1. **State the problem:** Solve the inequality $3x^2 - x - 2 > 0$. 2. **Find the roots of the quadratic equation:** To understand where the parabola is above zero, first solve $3x^2 - x - 2 = 0$. 3. **Use the quadratic formula:** For $ax^2 + bx + c = 0$, roots are given by $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a=3$, $b=-1$, and $c=-2$. 4. **Calculate the discriminant:** $$\Delta = (-1)^2 - 4 \times 3 \times (-2) = 1 + 24 = 25$$ 5. **Find the roots:** $$x = \frac{-(-1) \pm \sqrt{25}}{2 \times 3} = \frac{1 \pm 5}{6}$$ 6. **Evaluate roots:** - $$x_1 = \frac{1 - 5}{6} = \frac{-4}{6} = -\frac{2}{3}$$ - $$x_2 = \frac{1 + 5}{6} = \frac{6}{6} = 1$$ 7. **Analyze the inequality:** Since the parabola opens upwards ($a=3 > 0$), the quadratic is positive outside the roots and negative between them. 8. **Write the solution:** $$x < -\frac{2}{3} \quad \text{or} \quad x > 1$$ **Final answer:** The solution to $3x^2 - x - 2 > 0$ is $$\boxed{x < -\frac{2}{3} \text{ or } x > 1}$$.