Problem 41 - College Algebra

Question

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

Step-by-Step Solution

Answer

The solution is \(\frac{9x + 39}{(x+4)(x+5)}\)

Step 1: Find common denominator

The common denominator is the product of the two denominators. So, the common denominator is \( (x+4)(x+5) \).

Step 2: Convert the fractions

Convert the first given fraction, \(\frac{3}{x+4}\), to have the common denominator. Do this by multiplying both the numerator and the denominator by the missing factor from the common denominator, in this case, \(x+5\). Repeat for the second fraction, but multiply by \(x+4\). The result is: \(\frac{3(x+5)}{(x+4)(x+5)}+\frac{6(x+4)}{(x+4)(x+5)}\)

Step 3: Balance the fractions

Now, proceed with the addition: \(\frac{3(x+5) + 6(x+4)}{(x+4)(x+5)}\).

Step 4: Simplify

Simplify the fractions by expanding and collecting like terms. The result is: \(\frac{3x + 15 + 6x + 24}{(x+4)(x+5)}\) which simplifies further to \(\frac{9x + 39}{(x+4)(x+5)}\)