Addition and Subtraction of Algebraic Fractions

Example 6.2.1:

Simplify the following:

a. \( \frac{4}{5m} \: + \: \frac{5}{6m} \)

b. \( \frac{2}{x\:+\:1} \: + \: \frac{3}{2x\:+\:3} \)

c. \( \frac{2}{x\:+\:1} \: + \: \frac{3}{x^2\:-\:1} \)

d. \( \frac{m}{n} \: + \: \frac{m\:-\:1}{5n} \: - \: \frac{m\:-\:2}{10n}\)

e. \( \frac{x\:+\:4}{x^2 \: - \: 3x} \: - \: \frac{x \: - \: 1}{9 \: - \: x^2} \)

Solution

a. \( \frac{4}{5m} \: + \: \frac{5}{6m} \)

First step is to find the L.C.M of the denominators

L.C.M = 30mn

⇒ \( \frac{4}{5m} \: + \: \frac{5}{6n}\\ = \frac{24n \: + \: 25m}{30mn} \)

b. \( \frac{2}{x\:+\:1} \: + \: \frac{3}{2x\:+\:3} \)

⇒ \( \frac{2(2x\:+\:3)\:+\: 3(x \:+\:1)}{(x \:+\:1)(2x\:+\:3)} \)

⇒ \( \frac{4x\:+\:6\:+\: 3x \:+\:3}{(x \:+\:1)(2x\:+\:3)} \)

⇒ \( \frac{7x\:+\:9}{(x \:+\:1)(2x\:+\:3)} \)

c. \( \frac{2}{x\:+\:1} \: - \: \frac{3}{x^2\:-\:1} \)

⇒ \( \frac{2}{x\:+\:1} \: - \: \frac{3}{(x\:+\:1)(x\:-\:1)} \)

⇒ \( \frac{2(x\:-\:1) \: - \: 3}{(x\:+\:1)(x\:-\:1)} \)

⇒ \( \frac{2x\:-\:2 \: - \: 3}{(x\:+\:1)(x\:-\:1)} \)

⇒ \( \frac{2x\:-\: 5}{(x\:+\:1)(x\:-\:1)} \)

d. ...