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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 cannot be further simplified without specific values for \(x\). So, the answer is \(\frac{3*(3x+6)+2*(2x+4)}{(2x+4)*(3x+6)}.\)

Step by step solution

01

Find Common Denominator

Before fractions can be added or subtracted, they must have the same denominator. One way to find a common denominator is to multiply the denominators of the two fractions. The denominators here are \(2x + 4\) and \(3x + 6\). Multiplying these two together gives a common denominator of \((2x + 4) * (3x + 6).\)
02

Rewrite Fractions

The original fractions \(\frac{3}{2x+4}+\frac{2}{3x+6}\) should be rewritten using the common denominator. This is done by multiplying the numerator and denominator of the first fraction by \((3x+6)\) and the numerator and denominator of the second fraction by \((2x+4)\). The new fractions are \(\frac{3*(3x+6)}{(2x+4)*(3x+6)}\) and \(\frac{2*(2x+4)}{(2x+4)*(3x+6)}.\)
03

Add the Fractions

Now that both fractions have the same denominator, they can be added together. This gives \(\frac{3*(3x+6)+2*(2x+4)}{(2x+4)*(3x+6)}.\)
04

Simplify the Result

The last step is to simplify the numerator and denominator of the resulting fraction as much as possible. The final simplified form is the solution to the problem.

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