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

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

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