# Solving Algebraic Fractions

Solving algebraic fractions to its lowest term follow the step-by-step explanation given below.

1. Simplify the algebraic fractions: $$\frac{x - 2}{8} - \frac{2(2x + 3)}{3} + \frac{11x - 3}{6}$$

Solution:

$$\frac{x - 2}{8} - \frac{2(2x + 3)}{3} + \frac{11x - 3}{6}$$

$$\frac{3(x - 2) - 16(2x + 3) + 4(11x - 3)}{24}$$

$$\frac{3x - 6 - 32x - 48 + 44x - 12}{24}$$

$$\frac{15x - 66}{24}$$

$$\frac{3(5x - 22)}{24}$$

$$\frac{5x - 22}{8}$$

2. Reduce the algebraic fractions: $$\frac{2a}{a - 2} - \frac{a^{2}}{a^{2} - 4}$$

Solution:

$$\frac{2a}{a - 2} - \frac{a^{2}}{a^{2} - 4}$$

= $$\frac{2a}{a - 2} - \frac{a^{2}}{a^{2} - 2^{2}}$$

= $$\frac{2a}{a - 2} - \frac{a^{2}}{(a + 2) (a - 2)}$$

= $$\frac{2a(a + 2) - a^{2}}{(a + 2) (a - 2)}$$

= $$\frac{2a^{2} + 4a - a^{2}}{a^{2} - 4}$$

= $$\frac{a^{2} + 4a}{a^{2} - 4}$$

3. Reduce to lowest terms -- if possible: $$\frac{2}{a + b} - \frac{3}{a - b} + \frac{6a}{a^{2} - b^{2}}$$

Solution:

$$\frac{2}{a + b} - \frac{3}{a - b} + \frac{6a}{a^{2} - b^{2}}$$

= $$\frac{2(a - b) - 3(a + b) + 6a}{(a + b) (a - b)}$$

= $$\frac{2a - 2b - 3a - 3b + 6a}{a^{2} - b^{2}}$$

= $$\frac{5a - 5b}{a^{2} - b^{2}}$$

= $$\frac{5(a - b)}{(a + b)(a - b)}$$

= $$\frac{5}{(a + b)}$$

4. simplify and Reduce:  $$\frac{3x}{x^{2} - 9} + \frac{1}{x^{2} + 2x - 15}$$

Solution:

$$\frac{3x}{x^{2} - 9} + \frac{1}{x^{2} + 2x - 15}$$

Step 1: Factorize the polynomials separately first:

(i) x$$^{2}$$ – 9 = (x + 3) (x – 3)

(ii) x$$^{2}$$ + 2x – 15 = x$$^{2}$$ + 5x – 3x – 15

= x(x + 5) – 3(x + 5)

= (x + 5) (x – 3)

Step 2: Simplify by substituting with the factors:

$$\frac{3x}{x^{2} - 9} + \frac{1}{x^{2} + 2x - 15}$$

= $$\frac{3x}{(x + 3) (x - 3)} + \frac{1}{(x + 5) (x - 3)}$$

= $$\frac{3x(x + 5) + x + 3}{(x + 3)(x - 3)(x + 5)}$$

= $$\frac{3x^{2} + 15x + x + 3}{(x + 3)(x - 3)(x + 5)}$$

= $$\frac{3x^{2} + 16x + 3}{(x + 3)(x - 3)(x + 5)}$$

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