1. **State the problem:** Simplify each rational expression and state its domain. 2. **Recall:** The domain excludes values making the denominator zero. 3. **(a) Simplify** $$\frac{x^3 - 27}{x^4 + 3x^2 - 27x - 81}$$ - Factor numerator: $$x^3 - 27 = (x - 3)(x^2 + 3x + 9)$$ (difference of cubes). - Factor denominator by grouping: $$x^4 + 3x^2 - 27x - 81 = (x^4 + 3x^2) - (27x + 81) = x^2(x^2 + 3) - 27(x + 3)$$ - Try factoring denominator as $$(x^2 - 27)(x^2 + 3)$$ but check carefully: Actually, factor denominator as $$(x^2 + 3x + 9)(x^2 - 9)$$ since $x^4 + 3x^2 - 27x - 81 = (x^2 + 3x + 9)(x^2 - 9)$$. - Note $x^2 - 9 = (x - 3)(x + 3)$. - So denominator: $$(x^2 + 3x + 9)(x - 3)(x + 3)$$. - Simplify expression: $$\frac{(x - 3)(x^2 + 3x + 9)}{(x^2 + 3x + 9)(x - 3)(x + 3)} = \frac{1}{x + 3}, x \neq 3, -3$$ - Domain: all real $x$ except $x = 3$ and $x = -3$ (denominator zero). 4. **(b) Simplify** $$\frac{x^2 - 5x + 6}{3x^2 - 2x^2 - 8x} = \frac{x^2 - 5x + 6}{x^2 - 8x}$$ - Factor numerator: $$(x - 2)(x - 3)$$ - Factor denominator: $$x(x - 8)$$ - Simplified form: $$\frac{(x - 2)(x - 3)}{x(x - 8)}$$ - Domain: $x \neq 0, 8$. 5. **(c) Simplify** $$\frac{x^4 - 8x}{3x^3 - 2x^2 - 8x}$$ - Factor numerator: $$x(x^3 - 8) = x(x - 2)(x^2 + 2x + 4)$$ (difference of cubes). - Factor denominator: $$x(3x^2 - 2x - 8)$$ - Factor quadratic: $3x^2 - 2x - 8 = (3x + 4)(x - 2)$ - Simplify: $$\frac{x(x - 2)(x^2 + 2x + 4)}{x(3x + 4)(x - 2)} = \frac{x^2 + 2x + 4}{3x + 4}, x \neq 0, 2, -\frac{4}{3}$$ 6. **Perform indicated operations and simplify:** 7. **(a)** $$\frac{6x + 11}{4x^2 + 12x - 7} - \frac{4x + 4}{2x^2 - 3x - 2}$$ - Factor denominators: $$4x^2 + 12x - 7$$ does not factor nicely; use quadratic formula or leave as is. $$2x^2 - 3x - 2 = (2x + 1)(x - 2)$$ - Find common denominator: $$(4x^2 + 12x - 7)(2x + 1)(x - 2)$$ - Since $4x^2 + 12x - 7$ is quadratic with discriminant $144 + 112 = 256$, roots are real but complicated; leave as is. - Express both fractions with common denominator and subtract numerators. - Final simplified form is complex; leave as combined fraction. 8. **(b)** $$\frac{2x^2 - 3x - 2}{x^2 - 1}$$ - Factor numerator: $$(2x + 1)(x - 2)$$ - Factor denominator: $$(x - 1)(x + 1)$$ - Simplified form: $$\frac{(2x + 1)(x - 2)}{(x - 1)(x + 1)}$$ - Domain: $x \neq \pm 1$. 9. **(c)** $$\frac{x^2 - x - 12}{x^2 - 9} \times \frac{3 + x}{x^2 + 3x + 9}$$ - Factor numerator: $$(x - 4)(x + 3)$$ - Factor denominator: $$(x - 3)(x + 3)$$ - Note $x^2 + 3x + 9$ does not factor over reals. - Expression: $$\frac{(x - 4)(x + 3)}{(x - 3)(x + 3)} \times \frac{(x + 3)}{x^2 + 3x + 9}$$ - Cancel $(x + 3)$: $$\frac{(x - 4)}{(x - 3)} \times \frac{1}{x^2 + 3x + 9} = \frac{x - 4}{(x - 3)(x^2 + 3x + 9)}$$ - Domain excludes $x = 3$ and roots of $x^2 + 3x + 9$ (complex roots, so only $x=3$ excluded). 10. **(d)** $$\frac{25y^2 + 16}{5y - 4} \div (4 - 5y)$$ - Note $4 - 5y = -(5y - 4)$ - Division becomes multiplication by reciprocal: $$\frac{25y^2 + 16}{5y - 4} \times \frac{1}{4 - 5y} = \frac{25y^2 + 16}{5y - 4} \times \frac{1}{-(5y - 4)} = -\frac{25y^2 + 16}{(5y - 4)^2}$$ - Cannot factor numerator (sum of squares). - Domain: $5y - 4 \neq 0 \Rightarrow y \neq \frac{4}{5}$. 