Problem 92 - College Algebra Essentials

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

Step 1: Identify the Denominators

In this problem, the rational expressions have the denominators \(x+5\) and \(x+2\). We can see that these two expressions have no common factors.

Step 2: Find the Least Common Denominator (LCD)

The Least Common Denominator of two fractions is the least common multiple of their two denominators. In the context of algebraic expressions such as these, the Least Common Denominator is usually just the product of the two denominators. Therefore, the LCD of these two fractions is \((x+5)(x+2)\).

Step 3: Transform the fractions to have the LCD

To do this, multiply the first fraction by \(\frac{x+2}{x+2}\), and the second fraction by \(\frac{x+5}{x+5}\). This will give: \(\frac{3(x+2)}{(x+5)(x+2)}+\frac{7(x+5)}{(x+5)(x+2)}\)

Step 4: Add the Transformed Fractions

Now that the fractions have the same denominator, addition is simple. When adding fractions, one adds the numerators and keeps the common denominator. This gives: \(\frac{3(x+2)+7(x+5)}{(x+5)(x+2)}\)

Step 5: Simplify the Resulting Fraction

To do this, distribute the numbers in the numerators to get: \(\frac{3x+6+7x+35}{(x+5)(x+2)}\). Then, combine like terms to get: \(\frac{10x+41}{(x+5)(x+2)}\)