Question 468027
{{{((x-7)/(x-8))-((x+1)/(x+8))+((x-24)/(x^2-64))}}}
Factor {{{x^2 - 64}}}
This is an example of difference of squares
{{{x^2 -64 = x^2 - 8^2 = (x-8)(x+8)}}}
Next find common denominator
*************
Ex:
{{{(1/4)+(1/3)+(1/12)}}}
The common denominator is the least common multiple of 3,4,12
Since 3 and 4 are both factors of 12 then the LCM is 12.
***********
In the same way, (x-8) and (x+8) are factors of (x-8)(x+8)
Thus the LCM and common denominator is (x-8)(x+8).
However, the value of the expression cannot change, if the denominator changes then the numerator must change by the same factor.
***************
Ex:
{{{(1/4)+(1/3)+(1/12) = (3/12)+(4/12)+(1/12)}}}
Because 4*3 = 12, multiply 1st fraction by 3, 2nd by 4, and 3rd by 1.
**************
In the same way, multiply 1st fraction by (x+8), 2nd by (x-8), 3rd by 1.
{{{((x+8)(x-7)/(x+8)(x-8))-((x-8)(x+1)/(x-8)(x+8))+((x-24)/(x-8)(x+8))}}}
Since denominators are now the same, combine into one fraction
{{{((x+8)(x-7) - (x-8)(x+1) + (x-24))/((x-8)(x+8))}}}
Distribute on top and add like terms
***************
Note: 
(x+a)(x+b) = x^2 + (a+b)x + ab
*****************
{{{((x^2+x-56) - (x^2-7x-8) + (x-24))/((x-8)(x+8))}}}
**Be careful when subtracting and remember to distribute the negative sign**
{{{(x^2-x^2+x+7x+x-56+8-24)/((x-8)(x+8))}}}
{{{(9x - 72)/((x-8)(x+8))}}}
Factor the 9 since it is a common factor to both 9 and 72
{{{9(x-8)/((x-8)(x+8))}}}
Cancel like factors both on top and bottom of a fraction
{{{(x-8)/(x-8) = 1}}}
Therefore leaving the solution
--> {{{9/(x+8)}}}