Question 378101
Break up the number line into 4 regions using the critical points of the function.
Region 1:({{{-infinity}}},{{{-3}}})
Region 2:({{{-3}}},{{{0}}})
Region 3:({{{0}}},{{{4}}})
Region 4:({{{4}}},{{{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 - 4)^3) / (x(x + 3))<=0 }}}
{{{(-512)/((-4)(-1))<0 }}}
{{{-128<0 }}}
True, Region 1 is part of the solution region.
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Region 2:{{{x=-2}}}
{{{((x - 4)^3) / (x(x + 3))<=0 }}}
{{{(-216)/((-2)(1))<0 }}}
{{{108<0 }}}
False, Region 2 is not part of the solution region.
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Region 3:{{{x=1}}}
{{{((x - 4)^3) / (x(x + 3))<=0 }}}
{{{(-27)/((1)(4))<0 }}}
{{{-27/4<0 }}}
True, Region 3 is part of the solution region.
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Region 4:{{{x=5}}}
{{{((x - 4)^3) / (x(x + 3)) <=0 }}}
{{{(5)/((5)(8))<0 }}}
{{{1/8<0 }}}
False, Region 4 is not part of the solution region.
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Solution Region: ({{{-infinity}}},{{{-3}}})U({{{0}}},{{{4}}})
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Graphical verification: Look for regions where the function is below the x-axis ({{{y<0}}})
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{{{drawing(300,300,-8,8,-100,100,grid(1),blue(line(-3,500,-3,-500)),blue(line(4,500,4,-500)),graph(300,300,-8,8,-100,100,0,(x-4)^3/x/(x+3) ))}}}