Question 326849
{{{x/(x+20)>2/(x+8)}}}
{{{x/(x+20)-2/(x+8)>0}}}
{{{(x/(x+20))((x+8)/(x+8))-(2/(x+8))((x+20)/(x+20))>0}}}
{{{(x(x+8)-2(x+20))/((x+8)(x+20))>0}}}
{{{(x^2+8x-2x-40)/((x+20)(x+8))>0}}}
{{{(x^2+6x-40)/((x+20)(x+8))>0}}}
{{{((x+10)(x-4))/((x+20)(x+8))>0}}}
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Break up the number line into 5 regions.
Region 1:({{{-infinity}}},{{{-20}}})
Region 2:({{{-20}}},{{{-10}}})
Region 3:({{{-10}}},{{{-8}}})
Region 4:({{{-8}}},{{{4.0}}})
Region 5:({{{4.0}}},{{{infinity}}})
Pick a point in each region (not an end point).
Test the inequality.
If the point satisifes the inequality, the region is part of the solution.
Move on to the next region.
Region 1: Let {{{x=-25}}},
{{{((x+10)(x-4))/((x+20)(x+8))>0}}}
{{{((-25+10)(-25-4))/((-25+20)(-25+8))>0}}}
{{{435/85>0}}}
True, this region is part of the solution.
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Region 2: Let {{{x=-15}}},
{{{((x+10)(x-4))/((x+20)(x+8))>0}}}
{{{((-5)(-19))/((5)(-7))>0}}}
{{{-95/35>0}}}
False, this region is not part of the solution.
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Region 3: Let {{{x=-9}}},
{{{((x+10)(x-4))/((x+20)(x+8))>0}}}
{{{((1.0)(-13.0))/((11.0)(-1.0))>0}}}
{{{13/11>0}}}
True, this region is part of the solution.
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Region 4: Let {{{x=0}}},
{{{((x+10)(x-4))/((x+20)(x+8))>0}}}
{{{((10)(-4))/((20)(8))>0}}}
{{{-40/160>0}}}
False, this region is not part of the solution.
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Region 5: Let {{{x=5}}},
{{{((x+10)(x-4))/((x+20)(x+8))>0}}}
{{{((15)(1))/((25)(13))>0}}}
{{{(15.0/325.0)>0}}}
True, this region is part of the solution.
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({{{-infinity}}},{{{-20}}}) U ({{{-10}}},{{{-8}}}) U ({{{4.0}}},{{{infinity}}})
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Graphically, the red line is the left hand side of the inequality, the green line is the right hand side of the inequality. Shown wih two different y scales.
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{{{drawing(300,300,-24,8,-15,15,grid(1),blue(line(-10,20,-10,-20)),blue(line(4,20,4,-20)),blue(line(-20,20,-20,-20)), blue(line(-8,20,-8,-20)),graph(300,300,-24,8,-15,15,x/(x+20) , 2/(x+8)))}}}
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{{{drawing(300,300,-24,8,-2,2,grid(1),blue(line(-10,20,-10,-20)),blue(line(4,20,4,-20)),blue(line(-20,20,-20,-20)), blue(line(-8,20,-8,-20)),graph(300,300,-24,8,-2,2,x/(x+20) , 2/(x+8)))}}}