1. **State the problem:** Solve the quadratic inequality $$2x^2 \geq 7x + 15$$ in the set of real numbers. 2. **Rewrite the inequality:** Bring all terms to one side to set the inequality to zero: $$2x^2 - 7x - 15 \geq 0$$ 3. **Find the roots of the quadratic equation:** Solve $$2x^2 - 7x - 15 = 0$$ using the quadratic formula: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a=2$, $b=-7$, and $c=-15$. 4. **Calculate the discriminant:** $$\Delta = (-7)^2 - 4 \times 2 \times (-15) = 49 + 120 = 169$$ 5. **Calculate the roots:** $$x = \frac{-(-7) \pm \sqrt{169}}{2 \times 2} = \frac{7 \pm 13}{4}$$ 6. **Find the two roots:** - $$x_1 = \frac{7 - 13}{4} = \frac{-6}{4} = -\frac{3}{2}$$ - $$x_2 = \frac{7 + 13}{4} = \frac{20}{4} = 5$$ 7. **Determine the intervals:** The parabola opens upwards (since $a=2 > 0$), so the quadratic expression is: - Positive or zero outside the roots, - Negative between the roots. 8. **Write the solution:** $$x \in (-\infty, -\frac{3}{2}] \cup [5, \infty)$$ This means the inequality $$2x^2 - 7x - 15 \geq 0$$ holds for all real $x$ less than or equal to $-\frac{3}{2}$ and greater than or equal to $5$.