Question 318664
{{{f(x)= (x^2+x-6)/(x^2+8x+15)}}}
{{{f(x)= ((x+3)(x-2))/((x+3)(x+5))}}}
{{{f(x)= (x-2)/(x+5)}}}
Break up the number line into three parts.
Region 1: ({{{-infinity}}},{{{-5}}})
Region 2: ({{{-5}}},{{{2}}})
Region 3: ({{{2}}},{{{infinity}}})
Pick a point in each region (not at the endpoints) and test the function.
If it solves the equation, it's in the solution region.
Region 1: {{{x=-10}}}
{{{(x-2)/(x+5)>0}}}
{{{(-10-2)/(-10+5)>0}}}
{{{(-12)/(-5)>0}}}
{{{12/5>0}}}
True, Region 1 is part of the solution.
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Region 2: {{{x=0}}}
{{{(x-2)/(x+5)>0}}}
{{{(-2)/(5)>0}}}
{{{(-2)/(5)>0}}}
False, Region 2 is not part of the solution.
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Region 3: {{{x=5}}}
{{{(x-2)/(x+5)>0}}}
{{{(5-2)/(5+5)>0}}}
{{{(3)/(10)>0}}}
True, Region 3 is part of the solution.
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({{{-infinity}}},{{{-5}}}) U ({{{2}}},{{{infinity}}})
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Graphical verification of the results.
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{{{drawing(300,300,-10,10,-10,10,grid(1),blue(line(-5,-10,-5,10)),blue(line(2,-10,2,10)), graph(300,300,-10,10,-10,10, (x-2)/(x+5)))}}}