Question

simplify. enter a single, simplified rational expression or integer.
$1 + \dfrac{3}{1 + \dfrac{3}{x + 6}}$

Explanation:

Step1: Simplify the denominator of the complex fraction

First, simplify the inner fraction \(1+\frac{3}{x + 6}\). We find a common denominator, which is \(x+6\). So we have:
\[1+\frac{3}{x + 6}=\frac{x + 6}{x + 6}+\frac{3}{x + 6}=\frac{x+6 + 3}{x + 6}=\frac{x + 9}{x + 6}\]

Step2: Substitute the simplified inner fraction back

Now the original expression \(1+\frac{3}{1+\frac{3}{x + 6}}\) becomes \(1+\frac{3}{\frac{x + 9}{x + 6}}\). When dividing by a fraction, we multiply by its reciprocal, so \(\frac{3}{\frac{x + 9}{x + 6}}=3\times\frac{x + 6}{x + 9}=\frac{3(x + 6)}{x + 9}\)

Step3: Simplify the entire expression

Now we have \(1+\frac{3(x + 6)}{x + 9}\). Again, find a common denominator, which is \(x + 9\). So:
\[1+\frac{3(x + 6)}{x + 9}=\frac{x + 9}{x + 9}+\frac{3(x + 6)}{x + 9}=\frac{x + 9+3(x + 6)}{x + 9}\]
Expand the numerator: \(x + 9+3x+18=4x + 27\)

Answer:

\(\frac{4x + 27}{x + 9}\)