Question 85018

Start with the given system of inequalities

{{{3x+4y<=12}}}


{{{x+3y<=6}}}


{{{x>=0}}}


{{{y>=0}}}




In order to graph this system of inequalities, we need to graph each inequality one at a time.


So lets graph the first inequality


    In order to graph {{{3x+4y<=12}}} we need to graph the equation {{{3x+4y=12}}} (just replace the inequality sign with an equal sign). So lets graph the line {{{3x+4y=12}}} (note: if you need help with graphing, check out this <a href=http://www.algebra.com/algebra/homework/Linear-equations/graphing-linear-equations.solver>solver<a>)

    {{{graph( 400, 300, -10, 10, -10, 10,(12-3*x)/4)}}} graph of {{{3x+4y=12}}}

    Now lets pick a test point, say (0,0) (any point will work, but this point is the easiest to work with), and evaluate the inequality {{{3x+4y<=12}}}

     {{{3(0)+4(0)<=12}}} Plug in x=0, y=0


     {{{0<=10}}} Simplify



Since this inequality is true, we shade the entire region containing (0,0)



{{{drawing( 400, 300, -10, 10, -10, 10,graph( 400, 300, -10, 10, -10, 10,(12-3*x)/4, (12-3*x)/4-0),
graph( 400, 300, -10, 10, -10, 10,(12-3*x)/4, (12-3*x)/4-0.75),
graph( 400, 300, -10, 10, -10, 10,(12-3*x)/4, (12-3*x)/4-1.5),
graph( 400, 300, -10, 10, -10, 10,(12-3*x)/4, (12-3*x)/4-2.25),
graph( 400, 300, -10, 10, -10, 10,(12-3*x)/4, (12-3*x)/4-3),
graph( 400, 300, -10, 10, -10, 10,(12-3*x)/4, (12-3*x)/4-3.75),
graph( 400, 300, -10, 10, -10, 10,(12-3*x)/4, (12-3*x)/4-4.5),
graph( 400, 300, -10, 10, -10, 10,(12-3*x)/4, (12-3*x)/4-5.25),
graph( 400, 300, -10, 10, -10, 10,(12-3*x)/4, (12-3*x)/4-6),
graph( 400, 300, -10, 10, -10, 10,(12-3*x)/4, (12-3*x)/4-6.75),
graph( 400, 300, -10, 10, -10, 10,(12-3*x)/4, (12-3*x)/4-7.5),
graph( 400, 300, -10, 10, -10, 10,(12-3*x)/4, (12-3*x)/4-8.25),
graph( 400, 300, -10, 10, -10, 10,(12-3*x)/4, (12-3*x)/4-9),
graph( 400, 300, -10, 10, -10, 10,(12-3*x)/4, (12-3*x)/4-9.75))}}}

 Here is the graph of {{{3x+4y<=12}}} with the graph of the line({{{3x+4y=12}}}) in red and the shaded region in green

(note: In this case, the red line is a solid line. This means the boundaries are included in the region.)





Now lets graph the second inequality


    In order to graph {{{x+3y<=6}}} we need to graph the equation {{{x+3y=6}}} (just replace the inequality sign with an equal sign). So lets graph the line {{{x+3y=6}}} (note: if you need help with graphing, check out this <a href=http://www.algebra.com/algebra/homework/Linear-equations/graphing-linear-equations.solver>solver<a>)

    {{{graph( 400, 300, -10, 10, -10, 10,(6-1*x)/3)}}} graph of {{{x+3y=6}}}

    Now lets pick a test point, say (0,0) (any point will work, but this point is the easiest to work with), and evaluate the inequality {{{x+3y<=6}}}

     {{{(0)+3(0)<=6}}} Plug in x=0, y=0


     {{{0<=10}}} Simplify



Since this inequality is true, we shade the entire region containing (0,0)



