4.1.2 * Solve the inequality: $\sqrt{3 x+5} \geqslant x-2$.


Analysis one: For $x \geqslant-\frac{5}{3}$. When $-\frac{5}{3} \leqslant x<2$, the inequality obviously holds. When $x \geqslant 2$, the original inequality can be transformed into $(x-2)^{2} \leqslant 3 x+5$, solving which yields $2 \leqslant x \leqslant \frac{7+\sqrt{53}}{2}$. In summary, the solution set of the original inequality is $\left[-\frac{5}{3}, \frac{7+\sqrt{53}}{2}\right]$.

Analysis two: Solve using graphs. As shown in the figure, plot the functions $y=$ $\sqrt{3 x+5}, y=x-2$. The x-coordinate of the intersection point of the two graphs is $x=\frac{7+\sqrt{53}}{2}$. Therefore, the solution set of the original inequality is
$$
\left\{x \left\lvert\,-\frac{5}{3} \leqslant x \leqslant \frac{7+\sqrt{53}}{2}\right.\right\} .
$$