1. Problem: Solve the inequalities given. 2. Formula and rules: For quadratic inequalities of the form $ax^2 + bx + c > 0$, $\geq 0$, $< 0$, or $\leq 0$, first find the roots by solving $ax^2 + bx + c = 0$. Then analyze the sign of the quadratic in intervals determined by the roots. 3. a) Solve $x^2 - 3x + 2 > 0$: - Find roots: $x^2 - 3x + 2 = 0$ factors as $(x-1)(x-2)=0$ so roots are $x=1$ and $x=2$. - Since $a=1>0$, parabola opens upward. - Inequality $>0$ means where parabola is above x-axis. - Solution intervals: $(-\infty,1) \cup (2,\infty)$. 4. c) Solve $x^2 + 7x + 12 \geq 0$: - Roots: $x^2 + 7x + 12 = 0$ factors as $(x+3)(x+4)=0$, roots $x=-3$, $x=-4$. - Parabola opens upward. - Inequality $\geq 0$ means parabola is on or above x-axis. - Solution intervals: $(-\infty,-4] \cup [-3,\infty)$. 5. e) Solve $x^2 + 4x - 12 < 0$: - Roots: Solve $x^2 + 4x - 12=0$. - Use quadratic formula: $x=\frac{-4 \pm \sqrt{16 + 48}}{2} = \frac{-4 \pm 8}{2}$. - Roots: $x=2$ and $x=-6$. - Parabola opens upward. - Inequality $<0$ means between roots. - Solution interval: $(-6,2)$. 6. g) Solve $x^2 - x + 1 < 0$: - Discriminant: $\Delta = (-1)^2 - 4(1)(1) = 1 - 4 = -3 < 0$. - No real roots, parabola opens upward. - Since no roots and $a>0$, parabola always positive. - Inequality $<0$ has no solution. 7. i) Solve $m^2 + m - 20 \geq 0$: - Roots: Solve $m^2 + m - 20=0$. - Use quadratic formula: $m=\frac{-1 \pm \sqrt{1 + 80}}{2} = \frac{-1 \pm 9}{2}$. - Roots: $m=4$ and $m=-5$. - Parabola opens upward. - Inequality $\geq 0$ means outside roots. - Solution intervals: $(-\infty,-5] \cup [4,\infty)$. Final answers: - a) $x \in (-\infty,1) \cup (2,\infty)$ - c) $x \in (-\infty,-4] \cup [-3,\infty)$ - e) $x \in (-6,2)$ - g) No solution - i) $m \in (-\infty,-5] \cup [4,\infty)$