1. **State the problem:** Solve the compound inequality $$\frac{3t + 1}{2} \geq -1 \text{ or } t + 14 \leq 8$$ for $t$ and write the answer as a compound inequality with integers. 2. **Solve the first inequality:** $$\frac{3t + 1}{2} \geq -1$$ Multiply both sides by 2 to clear the denominator: $$\cancel{2} \times \frac{3t + 1}{\cancel{2}} \geq -1 \times 2$$ which simplifies to $$3t + 1 \geq -2$$ Subtract 1 from both sides: $$3t + 1 - 1 \geq -2 - 1$$ $$3t \geq -3$$ Divide both sides by 3: $$\frac{3t}{\cancel{3}} \geq \frac{-3}{\cancel{3}}$$ $$t \geq -1$$ 3. **Solve the second inequality:** $$t + 14 \leq 8$$ Subtract 14 from both sides: $$t + 14 - 14 \leq 8 - 14$$ $$t \leq -6$$ 4. **Combine the solutions:** Since the compound inequality uses "or", the solution is $$t \geq -1 \text{ or } t \leq -6$$ This means $t$ can be any number greater than or equal to $-1$, or any number less than or equal to $-6$. **Final answer:** $$t \leq -6 \text{ or } t \geq -1$$