{{{drawing( 400, 300, -10, 10, -10, 10,graph( 400, 300, -10, 10, -10, 10,(6-1*x)/3, (6-1*x)/3-0),
graph( 400, 300, -10, 10, -10, 10,(6-1*x)/3, (6-1*x)/3-0.75),
graph( 400, 300, -10, 10, -10, 10,(6-1*x)/3, (6-1*x)/3-1.5),
graph( 400, 300, -10, 10, -10, 10,(6-1*x)/3, (6-1*x)/3-2.25),
graph( 400, 300, -10, 10, -10, 10,(6-1*x)/3, (6-1*x)/3-3),
graph( 400, 300, -10, 10, -10, 10,(6-1*x)/3, (6-1*x)/3-3.75),
graph( 400, 300, -10, 10, -10, 10,(6-1*x)/3, (6-1*x)/3-4.5),
graph( 400, 300, -10, 10, -10, 10,(6-1*x)/3, (6-1*x)/3-5.25),
graph( 400, 300, -10, 10, -10, 10,(6-1*x)/3, (6-1*x)/3-6),
graph( 400, 300, -10, 10, -10, 10,(6-1*x)/3, (6-1*x)/3-6.75),
graph( 400, 300, -10, 10, -10, 10,(6-1*x)/3, (6-1*x)/3-7.5),
graph( 400, 300, -10, 10, -10, 10,(6-1*x)/3, (6-1*x)/3-8.25),
graph( 400, 300, -10, 10, -10, 10,(6-1*x)/3, (6-1*x)/3-9),
graph( 400, 300, -10, 10, -10, 10,(6-1*x)/3, (6-1*x)/3-9.75))}}}

 Here is the graph of {{{x+3y<=6}}} with the graph of the line({{{x+3y=6}}}) in red and the shaded region in green

(note: In this case, the red line is a solid line. This means the boundaries are included in the region.)





Now lets graph the third inequality


    In order to graph {{{x>=0}}} we need to graph the equation {{{x=0}}} (just replace the inequality sign with an equal sign). So lets graph the line {{{x=0}}} (note: if you need help with graphing, check out this <a href=http://www.algebra.com/algebra/homework/Linear-equations/graphing-linear-equations.solver>solver<a>)

    {{{graph( 400, 300, -10, 10, -10, 10,1000(x-10/1000))}}} graph of {{{x=0}}}

    Now lets pick a test point, say (1,0) (any point will work, but this point is the easiest to work with), and evaluate the inequality {{{x>=0}}}

     {{{1>=0}}} Plug in x=1


     {{{1>=0}}} Simplify



Since this inequality is  true, we shade the entire region that contains (1,0)



{{{drawing( 400, 300, -10, 10, -10, 10,graph( 400, 300, -10, 10, -10, 10,10000(x-10/1000), 10000(x-10/1000-0)),
graph( 400, 300, -10, 10, -10, 10,10000(x-10/1000), 10000(x-10/1000-0.75)),
graph( 400, 300, -10, 10, -10, 10,10000(x-10/1000), 10000(x-10/1000-1.5)),
graph( 400, 300, -10, 10, -10, 10,10000(x-10/1000), 10000(x-10/1000-2.25)),
graph( 400, 300, -10, 10, -10, 10,10000(x-10/1000), 10000(x-10/1000-3)),
graph( 400, 300, -10, 10, -10, 10,10000(x-10/1000), 10000(x-10/1000-3.75)),
graph( 400, 300, -10, 10, -10, 10,10000(x-10/1000), 10000(x-10/1000-4.5)),
graph( 400, 300, -10, 10, -10, 10,10000(x-10/1000), 10000(x-10/1000-5.25)),
graph( 400, 300, -10, 10, -10, 10,10000(x-10/1000), 10000(x-10/1000-6)),
graph( 400, 300, -10, 10, -10, 10,10000(x-10/1000), 10000(x-10/1000-6.75)),
graph( 400, 300, -10, 10, -10, 10,10000(x-10/1000), 10000(x-10/1000-7.5)),
graph( 400, 300, -10, 10, -10, 10,10000(x-10/1000), 10000(x-10/1000-8.25)),
graph( 400, 300, -10, 10, -10, 10,10000(x-10/1000), 10000(x-10/1000-9)),
graph( 400, 300, -10, 10, -10, 10,10000(x-10/1000), 10000(x-10/1000-9.75)))}}}

 Here is the graph of {{{x>=0}}} with the graph of the line({{{x=0}}}) in red and the shaded region in green

(note: In this case, the red line is a solid line. This means the boundaries are included in the region.)





