Problem 92 - 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

Therefore, \(\frac{3}{x+5}+\frac{7}{x+2} = \frac{10x+41}{(x+2)(x+5)}\)

Step 1: Identify and List the Denominators

In this instance, the denominators are \(x+5\) and \(x+2\). Write these two expressions out.

Step 2: Calculate the Least Common Denominator (LCD)

Since the two given expressions do not have any common factors, the LCD will be the product of the two denominators. Therefore, \(LCD = (x+5)(x+2)\).

Step 3: Re-write the Expression with the LCD

Re-write each fraction with the LCD as the denominator. Multiply the numerator and the denominator of each fraction by the missing factor from the LCD. This gives: \[\frac{3}{x+5} = \frac{3*(x+2)}{(x+5)(x+2)} = \frac{3x+6}{(x+5)(x+2)}\;\] and \[\frac{7}{x+2} = \frac{7*(x+5)}{(x+5)(x+2)} = \frac{7x+35}{(x+5)(x+2)}\].

Step 4: Perform the Addition

Now add the two fractions together: \[\frac{3x+6}{(x+2)(x+5)}+ \frac{7x+35}{(x+2)(x+5)} = \frac{3x + 6 + 7x + 35}{(x+2)(x+5)} = \frac{10x+41}{(x+2)(x+5)}\].