Question 315558
{{{(v-7)/(v-9) - (v+1)/(v+9) + (v-45)/(v^2-81)}}} Start with the given expression.



{{{(v-7)/(v-9) - (v+1)/(v+9) + (v-45)/((v-9)(v+9))}}} Factor the last denominator.



Take note that the LCD is {{{(v-9)(v+9)}}}



{{{((v-7)/(v-9))((v+9)/(v+9)) - (v+1)/(v+9) + (v-45)/((v-9)(v+9))}}} Multiply the first fraction by {{{(v+9)/(v+9)}}} to get that denominator to equal the LCD.



{{{((v-7)(v+9))/((v-9)(v+9)) - (v+1)/(v+9) + (v-45)/((v-9)(v+9))}}} Combine the fractions.



{{{(v^2+2v-63)/((v-9)(v+9)) - (v+1)/(v+9) + (v-45)/((v-9)(v+9))}}} FOIL the numerator of the first fraction.



{{{(v^2+2v-63)/((v-9)(v+9)) - ((v+1)/(v+9))((v-9)/(v-9)) + (v-45)/((v-9)(v+9))}}} Multiply the second fraction by {{{(v-9)/(v-9)}}} to get that denominator to equal the LCD.



{{{(v^2+2v-63)/((v-9)(v+9)) - ((v+1)(v-9))/((v+9)(v-9)) + (v-45)/((v-9)(v+9))}}} Combine the fractions.



{{{(v^2+2v-63)/((v-9)(v+9)) - (v^2-8v-9)/((v+9)(v-9)) + (v-45)/((v-9)(v+9))}}} FOIL the numerator of the second fraction.



Now the denominators are all equal to the LCD, we can finally combine the three fractions.



{{{((v^2+2v-63)- (v^2-8v-9)+ (v-45))/((v-9)(v+9))}}} Combine the three fractions by placing the numerators all over the LCD.



{{{(v^2+2v-63- v^2+8v+9+v-45)/((v-9)(v+9))}}} Distribute.



{{{(11v-99)/((v-9)(v+9))}}} Combine like terms.



{{{(11(v-9))/((v-9)(v+9))}}} Factor the numerator.



{{{(11*highlight((v-9)))/(highlight((v-9))(v+9))}}} Highlight the common terms.



{{{(11*cross((v-9)))/(cross((v-9))(v+9))}}} Cancel out the common terms.



{{{11/(v+9)}}} Simplify.



So {{{(v-7)/(v-9) - (v+1)/(v+9) + (v-45)/(v^2-81)}}} simplifies to {{{11/(v+9)}}}



In other words, {{{(v-7)/(v-9) - (v+1)/(v+9) + (v-45)/(v^2-81)=11/(v+9)}}} where {{{v<>-9}}} or {{{v<>9}}}