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Chapter 0: Problem 41

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

Short Answer

Expert verified
\\The added fraction is \(\frac{9x+39}{(x+4)(x+5)}\).

Step by step solution

01

Find the least common denominator (LCD)

To add or subtract fractions, we must have a common denominator. To find the LCD of fractions \(\frac{3}{x+4}\) and \(\frac{6}{x+5}\), we multiply the two denominators together. Thus, the LCD is \((x+4)(x+5)\).
02

Rewrite the fractions with the LCD

Now we rewrite each fraction with the LCD. Rewrite \(\frac{3}{x+4}\) as \(\frac{3(x+5)}{(x+4)(x+5)}\) and \(\frac{6}{x+5}\) as \(\frac{6(x+4)}{(x+4)(x+5)}\).
03

Perform the Addition

Now that both fractions have the same denominator, we can add them together. Put the common denominator \((x+4)(x+5)\) once and add the numerators. The result will be \(\frac{3(x+5)+6(x+4)}{(x+4)(x+5)}\).
04

Simplify the Numerator

Always try to simplify the numerator and denominator if possible. The numerator simplifies to \(3x+15+6x+24\) which further simplifies to \(9x+39\). Hence, the added fraction is \(\frac{9x+39}{(x+4)(x+5)}\)
05

Further Simplify if Possible

Since neither \(9x+39\) nor \((x+4)(x+5)\) can be factored, our final simplified version of the added fraction remains as \(\frac{9x+39}{(x+4)(x+5)}\).

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