make it simpler. For instance, if we had \(\frac{16x^2}{4x}\), we could simplify it to \(\frac{4x}{1}\) since both the numerator and denominator have a common factor of 4x.

Algebraic Expressions

Dealing with algebraic expressions in fractions can be daunting at first, but it follows the same principles we use with numerical fractions. An algebraic expression is a combination of numbers, variables (like x), and operators (plus, minus, multiply, divide) that represents a specific value. When adding algebraic fractions, we first find a common denominator that includes the variables, then we rewrite each fraction with the common denominator, and finally, we combine the numerators.

In our initial problem \(\frac{3}{x+4} + \frac{6}{x+5}\), we multiplied the denominators to get a common one, then adjusted the numerators accordingly. After that, we were able to combine them because the algebraic expressions in the numerators followed conventional arithmetic rules. Algebraic expressions allow us to work with unknown values in a concrete way, as we align the expressions to common terms for easy manipulation.