5. $[24,27]$.

From $\frac{4}{a}+\frac{3}{b} \leqslant 2 \Rightarrow b \geqslant \frac{3 a}{2 a-4}$.
Thus, the point $(a, b)$ is in the first quadrant and lies within the region bounded by the line $2 a+2 b=15$ and the hyperbola $b=\frac{3 a}{2 a-4}$.

Therefore, when $a=4, b=3$, $3 a+4 b$ reaches its minimum value of 24; when $a=3, b=4.5$, $3 a+4 b$ reaches its maximum value of 27.
Hence, the range of $3 a+4 b$ is $[24,27]$.
5. Given that $a$ and $b$ are positive real numbers, and
$$
2 a+2 b \leqslant 15, \frac{4}{a}+\frac{3}{b} \leqslant 2 \text {. }
$$