Answer
$x = 2$
Work Step by Step
Multiply each side of the equation by the LCD, which is $x(x + 2)$, to eliminate the fractions:
$$\begin{align*}
(x)(x+2) \left(\frac{1}{x}+\frac{x}{x+2}\right)&=1(x)(x+2)\\
\\(x)(x+2) \left(\frac{1}{x}\right)+x(x+2)\left(\frac{x}{x+2}\right)&=x(x+2)\\
\end{align*}$$
Simplify:
$$\require{cancel}
\begin{align*}
\cancel{(x)}(x+2) \left(\frac{1}{\cancel{x}}\right)+x\cancel{(x+2)}\left(\frac{x}{\cancel{x+2}}\right)&=x^2+2x\\
x+2+x(x)&=x^2+2x\\
x+2+x^2&=x^2+2x\\
\end{align*}$$
Move all terms to the left side of the equation:
$$\begin{align*}
x+2+x^2-(x^2+2x)&=0\\
x+2+x^2-x^2-2x&=0\\
-x+2&=0
\end{align*}
$$
Solve for $x$:
$$\begin{align*}
-x &= -2\\
x&=2
\end{align*}$$
To check the solution, plug in the value we just found for $x$ into the original equation:
$$\begin{align*}
\frac{1}{2} + \frac{2}{2 + 2} &\stackrel{?}{=} 1\\
\frac{1}{2}+\frac{2}{4}&\stackrel{?}{=}1\\
1&\stackrel{\checkmark}{=}1
\end{align*}$$