Now lets graph the fourth inequality


    In order to graph {{{y>=0}}} we need to graph the equation {{{y=0}}} (just replace the inequality sign with an equal sign). So lets graph the line {{{y=0}}} (note: if you need help with graphing, check out this <a href=http://www.algebra.com/algebra/homework/Linear-equations/graphing-linear-equations.solver>solver<a>)

    {{{graph( 400, 300, -10, 10, -10, 10,(10-1*x)/1000)}}} graph of {{{y=0}}}

    Now lets pick a test point, say (0,1) (any point will work, but this point is the easiest to work with), and evaluate the inequality {{{y>=0}}}

     {{{1>=0}}} Plug in y=1


     {{{1>=0}}} Simplify



Since this inequality is true, we shade the entire region that contains (0,1)



{{{drawing( 400, 300, -10, 10, -10, 10,graph( 400, 300, -10, 10, -10, 10,(10-1*x)/1000, (10-1*x)/1000+0),
graph( 400, 300, -10, 10, -10, 10,(10-1*x)/1000, (10-1*x)/1000+0.75),
graph( 400, 300, -10, 10, -10, 10,(10-1*x)/1000, (10-1*x)/1000+1.5),
graph( 400, 300, -10, 10, -10, 10,(10-1*x)/1000, (10-1*x)/1000+2.25),
graph( 400, 300, -10, 10, -10, 10,(10-1*x)/1000, (10-1*x)/1000+3),
graph( 400, 300, -10, 10, -10, 10,(10-1*x)/1000, (10-1*x)/1000+3.75),
graph( 400, 300, -10, 10, -10, 10,(10-1*x)/1000, (10-1*x)/1000+4.5),
graph( 400, 300, -10, 10, -10, 10,(10-1*x)/1000, (10-1*x)/1000+5.25),
graph( 400, 300, -10, 10, -10, 10,(10-1*x)/1000, (10-1*x)/1000+6),
graph( 400, 300, -10, 10, -10, 10,(10-1*x)/1000, (10-1*x)/1000+6.75),
graph( 400, 300, -10, 10, -10, 10,(10-1*x)/1000, (10-1*x)/1000+7.5),
graph( 400, 300, -10, 10, -10, 10,(10-1*x)/1000, (10-1*x)/1000+8.25),
graph( 400, 300, -10, 10, -10, 10,(10-1*x)/1000, (10-1*x)/1000+9),
graph( 400, 300, -10, 10, -10, 10,(10-1*x)/1000, (10-1*x)/1000+9.75))}}}

 Here is the graph of {{{y>=0}}} with the graph of the line({{{y=0}}}) in red and the shaded region in green

(note: In this case, the red line is a solid line. This means the boundaries are included in the region.)

So we essentially have these 4 regions

{{{drawing( 400, 300, -10, 10, -10, 10,graph( 400, 300, -10, 10, -10, 10,(12-3*x)/4, (12-3*x)/4-0),
graph( 400, 300, -10, 10, -10, 10,(12-3*x)/4, (12-3*x)/4-0.75),
graph( 400, 300, -10, 10, -10, 10,(12-3*x)/4, (12-3*x)/4-1.5),
graph( 400, 300, -10, 10, -10, 10,(12-3*x)/4, (12-3*x)/4-2.25),
graph( 400, 300, -10, 10, -10, 10,(12-3*x)/4, (12-3*x)/4-3),
graph( 400, 300, -10, 10, -10, 10,(12-3*x)/4, (12-3*x)/4-3.75),
graph( 400, 300, -10, 10, -10, 10,(12-3*x)/4, (12-3*x)/4-4.5),
graph( 400, 300, -10, 10, -10, 10,(12-3*x)/4, (12-3*x)/4-5.25),
graph( 400, 300, -10, 10, -10, 10,(12-3*x)/4, (12-3*x)/4-6),
graph( 400, 300, -10, 10, -10, 10,(12-3*x)/4, (12-3*x)/4-6.75),
graph( 400, 300, -10, 10, -10, 10,(12-3*x)/4, (12-3*x)/4-7.5),
graph( 400, 300, -10, 10, -10, 10,(12-3*x)/4, (12-3*x)/4-8.25),
graph( 400, 300, -10, 10, -10, 10,(12-3*x)/4, (12-3*x)/4-9),
graph( 400, 300, -10, 10, -10, 10,(12-3*x)/4, (12-3*x)/4-9.75))}}} Region #1 which is the graph of {{{3x+4y<=12}}}

