To solve the compound inequality $\frac{3}{4}x - 3 \le 3$ or $\frac{2}{5}(x + 10) \ge 0$, isolate the variable in each inequality. In the first inequality, adding $3$ results in $\frac{3}{4}x \le 6$, and multiplying by $\frac{4}{3}$ yields $x \le 8$. In the second inequality, multiplying by $\frac{5}{2}$ produces $x + 10 \ge 0$, which simplifies to $x \ge -10$. Combining the two statements with "or" means finding the union: $x \le 8$ or $x \ge -10$. Because all numbers are either less than or equal to $8$, or greater than or equal to $-10$, the combined inequalities encompass the entire number line. The solution is thus all real numbers, expressed in interval notation as $(-\infty, \infty)$.