Problem 49 - College Algebra

Question

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

Step-by-Step Solution

Answer

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 1: 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 \((2x + 4) * (3x + 6).\)

Step 2: 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 \(\frac{2*(2x+4)}{(2x+4)*(3x+6)}.\)

Step 3: Add the Fractions

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

Step 4: 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.