College Algebra, Chapter 5, 5.4, Section 5.4, Problem 70

Solve the inequality $x^2 e^x - 2e^x < 0$

$
\begin{equation}
\begin{aligned}
x^2 e^x - 2e^x &< 0 \\
\\
x^2 e^x &< 2e^x && \text{Add } 2e^x\\
\\
\ln x^2 e^x &< \ln 2e^x && \text{Take ln of each side}\\
\\
\ln x^2 + \ln e^x &< \ln 2 + \ln e^x && \text{Properties of ln } \ln(AB) = \ln A + \ln B\\
\\
2 \ln x + x \ln e &< \ln 2 + x \ln e && \text{Properties of ln } \ln A^c = C\ln A\\
\\
2 \ln x &< \ln 2 && \text{Subtract } x \ln e\\
\\
\ln x &< \frac{\ln 2}{2} && \text{Divide by 2}\\
\\
e^{\ln x} &< e^{\frac{\ln 2}{2}} && \text{Raise } e \text{ to each side}\\
\\
x &< e^{\frac{\ln 2}{2}} && \text{Property of ln}
\end{aligned}
\end{equation}
$