Question 356425
{{{x^2+10x+24>=0}}}
{{{(x+6)(x+4)>=0}}}
Break up the number line into 3 regions using the critical points of the function.
Region 1:({{{-infinity}}},{{{-6}}})
Region 2:({{{-6}}},{{{-4}}})
Region 3:({{{-4}}},{{{infinity}}})
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For each region, choose a point in the region (not an endpoint).
Test the inequality.
If the ineqaulity is satisfied, the region is part of the solution region.
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Region 1:{{{x=-7}}}
{{{(x+6)(x+4)>=0}}}
{{{(-7+6)(-7+4)>0 }}}
{{{(-1)(-3)>0 }}}
{{{3>0 }}}
True, Region 1 is part of the solution region.
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Region 2:{{{x=-5}}}
{{{(x+6)(x+4)>=0}}}
{{{(-5+6)(-5+4)>0 }}}
{{{(1)(-1)>0 }}}
{{{-1>0 }}}
False, Region 2 is not part of the solution region.
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Region 3:{{{x=0}}}
{{{(0+6)(0+4)>=0}}}
{{{24>0 }}}
True, Region 3 is part of the solution region.
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Solution Region: ({{{-infinity}}},{{{-6}}}) U ({{{-4}}},{{{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,-8,2,-4,10,grid(1),blue(line(-6,500,-6,-500)),blue(line(-4,500,-4,-500)),blue(line(-4,500,-4,-500)),graph(300,300,-8,2,-4,10,x^2+10x+24))}}}