1. **Problem 11:** Solve $$1 = \frac{v + 2}{v - 4} + \frac{7v - 42}{v - 4}$$ Since denominators are the same, combine numerators: $$1 = \frac{v + 2 + 7v - 42}{v - 4} = \frac{8v - 40}{v - 4}$$ Multiply both sides by $$v - 4$$: $$v - 4 = 8v - 40$$ Bring terms to one side: $$v - 4 - 8v + 40 = 0 \Rightarrow -7v + 36 = 0$$ Solve for $$v$$: $$7v = 36 \Rightarrow v = \frac{36}{7}$$ 2. **Problem 12:** Solve $$\frac{r - 4}{5r} = \frac{1}{5r} + 1$$ Multiply both sides by $$5r$$ (assuming $$r \neq 0$$): $$r - 4 = 1 + 5r$$ Bring terms to one side: $$r - 4 - 1 - 5r = 0 \Rightarrow -4r - 5 = 0$$ Solve for $$r$$: $$-4r = 5 \Rightarrow r = -\frac{5}{4}$$ 3. **Problem 13:** Solve $$1 + \frac{x^2 - 5x - 24}{3x} = \frac{x - 6}{3x}$$ Multiply both sides by $$3x$$ (assuming $$x \neq 0$$): $$3x + x^2 - 5x - 24 = x - 6$$ Simplify left side: $$x^2 - 2x - 24 = x - 6$$ Bring all terms to one side: $$x^2 - 2x - 24 - x + 6 = 0 \Rightarrow x^2 - 3x - 18 = 0$$ Factor or use quadratic formula: $$x = \frac{3 \pm \sqrt{9 + 72}}{2} = \frac{3 \pm 9}{2}$$ Solutions: $$x = 6 \text{ or } x = -3$$ 4. **Problem 14:** Solve $$1 = \frac{1}{x^2 + 2x} + \frac{x - 1}{x}$$ Rewrite denominator: $$x^2 + 2x = x(x + 2)$$ Multiply both sides by $$x(x + 2)$$ (assuming $$x \neq 0, x \neq -2$$): $$x(x + 2) = x + 2 + (x - 1)(x + 2)$$ Expand right side: $$x + 2 + x^2 + 2x - x - 2 = x^2 + 2x$$ Simplify right side: $$x^2 + 2x$$ Left side is: $$x^2 + 2x$$ So equation is: $$x^2 + 2x = x^2 + 2x$$ This is true for all $$x \neq 0, -2$$ 5. **Problem 15:** Solve $$\frac{n + 5}{n + 8} = 1 + \frac{6}{n + 1}$$ Rewrite right side: $$1 = \frac{n + 8}{n + 8}$$ So: $$\frac{n + 5}{n + 8} = \frac{n + 8}{n + 8} + \frac{6}{n + 1}$$ Multiply both sides by $$(n + 8)(n + 1)$$ (assuming $$n \neq -8, -1$$): $$(n + 5)(n + 1) = (n + 8)(n + 1) + 6(n + 8)$$ Expand: $$n^2 + n + 5n + 5 = n^2 + n + 8n + 8 + 6n + 48$$ Simplify: $$n^2 + 6n + 5 = n^2 + 15n + 56$$ Bring all terms to one side: $$n^2 + 6n + 5 - n^2 - 15n - 56 = 0 \Rightarrow -9n - 51 = 0$$ Solve for $$n$$: $$9n = -51 \Rightarrow n = -\frac{51}{9} = -\frac{17}{3}$$ 6. **Problem 16:** Solve $$\frac{r + 5}{r^2 - 2r} - 1 = \frac{1}{r^2 - 2r}$$ Rewrite denominator: $$r^2 - 2r = r(r - 2)$$ Multiply both sides by $$r(r - 2)$$ (assuming $$r \neq 0, 2$$): $$(r + 5) - r(r - 2) = 1$$ Expand: $$r + 5 - (r^2 - 2r) = 1$$ Simplify: $$r + 5 - r^2 + 2r = 1 \Rightarrow -r^2 + 3r + 5 = 1$$ Bring all terms to one side: $$-r^2 + 3r + 4 = 0$$ Multiply by $$-1$$: $$r^2 - 3r - 4 = 0$$ Factor: $$(r - 4)(r + 1) = 0$$ Solutions: $$r = 4 \text{ or } r = -1$$ 7. **Problem 17:** Solve $$\frac{1}{x^2 - 5x} = \frac{x + 7}{x} - 1$$ Rewrite denominator: $$x^2 - 5x = x(x - 5)$$ Rewrite right side: $$\frac{x + 7}{x} - 1 = \frac{x + 7 - x}{x} = \frac{7}{x}$$ So equation is: $$\frac{1}{x(x - 5)} = \frac{7}{x}$$ Multiply both sides by $$x(x - 5)$$ (assuming $$x \neq 0, 5$$): $$1 = 7(x - 5)$$ Expand: $$1 = 7x - 35$$ Bring terms to one side: $$7x - 36 = 0$$ Solve for $$x$$: $$7x = 36 \Rightarrow x = \frac{36}{7}$$ 8. **Problem 18:** Solve $$\frac{a - 2}{a + 3} - 1 = \frac{3}{a + 2}$$ Rewrite left side: $$\frac{a - 2}{a + 3} - \frac{a + 3}{a + 3} = \frac{a - 2 - (a + 3)}{a + 3} = \frac{-5}{a + 3}$$ So equation is: $$\frac{-5}{a + 3} = \frac{3}{a + 2}$$ Cross multiply (assuming $$a \neq -3, -2$$): $$-5(a + 2) = 3(a + 3)$$ Expand: $$-5a - 10 = 3a + 9$$ Bring terms to one side: $$-5a - 10 - 3a - 9 = 0 \Rightarrow -8a - 19 = 0$$ Solve for $$a$$: $$8a = -19 \Rightarrow a = -\frac{19}{8}$$ 9. **Problem 19:** Solve $$\frac{p + 5}{p^2 + p} = \frac{1}{p^2 + p} - \frac{p - 6}{p + 1}$$ Rewrite denominator: $$p^2 + p = p(p + 1)$$ Multiply both sides by $$p(p + 1)$$ (assuming $$p \neq 0, -1$$): $$(p + 5) = 1 - (p - 6)p$$ Expand right side: $$1 - p^2 + 6p$$ Bring all terms to one side: $$p + 5 - 1 + p^2 - 6p = 0 \Rightarrow p^2 - 5p + 4 = 0$$ Factor: $$(p - 4)(p - 1) = 0$$ Solutions: $$p = 4 \text{ or } p = 1$$ 10. **Problem 20:** Solve $$\frac{5}{n^3 + 5n^2} = \frac{4}{n + 5} + \frac{1}{n^2}$$ Rewrite denominator: $$n^3 + 5n^2 = n^2(n + 5)$$ Multiply both sides by $$n^2(n + 5)$$ (assuming $$n \neq 0, -5$$): $$5 = 4n^2 + (n + 5)$$ Expand right side: $$5 = 4n^2 + n + 5$$ Bring all terms to one side: $$0 = 4n^2 + n + 5 - 5 \Rightarrow 0 = 4n^2 + n$$ Factor: $$n(4n + 1) = 0$$ Solutions: $$n = 0 \text{ (excluded, denominator zero) or } n = -\frac{1}{4}$$ **Final answers:** 11) $$v = \frac{36}{7}$$ 12) $$r = -\frac{5}{4}$$ 13) $$x = 6, -3$$ 14) All $$x \neq 0, -2$$ 15) $$n = -\frac{17}{3}$$ 16) $$r = 4, -1$$ 17) $$x = \frac{36}{7}$$ 18) $$a = -\frac{19}{8}$$ 19) $$p = 4, 1$$ 20) $$n = -\frac{1}{4}$$