SUBTRACTION OF ALGEBRAIC FRACTIONS

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An algebraic fraction is one with algebra in the numerator and/or the denominator.

The subtraction of algebraic fractions is straightforward. As with a normal fraction, we take the L.C.M so that the denominator becomes the same.

๐Ÿ‘‰This post is a continuation of our series on algebraic fractions๐Ÿ‘ˆ

Example 1
Solve $\frac{4}{x+1}-\frac{3}{x-2}$

Solution:
Taking the L.C.M
$\frac{4(x-2)-3(x+1)}{(x+1)(x-2)}$
$\frac{4x-8-3x-3}{(x+1)(x-2)}$
$\frac{x-11}{(x+1)(x-2)}$

Example 2
Simplify $\frac{2}{2w+1}-\frac{1}{w+3}$

Solution:
Taking the L.C.M
$\frac{2(w+3)-1(2w+1)}{(2w+1)(w+3)}$
$\frac{2w+6-2w-1}{(2w+1)(w+3)}$
$\frac{5}{(2w+1)(w+3)}$

Example 3
Simplify $\frac{2}{x-1}-\frac{6}{(x-1)(2x+1)}$

Solution:
Taking the L.C.M
$\frac{2(2x+1)-6(1)}{(x-1)(2x+1)}$
$\frac{4x+2-6}{(x-1)(2x+1)}$
$\frac{4x-4}{(x-1)(2x+1)}$

This can be simplify further
$\frac{4\require{cancel}\bcancel{(x-1)}}{\require{cancel}\bcancel{(x-1)}(2x+1)}$
$\frac{4}{2x+1}$

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Example 4
Solve $\frac{4x}{t^2-9}-\frac{2}{t+3}$

Solution:
$t^2-9$ is a difference of two squares.
$\frac{4t}{(t+3)(t-3)}-\frac{2}{t+3}$

By taking the L.C.M
$\frac{4t(1)-2(t-3)}{(t+3)(t-3)}$
$\frac{4t-2t+6)}{(t+3)(t-3)}$
$\frac{2t+6}{(t+3)(t-3)}$
$\frac{2(t+3)}{(t+3)(t-3)}$
$\frac{2\require{cancel}\bcancel{(t+3)}}{\require{cancel}\bcancel{(t+3)}(t-3)}$
$\frac{2}{t-3}$

Example 5
Simplify $\frac{r}{r^2-9}-\frac{1}{r^2-4r+3}$

Solution:
$\frac{r}{(r-3)(r+3)}-\frac{1}{(r-3)(r-1)}$.

Taking the L.C.M
$\frac{r(r-1)-1(r+3)}{(r-3)(r+3)(r-1)}$
$\frac{r^2-r-r+3}{(r-3)(r+3)(r-1)}$
$\frac{r^2-2r+3}{(r-3)(r+3)(r-1)}$

Factorizing the numerator
$\frac{(r+1)(r-3)}{(r-3)(r+3)(r-1)}$
$\frac{(r+1)\require{cancel}\bcancel{(r-3)}}{\require{cancel}\bcancel{(r-3)}(r+3)(r-1)}$
$\frac{r+1}{(r+3)(r-1)}$

Example 6
Solve $\frac{5x^2-11x+9}{x^2+3x-10}-\frac{2x-3}{x-2}$

Solution:
The numerator of the right hand is not factorizable, but its denominator is factorizable. Hence we factorize.

$\frac{5x^2-11x+9}{(x-2)(x+5}-\frac{2x-3}{x-2}$

Taking the L.C.M
$\frac{5x^2-11x+9-(x+5)(2x-3)}{(x-2)(x+5)}$
$\frac{5x^2-11x+9-(2x^2+7x-15)}{(x-2)(x+5)}$

Opening the bracket at the numerator
$\frac{5x^2-11x+9-2x^2-7x+15}{(x-2)(x+5)}$
$\frac{3x^2-18x+24}{(x-2)(x+5)}$
$\frac{3x^2-6x-12x+24}{(x-2)(x+5)}$
$\frac{3x(x-2)-12(x-2)}{(x-2)(x+5)}$
$\frac{(3x-12)(x-2)}{(x-2)(x+5)}$
$\frac{(3x-12)\require{cancel}\bcancel{(x-2)}}{\require{cancel}\bcancel{(x-2)}(x+5)}$
$\frac{3x-12}{x+5}$

Related post
Example 7
Solve $\frac{4x}{25x^2-64}-\frac{x}{5x+8}-\frac{1}{5x-8}$

Solution:
$25^2-64$ is difference of two squares
$\frac{4x}{(5x+8)(5x-8)}-\frac{x}{5x+8}-\frac{1}{5x-8}$

Taking the L.C.M
$\frac{4x(1)-x(5x-8)-1(5x+8)}{(5x+8)(5x-8)}$
$\frac{4x-5x^2+8x-5x-8)}{(5x+8)(5x-8)}$
$\frac{-5x^2+7x-8)}{(5x+8)(5x-8)}$

In this next post, we will continue our series on algebraic fractions when we look at multiplication of algebraic fractions.

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