Question 364997
Break up the number line into 4 regions using the critical points of the function.
Region 1:({{{-infinity}}},{{{-8}}})
Region 2:({{{-8}}},{{{-3}}})
Region 3:({{{-3}}},{{{3}}})
Region 4:({{{3}}},{{{infinity}}})
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For each region, choose a point in the region (not an endpoint).
Test the inequality.
If the inequality is satisfied, the region is part of the solution region.
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Region 1:{{{x=-9}}}
{{{(x+8)(x+3)(x-3)>0}}}
{{{(-9+8)(-9+3)(-9-3)>0}}}
{{{(-1)(-6)(-12)>0 }}}
{{{-72>0 }}}
False, Region 1 is not part of the solution region.
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Region 2{{{x=-4}}
{{{(x+8)(x+3)(x-3) >0}}}
{{{(-4+8)(-4+3)(-4-3)>0}}}
{{{(4)(-1)(-7)>0 }}}
{{{28>0 }}}
True, Region 2 is part of the solution region.
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Region 3:{{{x=0}}}
{{{(x+8)(x+3)(x-3)>0}}}
{{{(8)(3)(-3)>0}}}
{{{-72>0 }}}
False, Region 4 is not part of the solution region.
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Region 4:{{{x=4}}}
{{{(x+8)(x+3)(x-3)>0}}}
{{{(4+8)(4+3)(4-3)>0}}}
{{{(12)(7)(1)>0 }}}
{{{84>0 }}}
True, Region 4 is part of the solution region.
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Solution Region:({{{-8}}},{{{-3}}}) U ({{{3}}},{{{infinity}}})
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Graphical verification: Look for regions where the function is above the x-axis ({{{y>0}}})
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{{{drawing(300,300,-12,4,-120,120,grid(1),blue(line(-8,500,-8,-500)),blue(line(-3,500,-3,-500)),blue(line(3,500,3,-500)),graph(300,300,-12,4,-120,120,(x+8)(x+3)(x-3)))}}}