Question 189553
# 1


{{{6/(x^2-1)+3/(x+1)}}} Start with the given expression.



{{{6/((x+1)(x-1))+3/(x+1)}}} 



{{{6/((x+1)(x-1))+(3(x-1))/((x+1)(x-1))}}} Multiply both the numerator and denominator of the second fraction by {{{x-1}}}



{{{6/((x+1)(x-1))+(3x-3)/((x+1)(x-1))}}} Distribute



{{{(6+3x-3)/((x+1)(x-1))}}} Combine the fractions



{{{(3x+3)/((x+1)(x-1))}}} Combine like terms.



{{{(3(x+1))/((x+1)(x-1))}}} Factor



{{{(3cross((x+1)))/(cross((x+1))(x-1))}}} Cancel out the common terms.



{{{3/(x-1)}}} Simplify



So {{{6/(x^2-1)+3/(x+1)=3/(x-1)}}} where {{{x<>-1}}} or {{{x<>1}}}



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# 2


{{{(x+2)/(x^2+11x+18)+5/(x^2-3x-10)}}} Start with the given expression.



{{{(x+2)/((x+9)(x+2))+5/(x^2-3x-10)}}} Factor the first denominator



{{{(x+2)/((x+9)(x+2))+5/((x-5)(x+2))}}} Factor the second denominator



{{{((x+2)(x-5))/((x+9)(x-5)(x+2))+5/((x-5)(x+2))}}} Multiply both the numerator and denominator of the second fraction by {{{x-5}}}



{{{(x^2-3x-10)/((x+9)(x-5)(x+2))+5/((x-5)(x+2))}}} FOIL 



{{{(x^2-3x-10)/((x+9)(x-5)(x+2))+(5(x+9))/((x+9)(x-5)(x+2))}}} Multiply both the numerator and denominator of the second fraction by {{{x+9}}}



{{{(x^2-3x-10)/((x+9)(x-5)(x+2))+(5x+45)/((x+9)(x-5)(x+2))}}} FOIL



{{{(x^2-3x-10+5x+45)/((x+9)(x-5)(x+2))}}} Combine the fractions.



{{{(x^2+2x+35)/((x+9)(x-5)(x+2))}}} Combine like terms.



{{{(x^2+2x+35)/((x+9)(x^2-3x-10))}}} FOIL



{{{(x^2+2x+35)/(x^3+6x^2-37x-90)}}} Expand



So {{{(x+2)/(x^2+11x+18)+5/(x^2-3x-10)=(x^2+2x+35)/(x^3+6x^2-37x-90)}}} where {{{x<>-9}}}, {{{x<>-2}}}, or {{{x<>5}}}




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# 3


{{{1/(x-5)-2/(x+5)}}} Start with the given expression.



{{{(1(x+5))/((x-5)(x+5))-2/(x+5)}}} Multiply both the numerator and denominator of the first fraction by {{{x+5}}}



{{{(x+5)/((x-5)(x+5))-2/(x+5)}}} Distribute



{{{(x+5)/((x-5)(x+5))-(2(x-5))/((x-5)(x+5))}}} Multiply both the numerator and denominator of the second fraction by {{{x-5}}}



{{{(x+5)/((x-5)(x+5))-(2x-10)/((x-5)(x+5))}}} Distribute



{{{(x+5-(2x-10))/((x-5)(x+5))}}} Subtract the fractions



{{{(x+5-2x+10)/((x-5)(x+5))}}} Distribute



{{{(-x+15)/((x-5)(x+5))}}} Combine like terms.



{{{(-x+15)/(x^2-25)}}} FOIL



So {{{1/(x-5)-2/(x+5)=(-x+15)/(x^2-25)}}} where {{{x<>-5}}} or {{{x<>5}}}




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# 4

 
{{{x/2+8/3=1/6}}} Start with the given equation.



{{{cross(6)^3(x/cross(2))+cross(6)^2(8/cross(3))=cross(6)(1/cross(6))}}} Multiply both sides by the LCD {{{6}}} to clear any fractions.



{{{3x+2(8)=1}}} Simplify



{{{3x+16=1}}} Multiply.



{{{3x=1-16}}} Subtract {{{16}}} from both sides.



{{{3x=-15}}} Combine like terms on the right side.



{{{x=(-15)/(3)}}} Divide both sides by {{{3}}} to isolate {{{x}}}.



{{{x=-5}}} Reduce.



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Answer:


So the answer is {{{x=-5}}}




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# 5


I'm assuming that the equation is {{{6/x +3 = x + 8}}}???



{{{6/x +3 = x + 8}}} Start with the given equation.



{{{cross(x)(6/cross(x)) +3x = x*x + 8*x}}} Multiply EVERY term by the LCD "x"



{{{6+3x = x^2 + 8x}}} Multiply and simplify



{{{0 = x^2 + 8x-3x-6}}} Subtract 3x from both sides. Subtract 6 from both sides.



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



{{{0=(x+6)(x-1)}}} Factor 



{{{x+6=0}}} or {{{x-1=0}}} Set each factor equal to zero



{{{x=-6}}} or {{{x=1}}} Solve for "x" in each case



So the solutions are {{{x=-6}}} or {{{x=1}}}



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OR...


Is the equation {{{6/(x+3) = x + 8}}} ????



{{{6/(x+3) = x + 8}}} Start with the given equation.



{{{6 = (x+8)(x+3)}}} Multiply both sides by {{{x+3}}}.



{{{6 = x^2+11x+24}}} FOIL



{{{0 = x^2+11x+24-6}}} Subtract 6 from both sides.



{{{0 = x^2+11x+18}}} Combine like terms.



{{{0 = (x+9)(x+2)}}} Factor



{{{x+9=0}}} or {{{x+2=0}}} Set each factor equal to zero



{{{x=-9}}} or {{{x=-2}}} Solve for "x" in each case



So the solutions are {{{x=-9}}} or {{{x=-2}}}