Question 97394
{{{(7-8x)/((1-x)(2-x))}}} Start with the given expression



{{{(7-8x)/((1-x)(2-x))=A/(1-x)+B/(2-x)}}} Break up the fraction into two smaller fractions



{{{(1-x)(2-x)((7-8x)/((1-x)(2-x)))=(1-x)(2-x)(A/(1-x)+B/(2-x))}}} Multiply both sides by the LCD {{{(1-x)(2-x)}}}. This will eliminate any fractions. 




{{{cross((1-x)(2-x))((7-8x)/(cross((1-x)(2-x))))=A*cross((1-x))(2-x)/cross((1-x))+B(1-x)*cross((2-x))/cross((2-x))}}} Distribute. Notice the denominators cancel.



{{{7-8x=A(2-x)+B(1-x)}}} Simplify



Now to find A, simply eliminate B. So to eliminate B, let x=1




{{{7-8(1)=A(2-1)+B(1-1)}}} plug in x=1


{{{7-8=A(2-1)+B(1-1)}}} multiply



{{{-1=A(1)+B(0)}}} Combine like terms




{{{-1=A}}} Multiply 




So our first value is {{{A=-1}}}



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Now to find B, simply eliminate A. So to eliminate A, let x=2




{{{7-8(2)=A(2-2)+B(1-2)}}} plug in x=2



{{{7-16=A(2-2)+B(1-2)}}} multiply



{{{-9=A(0)+B(-1)}}} Combine like terms


{{{-9=-B}}} Multiply and combine like terms


{{{9=B}}} Divide both sides by -1



So our second value is {{{B=9}}}




So our values are {{{A=-1}}} and {{{B=9}}}




So that means {{{(7-8x)/((1-x)(2-x))=-1/(1-x)+9/(2-x)}}} where {{{x<>1}}} or {{{x<>2}}}




Check: 



To verify the answer, simply compare the graphs of the original expression and the answer. You'll see that they are equivalent, so that verifies your answer.