Question 365114
Break up the number line into 4 regions using the critical points of the function.
Region 1:({{{-infinity}}},{{{-3}}})
Region 2:({{{-3}}},{{{-2}}})
Region 3:({{{-2}}},{{{16}}})
Region 4:({{{16}}},{{{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=-4}}}
{{{(x+3)(x-16)(x+2)>0}}}
{{{(-4+3)(-4-16)(-4+2)>0}}}
{{{(-1)(-20)(-2)>0 }}}
{{{-40>0 }}}
False, Region 1 is not part of the solution region.
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Region 2:{{{x=-5/2}}}
{{{(x+3)(x-16)(x+2)>0}}}
{{{(-5/2+3)(-5/2-16)(-5/2+2)>0}}}
{{{(1)(-37/2)(-1/2)>0 }}}
{{{37/4>0 }}}
True, Region 2 is part of the solution region.
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Region 3:{{{x=0}}}
{{{(x+3)(x-16)(x+2)>0}}}
{{{(-4+3)(-4-16)(-4+2)>0}}}
{{{(3)(-16)(2)>0 }}}
{{{-96>0 }}}
False, Region 3 is not part of the solution region.
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Region 4:{{{x=17}}}
{{{(x+3)(x-16)(x+2)>0}}}
{{{(17+3)(17-16)(17+2)>0}}}
{{{(20)(1)(19)>0 }}}
{{{380>0 }}}
True, Region 4 is part of the solution region.
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Solution Region:({{{-3}}},{{{-2}}}) U ({{{16}}},{{{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,-4,17,-100,100,grid(1),blue(line(-3,500,-3,-500)),blue(line(-2,500,-2,-500)),blue(line(16,500,16,-500)),graph(300,300,-4,17,-100,100,0, (x+3)(x-16)(x+2) ))}}}