[Tex/LaTex] How to properly divide beamer slide in two sections

Question

I want to divide my slide in two sections. On the left I want to embed an array of two equations. On the right I want to show an image. I have tried the following:

\usepackage{amsfonts, amsmath}

\documentclass{beamer}

\begin{document}
\begin{frame}

\begin{tabular}{cl}
    \begin{tabular}{c}
    \begin{align*}
    U(x) = \left\{ \begin{array}{cc}
            x^\alpha & \text{ if } x\geq 0 \\
            -\lambda(-x)^\beta & \text{ if } x\leq 0
            \end{array} \right. \\
        \Psi_\gamma(p) = \frac{p^\gamma}{(p^\gamma + (1 - p)^ \gamma)^{1/\gamma }}
    \end{align*}
    \end{tabular} &
    \begin{tabular}{l}
    \begin{figure}
    \includegraphics[scale = 0.3]{sample_image.png}
    \end{figure}
    \end{tabular}
\end{tabular}

\end{frame}
\end{document}

It seems to divide the slide properly when only the image is inputed but not when I try to write the equations.

Thanks for the help.

Answer 1

I'd suggest to use the already provided columns mechanism:

\documentclass{beamer}
\usepackage{amsfonts, amsmath}
\begin{document}
\begin{frame}

\begin{columns}
    \column{.5\linewidth}
    \begin{align*}
    U(x) = \left\{ \begin{array}{cc}
            x^\alpha & \text{ if } x\geq 0 \\
            -\lambda(-x)^\beta & \text{ if } x\leq 0
            \end{array} \right. \\
        \Psi_\gamma(p) = \frac{p^\gamma}{(p^\gamma + (1 - p)^ \gamma)^{1/\gamma }}
    \end{align*}
    \column{.5\linewidth}
    \begin{figure}
    \includegraphics[width = .9\linewidth]{example-image}
    \end{figure}
\end{columns}

\end{frame}
\end{document}