1. **State the problem:** Solve the compound inequality $$-71 \leq 7x - 1 \leq 69$$. 2. **Add 1 to all parts of the inequality** to isolate the term with $x$: $$-71 + 1 \leq 7x - 1 + 1 \leq 69 + 1$$ which simplifies to $$-70 \leq 7x \leq 70$$. 3. **Divide all parts by 7** to solve for $x$: $$\frac{-70}{7} \leq \frac{7x}{7} \leq \frac{70}{7}$$ 4. **Show cancellation explicitly:** $$\frac{\cancel{-70}}{\cancel{7}} \leq x \leq \frac{\cancel{70}}{\cancel{7}}$$ 5. **Simplify the fractions:** $$-10 \leq x \leq 10$$. 6. **Final answer:** The solution to the inequality is all $x$ such that $$-10 \leq x \leq 10$$. This means $x$ can be any number between $-10$ and $10$, inclusive.