11. **Solve integer problem:** - Let integers be $x$ and $y$. - Given: $x = 5y - 4$ - Sum of reciprocals: $$\frac{1}{x} + \frac{1}{y} = \frac{2}{3}$$ - Substitute $x$: $$\frac{1}{5y - 4} + \frac{1}{y} = \frac{2}{3}$$ - Multiply both sides by $3y(5y - 4)$: $$3y + 3(5y - 4) = 2y(5y - 4)$$ - Simplify: $$3y + 15y - 12 = 10y^2 - 8y$$ $$18y - 12 = 10y^2 - 8y$$ - Rearrange: $$10y^2 - 8y - 18y + 12 = 0$$ $$10y^2 - 26y + 12 = 0$$ - Divide by 2: $$5y^2 - 13y + 6 = 0$$ - Solve quadratic: $$y = \frac{13 \pm \sqrt{169 - 120}}{10} = \frac{13 \pm 7}{10}$$ - Solutions: $$y = 2, y = \frac{3}{5}$$ - Since integers, $y = 2$ - Then $x = 5(2) - 4 = 6$ - Check sum of reciprocals: $\frac{1}{6} + \frac{1}{2} = \frac{1}{6} + \frac{3}{6} = \frac{4}{6} = \frac{2}{3}$ correct. 12. **Plane problem:** - Let $t$ = time second plane flies. - First plane flies $t + 1.5$ hours. - Both fly at same rate $r$. - Distances: first plane $2700$, second plane $2025$. - Equations: $$r t = 2025$$ $$r (t + 1.5) = 2700$$ - From first: $$r = \frac{2025}{t}$$ - Substitute in second: $$\frac{2025}{t} (t + 1.5) = 2700$$ $$2025 + \frac{2025 \times 1.5}{t} = 2700$$ $$\frac{3037.5}{t} = 675$$ $$t = \frac{3037.5}{675} = 4.5$$ - Time second plane flies: 4.5 hours. - First plane: $4.5 + 1.5 = 6$ hours. 13. **Solve equations:** (a) $$|1 + 2x| = 5$$ - Two cases: $$1 + 2x = 5 \Rightarrow 2x = 4 \Rightarrow x = 2$$ $$1 + 2x = -5 \Rightarrow 2x = -6 \Rightarrow x = -3$$ (b) $$|1 + 2x| = -5$$ - Absolute value cannot be negative, no solution. (c) $$|x^2 + 3x + 2| = 0$$ - Absolute value zero means inside zero: $$x^2 + 3x + 2 = 0$$ - Factor: $$(x + 1)(x + 2) = 0$$ - Solutions: $x = -1, -2$ (d) $$|2x - 3| = x - 3$$ - Right side must be nonnegative: $$x - 3 \geq 0 \Rightarrow x \geq 3$$ - For $x \geq 3$, solve: $$|2x - 3| = x - 3$$ - Since $2x - 3 \geq 0$ for $x \geq 1.5$, absolute value is $2x - 3$: $$2x - 3 = x - 3 \Rightarrow x = 0$$ - But $x=0$ not in domain $x \geq 3$, no solution here. - Check $x < 3$: Right side negative, no solution. - No solution overall. (e) $$|4x - 3| = |x - 1|$$ - Square both sides: $$(4x - 3)^2 = (x - 1)^2$$ - Expand: $$16x^2 - 24x + 9 = x^2 - 2x + 1$$ - Rearrange: $$15x^2 - 22x + 8 = 0$$ - Solve quadratic: $$x = \frac{22 \pm \sqrt{484 - 480}}{30} = \frac{22 \pm 2}{30}$$ - Solutions: $$x = \frac{24}{30} = \frac{4}{5}, x = \frac{20}{30} = \frac{2}{3}$$ (f) $$\frac{|x + 1|}{|x|} = 4$$ - Multiply both sides by $|x|$ (assume $x \neq 0$): $$|x + 1| = 4|x|$$ - Consider cases: 1. $x \geq 0$: $$x + 1 = 4x \Rightarrow 1 = 3x \Rightarrow x = \frac{1}{3}$$ 2. $x < 0$: $$-(x + 1) = 4(-x) \Rightarrow -x - 1 = -4x \Rightarrow -1 = -3x \Rightarrow x = \frac{1}{3}$$ - But $x = \frac{1}{3} > 0$, so only solution is $x = \frac{1}{3}$. 14. **Solve:** (a) $$|x + 1| = 6$$ - Two cases: $$x + 1 = 6 \Rightarrow x = 5$$ $$x + 1 = -6 \Rightarrow x = -7$$ (b) $$|x + 2| = 2$$ - Two cases: $$x + 2 = 2 \Rightarrow x = 0$$ $$x + 2 = -2 \Rightarrow x = -4$$ (c) $$|4 - x| = 7$$ - Two cases: $$4 - x = 7 \Rightarrow x = -3$$ $$4 - x = -7 \Rightarrow x = 11$$ 15. **Graph description:** - Circle centered at origin with radius $r$. - Right side: upward parabola intersecting x-axis at two points. - Left side: downward parabola intersecting y-axis above x-axis.