{{{drawing( 400, 300, -10, 10, -10, 10,graph( 400, 300, -10, 10, -10, 10,(6-1*x)/3, (6-1*x)/3-0),
graph( 400, 300, -10, 10, -10, 10,(6-1*x)/3, (6-1*x)/3-0.75),
graph( 400, 300, -10, 10, -10, 10,(6-1*x)/3, (6-1*x)/3-1.5),
graph( 400, 300, -10, 10, -10, 10,(6-1*x)/3, (6-1*x)/3-2.25),
graph( 400, 300, -10, 10, -10, 10,(6-1*x)/3, (6-1*x)/3-3),
graph( 400, 300, -10, 10, -10, 10,(6-1*x)/3, (6-1*x)/3-3.75),
graph( 400, 300, -10, 10, -10, 10,(6-1*x)/3, (6-1*x)/3-4.5),
graph( 400, 300, -10, 10, -10, 10,(6-1*x)/3, (6-1*x)/3-5.25),
graph( 400, 300, -10, 10, -10, 10,(6-1*x)/3, (6-1*x)/3-6),
graph( 400, 300, -10, 10, -10, 10,(6-1*x)/3, (6-1*x)/3-6.75),
graph( 400, 300, -10, 10, -10, 10,(6-1*x)/3, (6-1*x)/3-7.5),
graph( 400, 300, -10, 10, -10, 10,(6-1*x)/3, (6-1*x)/3-8.25),
graph( 400, 300, -10, 10, -10, 10,(6-1*x)/3, (6-1*x)/3-9),
graph( 400, 300, -10, 10, -10, 10,(6-1*x)/3, (6-1*x)/3-9.75))}}} Region #2 which is the graph of {{{x+3y<=6}}}

{{{drawing( 400, 300, -10, 10, -10, 10,graph( 400, 300, -10, 10, -10, 10,10000(x-10/1000), 10000(x-10/1000-0)),
graph( 400, 300, -10, 10, -10, 10,10000(x-10/1000), 10000(x-10/1000-0.75)),
graph( 400, 300, -10, 10, -10, 10,10000(x-10/1000), 10000(x-10/1000-1.5)),
graph( 400, 300, -10, 10, -10, 10,10000(x-10/1000), 10000(x-10/1000-2.25)),
graph( 400, 300, -10, 10, -10, 10,10000(x-10/1000), 10000(x-10/1000-3)),
graph( 400, 300, -10, 10, -10, 10,10000(x-10/1000), 10000(x-10/1000-3.75)),
graph( 400, 300, -10, 10, -10, 10,10000(x-10/1000), 10000(x-10/1000-4.5)),
graph( 400, 300, -10, 10, -10, 10,10000(x-10/1000), 10000(x-10/1000-5.25)),
graph( 400, 300, -10, 10, -10, 10,10000(x-10/1000), 10000(x-10/1000-6)),
graph( 400, 300, -10, 10, -10, 10,10000(x-10/1000), 10000(x-10/1000-6.75)),
graph( 400, 300, -10, 10, -10, 10,10000(x-10/1000), 10000(x-10/1000-7.5)),
graph( 400, 300, -10, 10, -10, 10,10000(x-10/1000), 10000(x-10/1000-8.25)),
graph( 400, 300, -10, 10, -10, 10,10000(x-10/1000), 10000(x-10/1000-9)),
graph( 400, 300, -10, 10, -10, 10,10000(x-10/1000), 10000(x-10/1000-9.75)))}}} Region #3 which is the graph of {{{x>=0}}}

{{{drawing( 400, 300, -10, 10, -10, 10,graph( 400, 300, -10, 10, -10, 10,(10-1*x)/1000, (10-1*x)/1000+0),
graph( 400, 300, -10, 10, -10, 10,(10-1*x)/1000, (10-1*x)/1000+0.75),
graph( 400, 300, -10, 10, -10, 10,(10-1*x)/1000, (10-1*x)/1000+1.5),
graph( 400, 300, -10, 10, -10, 10,(10-1*x)/1000, (10-1*x)/1000+2.25),
graph( 400, 300, -10, 10, -10, 10,(10-1*x)/1000, (10-1*x)/1000+3),
graph( 400, 300, -10, 10, -10, 10,(10-1*x)/1000, (10-1*x)/1000+3.75),
graph( 400, 300, -10, 10, -10, 10,(10-1*x)/1000, (10-1*x)/1000+4.5),
graph( 400, 300, -10, 10, -10, 10,(10-1*x)/1000, (10-1*x)/1000+5.25),
graph( 400, 300, -10, 10, -10, 10,(10-1*x)/1000, (10-1*x)/1000+6),
graph( 400, 300, -10, 10, -10, 10,(10-1*x)/1000, (10-1*x)/1000+6.75),
graph( 400, 300, -10, 10, -10, 10,(10-1*x)/1000, (10-1*x)/1000+7.5),
graph( 400, 300, -10, 10, -10, 10,(10-1*x)/1000, (10-1*x)/1000+8.25),
graph( 400, 300, -10, 10, -10, 10,(10-1*x)/1000, (10-1*x)/1000+9),
graph( 400, 300, -10, 10, -10, 10,(10-1*x)/1000, (10-1*x)/1000+9.75))}}} Region #4 which is the graph of {{{y>=0}}}



