If the domain of the function $$\sin^{-1}\left(\frac{3x-22}{2x-19}\right) + \log_e\left(\frac{3x^2-8x+5}{x^2-3x-10}\right)$$ is $$(\alpha, \beta]$$, then $$3\alpha + 10\beta$$ is equal to:

For $$\sin^{-1}(u)$$: $$-1 \le \frac{3x-22}{2x-19} \le 1$$.

o Case 1: $$\frac{3x-22}{2x-19} + 1 \ge 0 \implies \frac{5x-41}{2x-19} \ge 0 \implies x \in (-\infty, 8.2] \cup (9.5, \infty)$$.

o Case 2: $$\frac{3x-22}{2x-19} - 1 \le 0 \implies \frac{x-3}{2x-19} \le 0 \implies x \in [3, 9.5)$$.

Intersection: $$x \in [3, 8.2]$$.

For $$\log(v)$$: $$\frac{(3x-5)(x-1)}{(x-5)(x+2)} > 0$$.

Critical points: $$-2, 1, 5/3, 5$$.

Intervals: $$(-\infty, -2) \cup (1, 5/3) \cup (5, \infty)$$.

Common Domain: Intersect $$[3, 8.2]$$ with the log domain $$\implies x \in (5, 8.2]$$.

$$\alpha = 5, \beta = 8.2 = \frac{41}{5}$$.

Calculate: $$3\alpha + 10\beta = 3(5) + 10(\frac{41}{5}) = 15 + 82 = 97$$.