Question 150026
{{{(x-4)/(x+1)-(10)/(x^2-1)=0}}} Start with the given equation.



{{{(x-4)/(x+1)-(10)/((x-1)(x+1))=0}}} Factor {{{x^2-1}}} to get {{{(x-1)(x+1)}}}



Notice how the LCD is {{{(x-1)(x+1)}}}



{{{((x-1)(x+1))((x-4)/(x+1)-(10)/((x-1)(x+1)))=(x-1)(x+1)(0)}}} Multiply both sides by the LCD  {{{(x-1)(x+1)}}} to clear the fractions.



{{{((x-1)cross((x+1)))((x-4)/cross((x+1)))-(cross((x-1)(x+1)))((10)/(cross((x-1)(x+1))))=(x-1)(x+1)(0)}}} Distribute. Notice how the denominators cancel.



{{{(x-1)(x-4)-10=(x-1)(x+1)(0)}}} Simplify



{{{(x-1)(x-4)-10=0}}} Multiply the terms on the right side to get 0



{{{x^2-5x+4-10=0}}} FOIL



{{{x^2-5x-6=0}}} Combine like terms.



Let's use the quadratic formula to solve for x



{{{x = (-b +- sqrt( b^2-4ac ))/(2a)}}} Start with the quadratic formula



{{{x = (-(-5) +- sqrt( (-5)^2-4(1)(-6) ))/(2(1))}}} Plug in  {{{a=1}}}, {{{b=-5}}}, and {{{c=-6}}}



{{{x = (5 +- sqrt( (-5)^2-4(1)(-6) ))/(2(1))}}} Negate {{{-5}}} to get {{{5}}}. 



{{{x = (5 +- sqrt( 25-4(1)(-6) ))/(2(1))}}} Square {{{-5}}} to get {{{25}}}. 



{{{x = (5 +- sqrt( 25--24 ))/(2(1))}}} Multiply {{{4(1)(-6)}}} to get {{{-24}}}



{{{x = (5 +- sqrt( 25+24 ))/(2(1))}}} Rewrite {{{sqrt(25--24)}}} as {{{sqrt(25+24)}}}



{{{x = (5 +- sqrt( 49 ))/(2(1))}}} Add {{{25}}} to {{{24}}} to get {{{49}}}



{{{x = (5 +- sqrt( 49 ))/(2)}}} Multiply {{{2}}} and {{{1}}} to get {{{2}}}. 



{{{x = (5 +- 7)/(2)}}} Take the square root of {{{49}}} to get {{{7}}}. 



{{{x = (5 + 7)/(2)}}} or {{{x = (5 - 7)/(2)}}} Break up the expression. 



{{{x = (12)/(2)}}} or {{{x =  (-2)/(2)}}} Combine like terms. 



{{{x = 6}}} or {{{x = -1}}} Simplify. 



So the possible answers are {{{x = 6}}} or {{{x = -1}}}



However, we need to check them first.


Check:


Let's test the first solution {{{x = 6}}}


{{{(x-4)/(x+1)-(10)/(x^2-1)=0}}} Start with the given equation.



{{{(6-4)/(6+1)-(10)/(6^2-1)=0}}} Plug in {{{x = 6}}}.



{{{(6-4)/(6+1)-(10)/(36-1)=0}}} Square 6 to get 36



{{{(2)/(7)-(10)/(35)=0}}} Combine like terms.



{{{(10)/(35)-(10)/(35)=0}}} Multiply {{{2/7}}} by {{{5/5}}}



{{{(0)/(35)=0}}} Combine the fractions.



{{{0=0}}} Reduce. Since the equation is true, this means that {{{x = 6}}} is a solution.



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Let's test the second solution {{{x = -1}}}


{{{(x-4)/(x+1)-(10)/(x^2-1)=0}}} Start with the given equation.



{{{(1-4)/(1+1)-(10)/((-1)^2-1)=0}}} Plug in {{{x = -1}}}.



{{{(1-4)/(1+1)-(10)/(1-1)=0}}} Square -1 to get 1



{{{(-3)/(2)-(10)/(0)=0}}} Combine like terms.

Since division by zero is undefined, this means that {{{x = -1}}} is <font size=4><b>not</b></font> a solution.



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Answer:


So the solution to {{{(x-4)/(x+1)-(10)/(x^2-1)=0}}} is {{{x=6}}}