Problem 41 - College Algebra Essentials

Question

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

Step-by-Step Solution

Answer

The sum of the two fractions \(\frac{3}{x+4}\) and \(\frac{6}{x+5}\) is \(\frac{9x+39}{x^2+9x+20}\)

Step 1: Find Common Denominator

The common denominator for the fractions \(\frac{3}{x+4}\) and \(\frac{6}{x+5}\) is the product of '(x+4)' and '(x+5)', which is '(x + 4)(x + 5)'.

Step 2: Rewrite Fractions

We rewrite each fraction with the common denominator identified in step 1. So, \(\frac{3}{x+4}\) becomes \(\frac{3(x+5)}{(x+4)(x+5)}\) and \(\frac{6}{x+5}\) becomes \(\frac{6(x+4)}{(x+4)(x+5)}\).

Step 3: Perform the Addition

We now add the two fractions: \[\frac{3(x+5)}{(x+4)(x+5)} + \frac{6(x+4)}{(x+4)(x+5)}\] After simplifying the expression above, we get \[\frac{3x+15+6x+24}{(x+4)(x+5)} = \frac{9x+39}{x^2+9x+20}\]