To solve the compound inequality $\frac{3}{5}x - 7 \le -1$ or $\frac{1}{3}(x + 6) \ge -2$, start by solving each inequality independently. For the first inequality, add $7$ to both sides to get $\frac{3}{5}x \le 6$, then multiply by $\frac{5}{3}$ to find $x \le 10$. For the second inequality, multiply by $3$ to obtain $x + 6 \ge -6$, then subtract $6$ to get $x \ge -12$. The solution to an "or" compound inequality is the union of these two results: $x \le 10$ or $x \ge -12$. Since these regions overlap and cover every possible value on the number line, the final solution includes all real numbers. In interval notation, this is written as $(-\infty, \infty)$.