Rational Expressions

1. Problem 30: Simplify and combine $$\frac{2}{x+1} + \frac{1}{x-2} - \frac{1}{x^2-1}$$. Note $$x^2-1 = (x+1)(x-1)$$. Find common denominator: $$(x+1)(x-2)(x-1)$$. Rewrite each fraction: $$\frac{2}{x+1} = \frac{2(x-2)(x-1)}{(x+1)(x-2)(x-1)}$$ $$\frac{1}{x-2} = \frac{1(x+1)(x-1)}{(x+1)(x-2)(x-1)}$$ $$\frac{1}{x^2-1} = \frac{1(x-2)}{(x+1)(x-1)(x-2)}$$ Sum numerator: $$2(x-2)(x-1) + (x+1)(x-1) - (x-2)$$ Expand terms: $$2(x^2 - 3x + 2) + (x^2 - 1) - (x - 2)$$ $$2x^2 - 6x + 4 + x^2 - 1 - x + 2 = 3x^2 - 7x + 5$$ Final result: $$\frac{3x^2 - 7x + 5}{(x-2)(x^2 - 1)}$$ Answer (c). 2. Problem 31: Simplify $$\frac{1}{x-3} + \frac{1}{x+3}$$. Common denominator: $$(x-3)(x+3) = x^2 - 9$$ Sum numerator: $$x+3 + x - 3 = 2x$$ Final: $$\frac{2x}{x^2 - 9}$$ Answer (b). 3. Problem 32: Simplify $$-\frac{2}{x} + \frac{1}{x-1} + \frac{1}{x+1}$$. Common denominator: $$x(x-1)(x+1) = x(x^2 - 1)$$ Rewrite each term: $$-\frac{2}{x} = -\frac{2(x-1)(x+1)}{x(x-1)(x+1)} = -\frac{2(x^2 -1)}{x(x^2 -1)}$$ $$\frac{1}{x-1} = \frac{x(x+1)}{x(x-1)(x+1)}$$ $$\frac{1}{x+1} = \frac{x(x-1)}{x(x-1)(x+1)}$$ Sum numerator: $$-2(x^2 -1) + x(x+1) + x(x -1) = -2x^2 + 2 + x^2 + x + x^2 - x = 2$$ Final fraction: $$\frac{2}{x(x^2 - 1)}$$ Answer (c). 4. Problem 33: Simplify $$\frac{x^2 - 3x + 2}{x^2 + 5x + 4} \times \frac{x+1}{x-1}$$. Factor: $$x^2 -3x + 2 = (x-1)(x-2)$$ $$x^2 + 5x + 4 = (x+1)(x+4)$$ Rewrite: $$\frac{(x-1)(x-2)}{(x+1)(x+4)} \times \frac{x+1}{x-1} = \frac{(x-1)(x-2)(x+1)}{(x+1)(x+4)(x-1)}$$ Cancel common factors: $$(x-1), (x+1)$$ Result: $$\frac{x-2}{x+4}$$ Answer (c). 5. Problem 34: Simplify $$\frac{4x^2 - 4}{x^2 + 2x + 1} \times \frac{x+1}{x-1}$$. Factor: $$4x^2 - 4 = 4(x^2 -1) = 4(x-1)(x+1)$$ $$x^2 + 2x +1 = (x+1)^2$$ Rewrite: $$\frac{4(x-1)(x+1)}{(x+1)^2} \times \frac{x+1}{x-1} = \frac{4(x-1)(x+1)(x+1)}{(x+1)^2 (x-1)}$$ Cancel $$(x-1)$$ and one $$(x+1)$$ Result: $$\frac{4x}{x+1}$$ not matching options, so check carefully: Actually canceling fully yields $$\frac{4x}{x+1}$$ but none of the options match. Check options: closest is none, so re-check steps: In numerator: $$4(x-1)(x+1)\times(x+1) = 4(x-1)(x+1)^2$$ Denominator: $$(x+1)^2 (x-1)$$ Cancel $$(x+1)^2 (x-1)$$ Result: $$4$$ Answer (c). 6. Problem 35: Simplify $$\frac{x^2 -9}{x^2 + 6x + 9} \times \frac{x+3}{x-9}$$. Factor: $$x^2 -9 = (x-3)(x+3)$$ $$x^2 + 6x + 9 = (x+3)^2$$ Rewrite: $$\frac{(x-3)(x+3)}{(x+3)^2} \times \frac{x+3}{x-9} = \frac{(x-3)(x+3)(x+3)}{(x+3)^2 (x-9)}$$ Cancel $(x+3)^2$ Result: $$\frac{x-3}{x-9}$$ Answer (a). 7. Problem 36: Simplify $$\frac{3x^2 - 12}{x+2} \times \frac{x}{x-2}$$. Factor numerator: $$3x^2 -12 = 3(x^2 -4) = 3(x-2)(x+2)$$ Rewrite: $$\frac{3(x-2)(x+2)}{x+2} \times \frac{x}{x-2} = 3(x-2)(x+2) \times \frac{x}{(x+2)(x-2)}$$ Cancel common factors: Result: $$3x$$ Answer (a). 8. Problem 37: Simplify $$\frac{2 - 3x - 2x^2}{x^2 + 3x} \div \frac{x^2 + 3x + 2}{x + 3}$$. Rewrite division as multiplication by reciprocal: $$\frac{2 - 3x - 2x^2}{x^2 + 3x} \times \frac{x+3}{x^2 + 3x + 2}$$ Factor denominators: $$x^2 + 3x = x(x+3)$$ $$x^2 + 3x + 2 = (x+1)(x+2)$$ Rewrite numerator of first fraction: $$2 - 3x - 2x^2 = -(2x^2 + 3x - 2)$$ Factor $$2x^2 + 3x -2 = (2x -1)(x+2)$$ so numerator is $$-(2x -1)(x+2)$$ Rewrite full expression: $$\frac{-(2x - 1)(x + 2)}{x(x + 3)} \times \frac{x + 3}{(x + 1)(x + 2)}$$ Cancel $$(x+3)$$ and $$(x + 2)$$ Result: $$\frac{-(2x-1)}{x(x+1)} = \frac{1 - 2x}{x(x+1)}$$ Answer (c). 9. Problem 38: Simplify $$\frac{x+2}{1 + \frac{2}{x}}$$. Rewrite denominator: $$1 + \frac{2}{x} = \frac{x+2}{x}$$ Rewrite fraction: $$\frac{x+2}{(x+2)/x} = (x+2) \times \frac{x}{x+2} = x$$ Answer (a).