Simplify.
$\frac{\frac{2+x}{8 x}}{\frac{3+x}{x}}$
The expression to simplify is given by the complex fraction:
\[ \frac{\frac{2+x}{8x}}{\frac{3+x}{x}} \]
First, we identify the domain of the expression. Since the denominators cannot be zero, we must have \(x
eq 0\) (from the expressions \(8x\) and \(x\)) and \(3+x
eq 0\), which means \(x
eq -3\). We are simplifying a quotient of two fractions, where the numerator is \(\frac{2+x}{8x}\) and the denominator is \(\frac{3+x}{x}\).
To simplify a quotient of fractions \(\frac{A/B}{C/D}\), we multiply the numerator by the reciprocal of the denominator:
\[ \frac{A}{B} \div \frac{C}{D} = \frac{A}{B} \times \frac{D}{C} \]
Applying this to our expression with \(A=2+x\), \(B=8x\), \(C=3+x\), and \(D=x\), we get:
\[ \frac{2+x}{8x} \times \frac{x}{3+x} \]
Now, we can multiply the numerators and the denominators:
\[ \frac{(2+x) \cdot x}{8x \cdot (3+x)} \]
Since \(x
eq 0\), we can cancel the common factor of \(x\) from the numerator and the denominator:
\[ \frac{2+x}{8(3+x)} \]
Expanding the denominator gives \(8 \times 3 + 8 \times x = 24 + 8x\). Thus, the simplified expression is:
\[ \frac{x+2}{8x+24} \]
Simplifying complex fractions is a fundamental skill in algebra with many real-world applications. For example, if you are calculating average speed over segments of a journey, you might represent the total distance divided by total time as a complex fraction. By simplifying these expressions, you can quickly analyze relationships between variables, such as how changing the duration of one segment of a trip affects the overall average, which is essential for efficient travel planning and logistics management.