1. **State the problem:** Solve the system of inequalities: $$|3x - 2| < 7$$ $$|x - 1| \geq 2$$ 2. **Recall the definition of absolute value inequalities:** - For $|A| < B$, where $B > 0$, the solution is $-B < A < B$. - For $|A| \geq B$, where $B > 0$, the solution is $A \leq -B$ or $A \geq B$. 3. **Solve the first inequality:** $$|3x - 2| < 7 \implies -7 < 3x - 2 < 7$$ Add 2 to all parts: $$-7 + 2 < 3x < 7 + 2$$ $$-5 < 3x < 9$$ Divide all parts by 3: $$-\frac{5}{3} < x < 3$$ 4. **Solve the second inequality:** $$|x - 1| \geq 2 \implies x - 1 \leq -2 \text{ or } x - 1 \geq 2$$ Solve each: $$x \leq -1 \text{ or } x \geq 3$$ 5. **Find the intersection of the two solution sets:** From the first inequality: $-\frac{5}{3} < x < 3$ From the second inequality: $x \leq -1$ or $x \geq 3$ The intersection is where both inequalities hold simultaneously: - For $x \leq -1$ and $-\frac{5}{3} < x < 3$, the overlap is $-\frac{5}{3} < x \leq -1$ - For $x \geq 3$ and $-\frac{5}{3} < x < 3$, there is no overlap because $x$ cannot be both $\geq 3$ and $< 3$ 6. **Final solution:** $$-\frac{5}{3} < x \leq -1$$