1. **State the problem:** Solve the inequality $$|x - 1| \geq 13$$ algebraically. 2. **Recall the definition of absolute value inequality:** For $$|A| \geq B$$ where $$B \geq 0$$, the solution is $$A \leq -B$$ or $$A \geq B$$. 3. **Apply the rule:** Here, $$A = x - 1$$ and $$B = 13$$, so $$x - 1 \leq -13 \quad \text{or} \quad x - 1 \geq 13$$ 4. **Solve each inequality separately:** - For $$x - 1 \leq -13$$: $$x \leq \cancel{1} - 13 + \cancel{1}$$ $$x \leq -12$$ - For $$x - 1 \geq 13$$: $$x \geq \cancel{1} + 13 - \cancel{1}$$ $$x \geq 14$$ 5. **Write the solution set:** $$x \in (-\infty, -12] \cup [14, \infty)$$ This means $$x$$ is less than or equal to $$-12$$ or greater than or equal to $$14$$. **Final answer:** $$x \leq -12 \quad \text{or} \quad x \geq 14$$