Question 177567
Wow, there are a lot of problems. I'll do the first three to get you started.


# 1


{{{((12)/(x-1))/(6/x)}}} Start with the given expression



{{{(x*cross(x-1)((12)/cross(x-1)))/(cross(x)(x-1)(6/cross(x)))}}} Multiply both numerator and denominator by the inner LCD {{{x(x-1)}}} to clear out the inner fractions.


{{{(12x)/(6(x-1))}}} Multiply and simplify.



{{{(2x)/(x-1)}}} Reduce



So {{{((12)/(x-1))/(6/x)=(2x)/(x-1)}}} where {{{x<>1}}} or {{{x<>0}}}



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# 2



{{{((k+1)/(2k))/((3k-1)/(4k))}}} Start with the given expression



{{{(4cross(k)((k+1)/(cross(2k))))/(cross(4k)((3k-1)/cross(4k)))}}} Multiply both numerator and denominator by the inner LCD {{{4k}}} to clear out the inner fractions.


{{{(2(k+1))/(3k-1)}}} Multiply and simplify.



{{{(2k+2)/(3k-1)}}} Distribute



So {{{((k+1)/(2k))/((3k-1)/(4k))=(2k+2)/(3k-1)}}} where {{{x<>0}}} or {{{x<>1/3}}}



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# 3



{{{(3/x+3/y)/(3/x-3/y)}}} Start with the given expression



{{{(cross(x)y(3/cross(x))+x*cross(y)(3/cross(y)))/(cross(x)y(3/cross(x))-x*cross(y)(3/cross(y)))}}} Multiply EVERY term by the inner LCD {{{xy}}} to clear out the inner fractions.


{{{(3y+3x)/(3y-3x)}}} Multiply and simplify.



{{{(3(x+y))/(3(x-y))}}} Factor out the GCF 3 from the numerator and denominator and rearrange the terms.



{{{(x+y)/(x-y)}}} Reduce


So {{{(3/x+3/y)/(3/x-3/y)=(x+y)/(x-y)}}} where {{{x<>0}}}, {{{y<>0}}}, or {{{x<>y}}}



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