Add.
$\frac{11}{3 x+6}+\frac{x}{x+2}$
Simplify your answer as much as possible.
The goal is to simplify the sum of two rational expressions: \[\frac{11}{3x+6} + \frac{x}{x+2}\]
First, we identify the domain of the expression. The denominators must not be zero:
Thus, the expression is valid for all real numbers \(x\) except \(x = -2\).
To add the two fractions, we need a common denominator. The first denominator is \(3x + 6\), which can be factored as \(3(x+2)\). The second denominator is \(x+2\).
The least common denominator is \(3(x+2)\).
We rewrite the second term to have this denominator:
\[\frac{x}{x+2} = \frac{x \cdot 3}{(x+2) \cdot 3} = \frac{3x}{3(x+2)}\]
Now we add the two fractions by combining the numerators over the common denominator:
\[\frac{11}{3(x+2)} + \frac{3x}{3(x+2)} = \frac{11 + 3x}{3(x+2)}\]
We check if the numerator \(3x + 11\) and the denominator \(3(x+2) = 3x + 6\) have any common factors. Since 3 does not divide 11, there are no common factors to simplify.
The simplified expression is \(\frac{3x+11}{3x+6}\).
Rational expressions are used in various fields, such as physics and economics, to represent rates or ratios that depend on a variable. For example, if you are calculating the total time taken to complete two parts of a journey where the speeds change based on a factor \(x\), you might end up with an expression like this. Simplifying these expressions allows you to better understand the behavior of the system and make calculations much easier. Knowing how to combine these terms ensures you can work with complex formulas efficiently in practical situations.