Question 336875
{{{x^2+2x>=35}}}
{{{x^2+2x-35>=0}}}
{{{(x+7)(x-5)>=0}}}
Break up the number line into three regions.
Region 1: ({{{-infinity}}},{{{-7}}})
Region 2: ({{{-7}}},{{{7}}})
Region 3: ({{{5}}},{{{infinity}}})
Choose a point in each region (not an endpoint) and test the inequality.
If the inequality is satisfied, that region is part of the solution.
Region 1:{{{x=-8}}}
{{{(x+7)(x-5)>=0}}}
{{{(-8+7)(-8-5)>=0}}}
{{{-1(-13)>=0}}}
{{{13>=0}}}
True, this region is part of the solution region.
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Region 2:{{{x=0}}}
{{{(x+7)(x-5)>=0}}}
{{{(0+7)(0-5)>=0}}}
{{{7(-5)>=0}}}
{{{-35>=0}}}
False, this region is not part of the solution region.
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Region 3:{{{x=6}}}
{{{(x+7)(x-5)>=0}}}
{{{(6+7)(6-5)>=0}}}
{{{13(1)>=0}}}
{{{13>=0}}}
True, this region is part of the solution region.
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Solution Region:({{{-infinity}}},{{{-7}}}) U ({{{5}}},{{{infinity}}})
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Graphical verification:
{{{x^2+2x-35>=0}}}
Look for the region where the function is above the x-axis.
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{{{drawing(300,300,-10,10,-20,20,grid(1),blue(line(-7,-100,-7,100)),blue(line(5,-100,5,100)),graph(300,300,-10,10,-20,20,x^2+2x-35))}}}