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:

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

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

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

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

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






8th Grade Math Practice

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