So these regions overlap to produce this region. It's a little hard to see, but after evenly shading each region, the intersecting region will be the most shaded in.

{{{

drawing( 400, 300, -10, 10, -10, 10, -10, 10,
graph( 400, 300, -10, 10, -10, 10,(12-3*x)/4, (12-3*x)/4-0),
graph( 400, 300, -10, 10, -10, 10,(12-3*x)/4, (12-3*x)/4-0.75),
graph( 400, 300, -10, 10, -10, 10,(12-3*x)/4, (12-3*x)/4-1.5),
graph( 400, 300, -10, 10, -10, 10,(12-3*x)/4, (12-3*x)/4-2.25),
graph( 400, 300, -10, 10, -10, 10,(12-3*x)/4, (12-3*x)/4-3),
graph( 400, 300, -10, 10, -10, 10,(12-3*x)/4, (12-3*x)/4-3.75),
graph( 400, 300, -10, 10, -10, 10,(12-3*x)/4, (12-3*x)/4-4.5),
graph( 400, 300, -10, 10, -10, 10,(12-3*x)/4, (12-3*x)/4-5.25),
graph( 400, 300, -10, 10, -10, 10,(12-3*x)/4, (12-3*x)/4-6),
graph( 400, 300, -10, 10, -10, 10,(12-3*x)/4, (12-3*x)/4-6.75),
graph( 400, 300, -10, 10, -10, 10,(12-3*x)/4, (12-3*x)/4-7.5),
graph( 400, 300, -10, 10, -10, 10,(12-3*x)/4, (12-3*x)/4-8.25),
graph( 400, 300, -10, 10, -10, 10,(12-3*x)/4, (12-3*x)/4-9),
graph( 400, 300, -10, 10, -10, 10,(12-3*x)/4, (12-3*x)/4-9.75),
graph( 400, 300, -10, 10, -10, 10,(6-1*x)/3, (6-1*x)/3-0),
graph( 400, 300, -10, 10, -10, 10,(6-1*x)/3, (6-1*x)/3-0.75),
graph( 400, 300, -10, 10, -10, 10,(6-1*x)/3, (6-1*x)/3-1.5),
graph( 400, 300, -10, 10, -10, 10,(6-1*x)/3, (6-1*x)/3-2.25),
graph( 400, 300, -10, 10, -10, 10,(6-1*x)/3, (6-1*x)/3-3),
graph( 400, 300, -10, 10, -10, 10,(6-1*x)/3, (6-1*x)/3-3.75),
graph( 400, 300, -10, 10, -10, 10,(6-1*x)/3, (6-1*x)/3-4.5),
graph( 400, 300, -10, 10, -10, 10,(6-1*x)/3, (6-1*x)/3-5.25),
graph( 400, 300, -10, 10, -10, 10,(6-1*x)/3, (6-1*x)/3-6),
graph( 400, 300, -10, 10, -10, 10,(6-1*x)/3, (6-1*x)/3-6.75),
graph( 400, 300, -10, 10, -10, 10,(6-1*x)/3, (6-1*x)/3-7.5),
graph( 400, 300, -10, 10, -10, 10,(6-1*x)/3, (6-1*x)/3-8.25),
graph( 400, 300, -10, 10, -10, 10,(6-1*x)/3, (6-1*x)/3-9),
graph( 400, 300, -10, 10, -10, 10,(6-1*x)/3, (6-1*x)/3-9.75),
graph( 400, 300, -10, 10, -10, 10,10000(x-10/1000), 10000(x-10/1000-0)),
graph( 400, 300, -10, 10, -10, 10,10000(x-10/1000), 10000(x-10/1000-0.75)),
graph( 400, 300, -10, 10, -10, 10,10000(x-10/1000), 10000(x-10/1000-1.5)),
graph( 400, 300, -10, 10, -10, 10,10000(x-10/1000), 10000(x-10/1000-2.25)),
graph( 400, 300, -10, 10, -10, 10,10000(x-10/1000), 10000(x-10/1000-3)),
graph( 400, 300, -10, 10, -10, 10,10000(x-10/1000), 10000(x-10/1000-3.75)),
