To solve the compound inequality $\frac{2}{3}x - 4 \le 3$ or $\frac{1}{4}(x + 8) \ge -1$, start by solving each inequality independently. For the first, multiply both sides by $3$ to produce $2x - 12 \le 9$. Adding $12$ to both sides and then dividing by $2$ yields $x \le \frac{21}{2}$. For the second inequality, multiplying by $4$ gives $x + 8 \ge -4$, which simplifies to $x \ge -12$. The solution to the compound inequality is the union of the two results: $x \le \frac{21}{2}$ or $x \ge -12$. Because every real number is either less than or equal to $\frac{21}{2}$ or greater than or equal to $-12$, the union spans the entire number line. The final solution is all real numbers, and the interval notation is $(-\infty, \infty)$.