Question

Simplify the complex rational expression by using the LCD:

$\frac{\frac{x}{x+3}}{1+\frac{1}{x+3}}$

Short answer

The simplified expression is$\frac{x}{x+4}$.

Step-by-step solution

Step 1. Given Information.

We have:

$\frac{\frac{x}{x+3}}{1+\frac{1}{x+3}}$

Step 2. Multiply the numerator and denominator by the LCD. 

The least common divisor (LCD) is $\left(x+3\right).$

Multiply the numerator and denominator by $\left(x+3\right):$

$=\frac{\left(x+3\right)\frac{x}{x+3}}{\left(x+3\right)\left(1+\frac{1}{x+3}\right)}$

Step 3. Simplify the expression.

Distribute in the denominator and simplify:

$$\begin{gathered} =\frac{\bcancel{\left(x+3\right)}\frac{x}{\bcancel{x+3}}}{1\left(x+3\right)+\bcancel{\left(x+3\right)}\frac{1}{\bcancel{x+3}}} \\ =\frac{x}{x+3+1} \\ =\frac{x}{x+4} \end{gathered}$$