Question 343829
Break the number line up into 4 regions.
Region 1: ({{{-infinity}}},{{{5}}})
Region 2: ({{{5}}},{{{6}}})
Region 3: ({{{6}}},{{{7}}})
Region 4: ({{{7}}},{{{infinity}}})
Choose a point in each region (not an endpoint).
Test the inequality.
If the inequality is satisfied, that region is part of the solution.
.
.
.
Region 1:{{{x=0}}}
{{{(x-5)(x-6)(x-7)>=0}}}
{{{(-5)(-6)(-7)>=0}}}
{{{-210>=0}}}
False, this region is not part of the solution region.
.
.
.
Region 2:{{{x=5.5}}}
{{{(x-5)(x-6)(x-7)>=0}}}
{{{(5.5-5)(5.5-6)(5.5-7)>=0}}}
{{{0.5(-0.5)(-1.5)>=0}}}
{{{0.375>=0}}}
True, this region is part of the solution region.
.
.
.
Region 3:{{{x=6.5}}}
{{{(x-5)(x-6)(x-7)>=0}}}
{{{(6.5-5)(6.5-6)(6.5-7)>=0}}}
{{{1.5(0.5)(-0.5)>=0}}}
{{{-0.375>=0}}}
False, this region is not part of the solution region.
.
.
.
Region 4:{{{x=10}}}
{{{(x-5)(x-6)(x-7)>=0}}}
{{{(10-5)(10-6)(10-7)>=0}}}
{{{5(4)(3)?=0}}}
{{{60>=0}}}
True, this region is part of the solution region.
.
.
.
Solution:({{{5}}},{{{6}}}) U ({{{7}}},{{{infinity}}})
.
.
.
You can verify it graphically by plotting the function and checking the regions where the function is greater than zero (above the x-axis).
.
.
.
{{{drawing(300,300,-2,10,-6,6,grid(1),blue(line(5,-10,5,10)),blue(line(6,10,6,-10)),blue(line(7,10,7,-10)),graph(300,300,-2,10,-6,6,(x-5)(x-6)(x-7)))}}}