Question 96882
{{{(x+1)(2x-1)/(2x-3)(x-3)-(x-3)(x+1)/(3-x)(3-2x)+(2x+1)(x+3)/(3-2x)(x-3)}}} Start with the given expression



 



{{{(x+1)(2x-1)/(2x-3)(x-3)-(x-3)(x+1)/(x-3)(2x-3)+(2x+1)(x+3)/(-1(2x-3)(x-3))}}} Rewrite {{{3-x}}} as {{{-(x-3)}}} and {{{(3-2x)}}} as {{{-(2x-3)}}} (note: the 2nd fraction will have two negatives multiplying to a positive )




Since we are dealing with fractions, we need to find the LCD so we can combine them.


{{{-(x+1)(2x-1)/(-1(2x-3)(x-3))+(x-3)(x+1)/(-1(x-3)(2x-3))+(2x+1)(x+3)/(-1(2x-3)(x-3))}}} Now multiply the first two fractions by {{{-1/-1}}}. This will make every fraction have the same denominator.




{{{(-(x+1)(2x-1)+(x-3)(x+1)+(2x+1)(x+3))/(-1(2x-3)(x-3))}}} Now combine the fractions.



{{{(-(2x^2+x-1)+(x^2-2x-3)+(2x^2+7x+3))/(-1(2x-3)(x-3))}}} Foil the parenthesis in the numerator



{{{(-2x^2-x+1+x^2-2x-3+2x^2+7x+3)/(-1(2x-3)(x-3))}}} Distribute the negative



{{{(-2x^2-x+1-x^2+2x+3+2x^2+7x+3)/(-1(2x^2-9x+9))}}} Foil the terms in the denominator




{{{(-2x^2-x+1-x^2+2x+3+2x^2+7x+3)/(-2x^2+9x-9)}}} Distribute




{{{(x^2+4x+1)/(-2x^2+9x-9)}}} Combine like terms in the numerator




So {{{(x+1)(2x-1)/(2x-3)(x-3)-(x-3)(x+1)/(3-x)(3-2x)+(2x+1)(x+3)/(3-2x)(x-3)}}} simplifies to {{{(x^2+4x+1)/(-2x^2+9x-9)}}}



Check:


If you graph the original expression and the reduced expression on the same plot, they will line up perfectly. Since they line up, they are equivalent and this verifies our answer.