Solutions to Problem Set 18

Math 101-06

10-20-2017

[Adding fractions]

1. Add the fractions and simplify:

$$\dfrac{1}{x^2 + 6 x} + \dfrac{2}{x^2 + 3 x - 18}.$$

$$\dfrac{1}{x^2 + 6 x} + \dfrac{2}{x^2 + 3 x - 18} = \dfrac{1}{x(x + 6)} + \dfrac{2}{(x - 3)(x + 6)} = \dfrac{1}{x(x + 6)} \cdot \dfrac{x - 3}{x - 3} + \dfrac{2}{(x - 3)(x + 6)} \cdot \dfrac{x}{x} =$$

$$\dfrac{(x - 3) + 2 x}{x(x - 3)(x + 6)} = \dfrac{3 x - 3}{x(x - 3)(x + 6)} = \dfrac{3(x - 1)}{x(x - 3)(x + 6)}.\quad\halmos$$


2. Add the fractions and simplify:

$$\dfrac{2}{x^2 + 2 x} - \dfrac{1}{x^2 + 5 x + 6} + \dfrac{3}{x^2 + 3 x}.$$

$$\dfrac{2}{x^2 + 2 x} - \dfrac{1}{x^2 + 5 x + 6} + \dfrac{3}{x^2 + 3 x} = \dfrac{2}{x(x + 2)} - \dfrac{1}{(x + 2)(x + 3)} + \dfrac{3}{x(x + 3)} =$$

$$\dfrac{2}{x(x + 2)} \cdot \dfrac{x + 3}{x + 3} - \dfrac{1}{(x + 2)(x + 3)} \cdot \dfrac{x}{x} + \dfrac{3}{x(x + 3)} \cdot \dfrac{x + 2}{x + 2} = \dfrac{2(x + 3) - x + 3(x + 2)}{x(x + 2)(x + 3)} = \dfrac{2 x + 6 - x + 3 x + 6}{x(x + 2)(x + 3)} =$$

$$\dfrac{4 x + 12}{x(x + 2)(x + 3)} = \dfrac{4(x + 3)}{x(x + 2)(x + 3)} = \dfrac{4}{x(x + 2)}.\quad\halmos$$


[Complex fractions]

3. Simplify:

$$\dfrac{4 + \dfrac{1}{x}}{7 + \dfrac{2}{x}}.$$

$$\dfrac{4 + \dfrac{1}{x}}{7 + \dfrac{2}{x}} = \dfrac{4 + \dfrac{1}{x}}{7 + \dfrac{2}{x}} \cdot \dfrac{\vphantom{\dfrac{x}{x}} x}{\vphantom{\dfrac{x}{x}} x} = \dfrac{4 x + \dfrac{x}{x}}{7 x + \dfrac{2 x}{x}} = \dfrac{4 x + 1}{7 x + 2}.\quad\halmos$$


4. Simplify:

$$\dfrac{1 - \dfrac{3}{x} + \dfrac{2}{x^2}}{1 - \dfrac{4}{x} + \dfrac{3}{x^2}}.$$

$$\dfrac{1 - \dfrac{3}{x} + \dfrac{2}{x^2}}{1 - \dfrac{4}{x} + \dfrac{3}{x^2}} = \dfrac{1 - \dfrac{3}{x} + \dfrac{2}{x^2}}{1 - \dfrac{4}{x} + \dfrac{3}{x^2}} \cdot \dfrac{\vphantom{\dfrac{x}{x}} x^2}{\vphantom{\dfrac{x}{x}} x^2} = \dfrac{x^2 - \dfrac{3 x^2}{x} + \dfrac{2 x^2}{x^2}} {x^2 - \dfrac{4 x^2}{x} + \dfrac{3 x^2}{x^2}} = \dfrac{x^2 - 3 x + 2}{x^2 - 4 x + 3} = \dfrac{(x - 1)(x - 2)}{(x - 1)(x - 3)} = \dfrac{x - 2}{x - 3}.\quad\halmos$$


5. Simplify:

$$\dfrac{1 - \dfrac{3}{x^2 - 1}}{\dfrac{3}{x - 1} - \dfrac{1}{x + 1}}.$$

$$\dfrac{1 - \dfrac{3}{x^2 - 1}}{\dfrac{3}{x - 1} - \dfrac{1}{x + 1}} = \dfrac{1 - \dfrac{3}{(x - 1)(x + 1)}}{\dfrac{3}{x - 1} - \dfrac{1}{x + 1}} = \dfrac{1 - \dfrac{3}{(x - 1)(x + 1)}}{\dfrac{3}{x - 1} - \dfrac{1}{x + 1}} = \dfrac{\vphantom{\dfrac{x}{x}} (x - 1)(x + 1)}{\vphantom{\dfrac{x}{x}} (x - 1)(x + 1)} =$$

$$\dfrac{(x - 1)(x + 1) \dfrac{3 (x - 1)(x + 1)}{(x - 1)(x + 1)}} {\dfrac{3(x - 1)(x + 1)}{x - 1} - \dfrac{(x - 1)(x + 1)}{x + 1}} = \dfrac{(x - 1)(x + 1) - 3}{3(x + 1) - (x - 1)} = \dfrac{x^2 - 1 - 3}{3 x + 3 - x + 1} = \dfrac{x^2 - 4}{2 x + 4} =$$

$$\dfrac{(x - 2)(x + 2)}{2(x + 2)} = \dfrac{x - 2}{2}.\quad\halmos$$


6. Simplify:

$$\dfrac{\dfrac{5}{x - 3} - \dfrac{1}{x + 1}}{1 - \dfrac{5}{x^2 - 2 x - 3}}.$$

$$\dfrac{\dfrac{5}{x - 3} - \dfrac{1}{x + 1}}{1 - \dfrac{5}{x^2 - 2 x - 3}} = \dfrac{\dfrac{5}{x - 3} - \dfrac{1}{x + 1}}{1 - \dfrac{5}{(x - 3)(x + 1)}} = \dfrac{\dfrac{5}{x - 3} - \dfrac{1}{x + 1}}{1 - \dfrac{5}{(x - 3)(x + 1)}} = \dfrac{\vphantom{\dfrac{x}{x}} (x - 3)(x + 1)}{\vphantom{\dfrac{x}{x}} (x - 3)(x + 1)} =$$

$$\dfrac{\dfrac{5(x - 3)(x + 1)}{x - 3} - \dfrac{(x - 3)(x + 1)}{x + 1}} {(x - 3)(x + 1) - \dfrac{5(x - 3)(x + 1)}{(x - 3)(x + 1)}} = \dfrac{5(x + 1) - (x - 3)}{(x - 3)(x + 1) - 5} = \dfrac{5 x + 5 - x + 3}{x^2 - 2 x - 3 - 5} = \dfrac{4 x + 8}{x^2 - 2 x - 8} =$$

$$\dfrac{4(x + 2)}{(x - 4)(x + 2)} = \dfrac{4}{x - 4}.\quad\halmos$$


All action is involved in imperfection, like fire in smoke. - The Bhagavad Gita


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