Solutions to Problem Set 17

Math 101-06

10-18-2017

[Adding fractions]

1. Add the fractions and simplify:

$$\dfrac{1}{x} + \dfrac{1}{y} - \dfrac{2}{x y}.$$

$$\dfrac{1}{x} + \dfrac{1}{y} - \dfrac{2}{x y} = \dfrac{1}{x} \cdot \dfrac{y}{y} + \dfrac{1}{y} \cdot \dfrac{x}{x} - \dfrac{2}{x y} = \dfrac{y + x - 2}{x y}.\quad\halmos$$


2. Add the fractions and simplify:

$$\dfrac{4}{x} - \dfrac{5}{x^2}.$$

$$\dfrac{4}{x} - \dfrac{5}{x^2} = \dfrac{4}{x} \cdot \dfrac{x}{x} - \dfrac{5}{x^2} = \dfrac{4 x - 5}{x^2}.\quad\halmos$$


3. Add the expressions and simplify:

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

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

$$\dfrac{4(x + 3) + (x - 2)}{(x - 2)(x + 3)} = \dfrac{4 x + 12 + x - 2}{(x - 2)(x + 3)} = \dfrac{5 x + 10}{(x - 2)(x + 3)} = \dfrac{5(x + 2)}{(x - 2)(x + 3)}.\quad\halmos$$


4. Add the fractions and simplify:

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

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

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


5. Add the fractions and simplify:

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

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


6. Add the fractions and simplify:

$$\dfrac{3}{x^2 - 5 x + 4} - \dfrac{1}{x^2 - 7 x + 12}.$$

$$\dfrac{3}{x^2 - 5 x + 4} - \dfrac{1}{x^2 - 7 x + 12} = \dfrac{3}{(x - 1)(x - 4)} - \dfrac{1}{(x - 3)(x - 4)} =$$

$$\dfrac{3}{(x - 1)(x - 4)} \cdot \dfrac{x - 3}{x - 3} - \dfrac{1}{(x - 3)(x - 4)} \cdot \dfrac{x - 1}{x - 1} = \dfrac{3(x - 3) - (x - 1)}{(x - 1)(x - 3)(x - 4)} =$$

$$\dfrac{3 x - 9 - x + 1}{(x - 1)(x - 3)(x - 4)} = \dfrac{2 x - 8}{(x - 1)(x - 3)(x - 4)} = \dfrac{2(x - 4)}{(x - 1)(x - 3)(x - 4)} = \dfrac{2}{(x - 1)(x - 3)}.\quad\halmos$$


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