Solution — 2023 Question 1 Exponential & Logarithmic Functions| Quadratic Equations & Inequalities

$\begin{matrix} 3 e^{x} & \geq 4 e^{- x} - 4 \\ 3 e^{x} & \geq \frac{4}{e^{x}} - 4 \end{matrix}$

Let $u = e^{x} .$ Substituting into the inequality,

$ ParseError: Can't use function '$' in math mode at position 1: $̲ 3u \geq \frac{4}{u} - 4

Multiplying both sides by $u$ (since $u = e^{x} > 0$ ),

\begin{align*} 3u^2 &\geq 4 - 4u \\ 3u^2 + 4u - 4 &\geq 0 \\ \left( u + 2 \right)\left( 3u - 2 \right) &\geq 0 \end{align*}$$ ParseError: {align*} can be used only in display mode.

$u \leq - 2 or u \geq \frac{2}{3}$

Since $u = e^{x} > 0,$ we reject $u \leq - 2$

\begin{align*} \mathrm{e}^x &\geq \frac{2}{3} \\ \ln \mathrm{e}^x &\geq \ln \frac{2}{3} \\ x &\geq \ln \frac{2}{3}\; \QED \end{align*}$$