1. **Problem 1: List subsets of the set \{4, 5, 6\}** The subsets of a set include the empty set, single-element sets, two-element sets, and the full set itself. 2. **Subsets of \{4, 5, 6\}:** - \{\}\ (empty set) - \{4\}, \{5\}, \{6\} - \{4, 5\}, \{4, 6\}, \{5, 6\} - \{4, 5, 6\} --- 3. **Problem 2: Graph the inequality $y > -\frac{1}{3}x - 2$** - The boundary line is $y = -\frac{1}{3}x - 2$. - The slope is $-\frac{1}{3}$, and the y-intercept is $-2$. - The inequality $y > -\frac{1}{3}x - 2$ means the region above this line is shaded. 4. **Check if point $(0, 3)$ fits the inequality:** Substitute $x=0$ and $y=3$: $$3 > -\frac{1}{3} \times 0 - 2 = -2$$ Since $3 > -2$ is true, the point $(0, 3)$ satisfies the inequality. --- 5. **Problem 3: Match lines and rotate 90° around (0,0)** - Original lines: - $y = 3x$ (line with slope 3) - $y = 3$ (horizontal line) - $x = 3$ (vertical line) 6. **Rotate $y = 3x$ by 90° around (0,0):** - The slope of the original line is $m = 3$. - The slope of the rotated line is $m' = -\frac{1}{m} = -\frac{1}{3}$ (negative reciprocal). - The rotated line passes through (0,0), so its equation is: $$y = -\frac{1}{3}x$$ --- 7. **Problem 4: Slope and translation of line through points (-2,5) and (4,2)** - Slope formula: $$m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{2 - 5}{4 - (-2)} = \frac{-3}{6} = -\frac{1}{2}$$ - Equation of the line in point-slope form using point (-2,5): $$y - 5 = -\frac{1}{2}(x + 2)$$ - Simplify: $$y - 5 = -\frac{1}{2}x - 1$$ $$y = -\frac{1}{2}x - 1 + 5$$ $$y = -\frac{1}{2}x + 4$$ - Translate the line up by 5 units: $$y_{new} = y + 5 = -\frac{1}{2}x + 4 + 5 = -\frac{1}{2}x + 9$$ --- **Summary:** - Subsets of \{4,5,6\} listed. - Graph and inequality $y > -\frac{1}{3}x - 2$ explained. - Point (0,3) satisfies the inequality. - Rotation of line $y=3x$ by 90° is $y = -\frac{1}{3}x$. - Slope of line through (-2,5) and (4,2) is $-\frac{1}{2}$. - New equation after translation up 5 units is $y = -\frac{1}{2}x + 9$.