Problem 76 - College Algebra

Question

Explain how to add rational expressions having no common factors in their denominators. Use \(\frac{3}{x+5}+\frac{7}{x+2}\) in your explanation.

Step-by-Step Solution

Answer

The sum of the rational expressions \( \frac{3}{x+5} + \frac{7}{x+2} \) is \( \frac{10x+41}{(x+5)(x+2)} \).

Step 1: Identifying the Denominators

The first step is identifying the denominators in the rational expressions. Here they are \(x+5\) and \(x+2\). For adding or subtracting fractions, we need a common denominator.

Step 2: Finding the Common Denominator

Now, we will find the common denominator. When the fractional expressions do not have common factors, their common denominator will be the product of their denominators. Here, the common denominator would be \((x+5)(x+2)\).

Step 3: Forming Equivalent Fractions

Let’s rewrite the rational expressions into equivalent fractions using the common denominator. The first expression \( \frac{3}{x+5} \) multiplies by \( \frac{x+2}{x+2} \) to become \( \frac{3(x+2)}{(x+5)(x+2)} \). The second expression \( \frac{7}{x+2} \) multiplies by \( \frac{x+5}{x+5} \) to become \( \frac{7(x+5)}{(x+5)(x+2)} \).

Step 4: Adding the Rational Expressions

Now that the two expressions have the same denominator, they can be added together. By adding the numerators we get:\( \frac{3(x+2)+7(x+5)}{(x+5)(x+2)} \).

Step 5: Simplifying the Result

The final step is to simplify the result. Distribute the values within the parentheses in the numerator to get \( \frac{3x+6+7x+35}{(x+5)(x+2)} \), which further simplifies to \( \frac{10x+41}{(x+5)(x+2)} \).