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Chapter 0: Problem 49

Add or subtract as indicated. $$\frac{3}{2 x+4}+\frac{2}{3 x+6}$$

Short Answer

Expert verified
The result of the operation is \(\frac{13}{6(x+2)}\).

Step by step solution

01

Identify the denominators

The denominators are \(2x + 4\) and \(3x + 6\). We need to find the least common multiple (LCM) of these two expressions.
02

Determine the LCM

We can factorize the denominators, \(2x + 4 = 2( x + 2)\) and \(3x + 6 = 3( x + 2)\). We see that the common factor is \(x+2\). Therefore, the LCM is \((x+2)\times2\times3 = 6(x+2)\).
03

Adjust the fractions

Now we adjust the original fractions to have the LCM as the denominator. To do this, we multiply each fraction by a form of 1 that will result in the LCM as the denominator. So our fractions become: \(\frac{3}{2x+4} \times \frac{3}{3} = \frac{9}{6(x+2)}\) and \(\frac{2}{3x+6} \times \frac{2}{2} = \frac{4}{6(x+2)}\)
04

Perform the addition

We can now add the two fractions together as they have the same denominator. This gives us the final result: \(\frac{9}{6(x+2)} + \frac{4}{6(x+2)} = \frac{9 + 4}{6(x+2)} = \frac{13}{6(x+2)}\)

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