1. **State the problem:** Solve the inequality $2x^2 - 5x - 3 < 0$. 2. **Formula and rules:** To solve a quadratic inequality, first find the roots of the quadratic equation $2x^2 - 5x - 3 = 0$ by using the quadratic formula: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a=2$, $b=-5$, and $c=-3$. 3. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = (-5)^2 - 4 \times 2 \times (-3) = 25 + 24 = 49$$ 4. **Find the roots:** $$x = \frac{-(-5) \pm \sqrt{49}}{2 \times 2} = \frac{5 \pm 7}{4}$$ 5. **Evaluate roots:** - For $+$ sign: $$x = \frac{5 + 7}{4} = \frac{12}{4} = 3$$ - For $-$ sign: $$x = \frac{5 - 7}{4} = \frac{-2}{4} = -\frac{1}{2}$$ 6. **Analyze the inequality:** The parabola opens upwards since $a=2 > 0$. 7. **Solution:** The quadratic is less than zero between the roots: $$-\frac{1}{2} < x < 3$$ **Final answer:** $$\boxed{-\frac{1}{2} < x < 3}$$