5. Given three positive real numbers $a, b, c$ satisfying $a \leqslant b+c \leqslant 2 a, b \leqslant a+c \leqslant 2 b$, then the range of $\frac{b}{a}$ is
5. $\left[\frac{2}{3}, \frac{3}{2}\right]$ Analysis: Draw the feasible region of $a, b$ as the shaded area shown in the figure. By the slope $k=\frac{b-0}{a-0}$ and combining with the graph, we get $k_{\mathrm{OV}} \leqslant k \leqslant k_{\mathrm{OM}}$ $\Rightarrow \frac{2}{3} \leqslant k \leqslant \frac{3}{2}$.