1. **Graph the inequality** $2x + y \geq 8$. The inequality can be rewritten as $y \geq 8 - 2x$. This represents all points on or above the line $y = 8 - 2x$. The boundary line has slope $-2$ and y-intercept $8$. Since the inequality is $\geq$, the line is solid and included in the solution. 2. **Divide** $\frac{x^2 + x - 2}{3x^2 + 9x + 6}$ **by** $x - 1$. First, factor numerator and denominator: Numerator: $x^2 + x - 2 = (x + 2)(x - 1)$ Denominator: $3x^2 + 9x + 6 = 3(x^2 + 3x + 2) = 3(x + 1)(x + 2)$ So original fraction is $\frac{(x + 2)(x - 1)}{3(x + 1)(x + 2)}$. Dividing by $x - 1$ is multiplying by $\frac{1}{x - 1}$: $$\frac{(x + 2)(x - 1)}{3(x + 1)(x + 2)} \times \frac{1}{x - 1} = \frac{(x + 2)(x - 1)}{3(x + 1)(x + 2)(x - 1)}$$ Cancel $(x - 1)$ and $(x + 2)$: $$= \frac{1}{3(x + 1)}$$ 3. **Sketch the graph of** $y = x^2 - 2x - 3$. This is a quadratic function with: - Vertex at $x = -\frac{b}{2a} = -\frac{-2}{2 \times 1} = 1$. - Evaluate $y$ at $x=1$: $1^2 - 2(1) - 3 = 1 - 2 - 3 = -4$. - So vertex is at $(1, -4)$. - Find x-intercepts by solving $x^2 - 2x - 3 = 0$: $$x = \frac{2 \pm \sqrt{(-2)^2 - 4 \times 1 \times (-3)}}{2} = \frac{2 \pm \sqrt{4 + 12}}{2} = \frac{2 \pm 4}{2}$$ So $x=3$ or $x=-1$. - y-intercept at $x=0$ is $y = -3$. The parabola opens upwards. 4. **Find horizontal and vertical components of velocity** for a ball thrown at $25$ m/s at $45^\circ$. - Horizontal component: $v_x = v \cos \theta = 25 \times \cos 45^\circ = 25 \times \frac{\sqrt{2}}{2} = \frac{25\sqrt{2}}{2} \approx 17.68$ m/s. - Vertical component: $v_y = v \sin \theta = 25 \times \sin 45^\circ = 25 \times \frac{\sqrt{2}}{2} = \frac{25\sqrt{2}}{2} \approx 17.68$ m/s. 5. **Find composite functions** $f(g(x))$ and $g(f(x))$ where $f(x) = \sin x$ and $g(x) = 1 - x^2$. - $f(g(x)) = \sin(1 - x^2)$. - $g(f(x)) = 1 - (\sin x)^2 = 1 - \sin^2 x$. 6. **Find scalar coefficients** $m$ and $n$ such that $\vec{C} = m \vec{A} + n \vec{B}$ where $\vec{A} = (4,1)$, $\vec{B} = (-2,3)$, $\vec{C} = (10,11)$. Set up equations: $$4m - 2n = 10$$ $$m + 3n = 11$$ From second equation: $m = 11 - 3n$. Substitute into first: $$4(11 - 3n) - 2n = 10 \Rightarrow 44 - 12n - 2n = 10 \Rightarrow 44 - 14n = 10$$ $$-14n = 10 - 44 = -34 \Rightarrow n = \frac{34}{14} = \frac{17}{7}$$ Then $$m = 11 - 3 \times \frac{17}{7} = 11 - \frac{51}{7} = \frac{77}{7} - \frac{51}{7} = \frac{26}{7}$$ **Final answers:** - Inequality graph: $y \geq 8 - 2x$ - Division result: $\frac{1}{3(x + 1)}$ - Parabola vertex: $(1, -4)$, roots: $x = -1, 3$ - Velocity components: $v_x = v_y = \frac{25\sqrt{2}}{2} \approx 17.68$ m/s - Composite functions: $f(g(x)) = \sin(1 - x^2)$, $g(f(x)) = 1 - \sin^2 x$ - Scalars: $m = \frac{26}{7}$, $n = \frac{17}{7}$