graph( 400, 300, -10, 10, -10, 10,10000(x-10/1000), 10000(x-10/1000-4.5)),
graph( 400, 300, -10, 10, -10, 10,10000(x-10/1000), 10000(x-10/1000-5.25)),
graph( 400, 300, -10, 10, -10, 10,10000(x-10/1000), 10000(x-10/1000-6)),
graph( 400, 300, -10, 10, -10, 10,10000(x-10/1000), 10000(x-10/1000-6.75)),
graph( 400, 300, -10, 10, -10, 10,10000(x-10/1000), 10000(x-10/1000-7.5)),
graph( 400, 300, -10, 10, -10, 10,10000(x-10/1000), 10000(x-10/1000-8.25)),
graph( 400, 300, -10, 10, -10, 10,10000(x-10/1000), 10000(x-10/1000-9)),
graph( 400, 300, -10, 10, -10, 10,10000(x-10/1000), 10000(x-10/1000-9.75)),
graph( 400, 300, -10, 10, -10, 10,(10-1*x)/1000, (10-1*x)/1000+0),
graph( 400, 300, -10, 10, -10, 10,(10-1*x)/1000, (10-1*x)/1000+0.75),
graph( 400, 300, -10, 10, -10, 10,(10-1*x)/1000, (10-1*x)/1000+1.5),
graph( 400, 300, -10, 10, -10, 10,(10-1*x)/1000, (10-1*x)/1000+2.25),
graph( 400, 300, -10, 10, -10, 10,(10-1*x)/1000, (10-1*x)/1000+3),
graph( 400, 300, -10, 10, -10, 10,(10-1*x)/1000, (10-1*x)/1000+3.75),
graph( 400, 300, -10, 10, -10, 10,(10-1*x)/1000, (10-1*x)/1000+4.5),
graph( 400, 300, -10, 10, -10, 10,(10-1*x)/1000, (10-1*x)/1000+5.25),
graph( 400, 300, -10, 10, -10, 10,(10-1*x)/1000, (10-1*x)/1000+6),
graph( 400, 300, -10, 10, -10, 10,(10-1*x)/1000, (10-1*x)/1000+6.75),
graph( 400, 300, -10, 10, -10, 10,(10-1*x)/1000, (10-1*x)/1000+7.5),
graph( 400, 300, -10, 10, -10, 10,(10-1*x)/1000, (10-1*x)/1000+8.25),
graph( 400, 300, -10, 10, -10, 10,(10-1*x)/1000, (10-1*x)/1000+9),
graph( 400, 300, -10, 10, -10, 10,(10-1*x)/1000, (10-1*x)/1000+9.75))}}}

Here is a cleaner look at the intersection of regions


{{{drawing( 400, 300, -10, 10, -10, 10,
  graph( 400, 300, -10, 10, -10, 10,(12-3*x)/4,(6-1*x)/3,(10-1000*x)/1,(10-1*x)/1000),circle(0.5,0.5,0.05),
 circle(0.5,0.5,0.08),circle(0.5,1,0.05),
 circle(0.5,1,0.08),circle(0.5,1.5,0.05),
 circle(0.5,1.5,0.08),circle(1,0.5,0.05),
 circle(1,0.5,0.08),circle(1,1,0.05),
 circle(1,1,0.08),circle(1,1.5,0.05),
 circle(1,1.5,0.08),circle(1.5,0.5,0.05),
 circle(1.5,0.5,0.08),circle(1.5,1,0.05),
 circle(1.5,1,0.08),circle(2,0.5,0.05),
 circle(2,0.5,0.08),circle(2,1,0.05),
 circle(2,1,0.08),circle(2.5,0.5,0.05),
 circle(2.5,0.5,0.08),circle(2.5,1,0.05),
 circle(2.5,1,0.08),circle(3,0.5,0.05),
 circle(3,0.5,0.08))}}} Here is the intersection of the 4 regions represented by the dots