Solution — 2016 Question 2 Exponential & Logarithmic Functions| Quadratic Equations & Inequalities

$\begin{matrix} 2 e^{2 x} & \geq 9 - 3 e^{x} \end{matrix}$

Let $u = e^{x} .$ Since $e^{2 x} = u^{2},$ substituting into the inequality,

$\begin{matrix} 2 u^{2} & \geq 9 - 3 u \\ 2 u^{2} + 3 u - 9 & \geq 0 \\ (u + 3) (2 u - 3) & \geq 0 \end{matrix}$

$ ParseError: Can't use function '$' in math mode at position 1: $̲ u \leq -3 \quad \text{ or } \quad u \geq \frac{3}{2}

Since $u = e^{x} > 0,$ we reject $u \leq - 3$

\begin{align*} \mathrm{e}^x &\geq \frac{3}{2} \\ \ln \mathrm{e}^x &\geq \ln \frac{3}{2} \\ x &\geq \ln \frac{3}{2}\; \QED \end{align*}$$ ParseError: {align*} can be used only in display mode.