Question 90877
{{{x/(x-3)<=-8/(x-6)}}}



{{{cross((x-3))(x-6)(x/cross((x-3)))<=(x-3)cross((x-6))(-8/cross((x-6)))}}} Multiply both sides by the LCD



{{{x(x-6)<=-8(x-3)}}} Simplify



{{{x^2-6x<=-8x+24}}} Distribute


{{{x^2+2x-24<=0}}} Get everything to one side


{{{(x+6)(x-4)<=0}}} Factor the left side (note: if you need help with factoring, check out this <a href=http://www.algebra.com/algebra/homework/playground/change-this-name4450.solver>solver</a>)


So our zeros are {{{x=-6}}} and {{{x=4}}}


This means we can plug in values near {{{x=-6}}} and {{{x=4}}} to find out if they produce y values that are less than zero:




Now lets test everything from negative infinity to x=-6


Plug in {{{x=-7}}} (any point less than x=-6 will work)



{{{f(x)=x^2+2x-24}}} Start with the given polynomial



{{{f(-7)=(-7)^2+2(-7)-24}}} Plug in {{{x=-7}}}



{{{f(-7)=(49)+2(-7)-24}}} Raise -7 to the second power to get 49



{{{f(-7)=(49)+-14-24}}} Multiply 2 by -7 to get -14



{{{f(-7)=11}}} Now combine like terms



So x=-7 produces a y value greater than zero


So everything less than x=-6 will produce a y value greater than zero. So lets ignore this range


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Now lets test everything from x=4 to infinity



Lets evaluate {{{f(5)}}} (any point greater than x=4 will work)


Plug in {{{x=5}}}



{{{f(x)=x^2+2x-24}}} Start with the given polynomial



{{{f(5)=(5)^2+2(5)-24}}} Plug in {{{x=5}}}



{{{f(5)=(25)+2(5)-24}}} Raise 5 to the second power to get 25



{{{f(5)=(25)+10-24}}} Multiply 2 by 5 to get 10



{{{f(5)=11}}} Now combine like terms



So everything greater than x=4 will produce a y value greater than zero. So lets ignore this range.


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Now lets test everything from x=-6 to x=4



Plug in {{{x=-5}}} (any point in between x=-6 and x=4 will work)


Lets evaluate {{{f(-5)}}}


{{{f(x)=x^2+2x-24}}} Start with the given polynomial



{{{f(-5)=(-5)^2+2(-5)-24}}} Plug in {{{x=-5}}}



{{{f(-5)=(25)+2(-5)-24}}} Raise -5 to the second power to get 25



{{{f(-5)=(25)+-10-24}}} Multiply 2 by -5 to get -10



{{{f(-5)=-9}}} Now combine like terms


So everything from x=-6 to x=4 will produce a y value less than zero



So the solution set is 


{{{-6<=x<=4}}}



which looks like this in interval notation


*[Tex \LARGE \left[-6,4\right]]




If we graph  {{{y=x^2+2x-24}}}, we can see the range that is below y=0


{{{ graph( 500, 500, -10, 10, -10, 10, x^2+2x-24) }}}