1. **Problem 1: Solve the rational inequality** Given: $$\frac{x+2}{x-4} \leq 3$$ 2. **Rewrite the inequality:** Bring all terms to one side: $$\frac{x+2}{x-4} - 3 \leq 0$$ 3. **Find a common denominator and combine:** $$\frac{x+2}{x-4} - \frac{3(x-4)}{x-4} \leq 0$$ $$\frac{x+2 - 3(x-4)}{x-4} \leq 0$$ 4. **Simplify the numerator:** $$x + 2 - 3x + 12 = -2x + 14$$ So inequality becomes: $$\frac{-2x + 14}{x-4} \leq 0$$ 5. **Factor numerator:** $$\frac{-2(x - 7)}{x-4} \leq 0$$ 6. **Determine critical points:** Numerator zero at $$x=7$$ Denominator zero at $$x=4$$ (excluded from domain) 7. **Test intervals around critical points:** Intervals: $$(-\infty,4), (4,7), (7,\infty)$$ - For $$x=0$$ in $$(-\infty,4)$$: $$\frac{-2(0-7)}{0-4} = \frac{-2(-7)}{-4} = \frac{14}{-4} = -3.5 \leq 0$$ True - For $$x=5$$ in $$(4,7)$$: $$\frac{-2(5-7)}{5-4} = \frac{-2(-2)}{1} = 4 > 0$$ False - For $$x=8$$ in $$(7,\infty)$$: $$\frac{-2(8-7)}{8-4} = \frac{-2(1)}{4} = -\frac{1}{2} \leq 0$$ True 8. **Check inclusion of critical points:** At $$x=7$$ numerator zero, fraction zero, so include $$x=7$$. At $$x=4$$ denominator zero, undefined, exclude $$x=4$$. 9. **Solution:** $$(-\infty,4) \cup [7,\infty)$$ --- 10. **Problem 2: Analyze the reciprocal function** Given: $$f(x) = \frac{1}{3x + 9}$$ **a) Find vertical and horizontal asymptotes:** - Vertical asymptote where denominator zero: $$3x + 9 = 0 \Rightarrow x = -3$$ - Horizontal asymptote as $$x \to \pm \infty$$: Since numerator is constant and denominator grows large, horizontal asymptote is: $$y = 0$$ **b) Find intercepts:** - x-intercept: set $$f(x) = 0$$ $$\frac{1}{3x + 9} = 0$$ No solution since numerator 1 never zero, so no x-intercept. - y-intercept: evaluate at $$x=0$$ $$f(0) = \frac{1}{3(0) + 9} = \frac{1}{9}$$ So y-intercept is $$\left(0, \frac{1}{9}\right)$$ **c) Sketch description:** - Vertical asymptote at $$x = -3$$ - Horizontal asymptote at $$y = 0$$ - No x-intercept - y-intercept at $$\left(0, \frac{1}{9}\right)$$ - Graph is a hyperbola with branches in top-left and bottom-right relative to vertical asymptote. **Final answers:** - Rational inequality solution: $$(-\infty,4) \cup [7,\infty)$$ - Vertical asymptote: $$x = -3$$ - Horizontal asymptote: $$y = 0$$ - x-intercept: none - y-intercept: $$\left(0, \frac{1}{9}\right)$$