Answer
$3$ or $\displaystyle \frac{1}{3}$
Work Step by Step
$\left[\begin{array}{lll}
\text{The sum} & \rightarrow & ..+...\\\\
\text{of 2 times a number and } & \rightarrow & 2x+...\\\\
\text{... twice its reciprocal } & \rightarrow & 2x+2\cdot\frac{1}{x}\\ \\
\text{... is }\frac{20}{3} & \rightarrow & 2x+2\cdot\frac{1}{x}=\frac{20}{3}
\end{array}\right]$
$2x+2\displaystyle \cdot\frac{1}{x}=\frac{20}{3}\qquad.../\times 3x$
$6x^{2}+6=20x\qquad.../\div 2$
$3x^{2}+3=10x\qquad.../-10x$
$3x^{2}-10x+3=0$
... Trying with synthetic division, we find:
$$ \begin{array}{rrrrrr}
3 & | & 3 & -10 & 3 & \\
& & & 9 & -3 & \\
& & -- & -- & -- & \\
& & 3 & -1 & 0 &
\end{array}$$
$0=(x-3)(3x-1)$
$x=3$ or $x=\displaystyle \frac{1}{3}$
