Question 85010
Start with the given system of inequalities

{{{x-2y<=4}}}


{{{x>=1}}}




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 {{{x-2y<=4}}} we need to graph the equation {{{x-2y=4}}} (just replace the inequality sign with an equal sign). So lets graph the line {{{x-2y=4}}} (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,(4-1*x)/-2)}}} graph of {{{x-2y=4}}}

    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-2y<=4}}}

     {{{(0)-2(0)<=4}}} Plug in x=0, y=0


     {{{0<=4}}} 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,(4-1*x)/-2, (4-1*x)/-2+0),
graph( 400, 300, -10, 10, -10, 10,(4-1*x)/-2, (4-1*x)/-2+0.75),
graph( 400, 300, -10, 10, -10, 10,(4-1*x)/-2, (4-1*x)/-2+1.5),
graph( 400, 300, -10, 10, -10, 10,(4-1*x)/-2, (4-1*x)/-2+2.25),
graph( 400, 300, -10, 10, -10, 10,(4-1*x)/-2, (4-1*x)/-2+3),
graph( 400, 300, -10, 10, -10, 10,(4-1*x)/-2, (4-1*x)/-2+3.75),
graph( 400, 300, -10, 10, -10, 10,(4-1*x)/-2, (4-1*x)/-2+4.5),
graph( 400, 300, -10, 10, -10, 10,(4-1*x)/-2, (4-1*x)/-2+5.25),
graph( 400, 300, -10, 10, -10, 10,(4-1*x)/-2, (4-1*x)/-2+6),
graph( 400, 300, -10, 10, -10, 10,(4-1*x)/-2, (4-1*x)/-2+6.75),
graph( 400, 300, -10, 10, -10, 10,(4-1*x)/-2, (4-1*x)/-2+7.5),
graph( 400, 300, -10, 10, -10, 10,(4-1*x)/-2, (4-1*x)/-2+8.25),
graph( 400, 300, -10, 10, -10, 10,(4-1*x)/-2, (4-1*x)/-2+9),
graph( 400, 300, -10, 10, -10, 10,(4-1*x)/-2, (4-1*x)/-2+9.75))}}}

 Here is the graph of {{{x-2y<=4}}} with the graph of the line({{{x-2y=4}}}) 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>=1}}} we need to graph the equation {{{x=1}}} (just replace the inequality sign with an equal sign). So lets graph the line {{{x=1}}} (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-1000/1000))}}} graph of {{{x=1}}}

    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>=1}}}

     {{{(0)>=1}}} Plug in x=0, y=0


     {{{0>=1}}} Simplify



Since this inequality is <b>not</b> true, we shade the entire region that <b>doesn't</b> contain (0,0)



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

 Here is the graph of {{{x>=1}}} with the graph of the line({{{x=1}}}) 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 2 regions

{{{drawing( 400, 300, -10, 10, -10, 10,graph( 400, 300, -10, 10, -10, 10,(4-1*x)/-2, (4-1*x)/-2+0),
graph( 400, 300, -10, 10, -10, 10,(4-1*x)/-2, (4-1*x)/-2+0.75),
graph( 400, 300, -10, 10, -10, 10,(4-1*x)/-2, (4-1*x)/-2+1.5),
graph( 400, 300, -10, 10, -10, 10,(4-1*x)/-2, (4-1*x)/-2+2.25),
graph( 400, 300, -10, 10, -10, 10,(4-1*x)/-2, (4-1*x)/-2+3),
graph( 400, 300, -10, 10, -10, 10,(4-1*x)/-2, (4-1*x)/-2+3.75),
graph( 400, 300, -10, 10, -10, 10,(4-1*x)/-2, (4-1*x)/-2+4.5),
graph( 400, 300, -10, 10, -10, 10,(4-1*x)/-2, (4-1*x)/-2+5.25),
graph( 400, 300, -10, 10, -10, 10,(4-1*x)/-2, (4-1*x)/-2+6),
graph( 400, 300, -10, 10, -10, 10,(4-1*x)/-2, (4-1*x)/-2+6.75),
graph( 400, 300, -10, 10, -10, 10,(4-1*x)/-2, (4-1*x)/-2+7.5),
graph( 400, 300, -10, 10, -10, 10,(4-1*x)/-2, (4-1*x)/-2+8.25),
graph( 400, 300, -10, 10, -10, 10,(4-1*x)/-2, (4-1*x)/-2+9),
graph( 400, 300, -10, 10, -10, 10,(4-1*x)/-2, (4-1*x)/-2+9.75))}}} Region #1 which is the graph of {{{x-2y<=4}}}

{{{drawing( 400, 300, -10, 10, -10, 10,graph( 400, 300, -10, 10, -10, 10,10000(x-1000/1000), 10000(x-1000/1000-0)),
graph( 400, 300, -10, 10, -10, 10,10000(x-1000/1000), 10000(x-1000/1000-0.75)),
graph( 400, 300, -10, 10, -10, 10,10000(x-1000/1000), 10000(x-1000/1000-1.5)),
graph( 400, 300, -10, 10, -10, 10,10000(x-1000/1000), 10000(x-1000/1000-2.25)),
graph( 400, 300, -10, 10, -10, 10,10000(x-1000/1000), 10000(x-1000/1000-3)),
graph( 400, 300, -10, 10, -10, 10,10000(x-1000/1000), 10000(x-1000/1000-3.75)),
graph( 400, 300, -10, 10, -10, 10,10000(x-1000/1000), 10000(x-1000/1000-4.5)),
graph( 400, 300, -10, 10, -10, 10,10000(x-1000/1000), 10000(x-1000/1000-5.25)),
graph( 400, 300, -10, 10, -10, 10,10000(x-1000/1000), 10000(x-1000/1000-6)),
graph( 400, 300, -10, 10, -10, 10,10000(x-1000/1000), 10000(x-1000/1000-6.75)),
graph( 400, 300, -10, 10, -10, 10,10000(x-1000/1000), 10000(x-1000/1000-7.5)),
graph( 400, 300, -10, 10, -10, 10,10000(x-1000/1000), 10000(x-1000/1000-8.25)),
graph( 400, 300, -10, 10, -10, 10,10000(x-1000/1000), 10000(x-1000/1000-9)),
graph( 400, 300, -10, 10, -10, 10,10000(x-1000/1000), 10000(x-1000/1000-9.75)))}}} Region #2 which is the graph of {{{x>=1}}}



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,(4-1*x)/-2, (4-1*x)/-2+0),
graph( 400, 300, -10, 10, -10, 10,(4-1*x)/-2, (4-1*x)/-2+0.75),
graph( 400, 300, -10, 10, -10, 10,(4-1*x)/-2, (4-1*x)/-2+1.5),
graph( 400, 300, -10, 10, -10, 10,(4-1*x)/-2, (4-1*x)/-2+2.25),
graph( 400, 300, -10, 10, -10, 10,(4-1*x)/-2, (4-1*x)/-2+3),
graph( 400, 300, -10, 10, -10, 10,(4-1*x)/-2, (4-1*x)/-2+3.75),
graph( 400, 300, -10, 10, -10, 10,(4-1*x)/-2, (4-1*x)/-2+4.5),
graph( 400, 300, -10, 10, -10, 10,(4-1*x)/-2, (4-1*x)/-2+5.25),
graph( 400, 300, -10, 10, -10, 10,(4-1*x)/-2, (4-1*x)/-2+6),
graph( 400, 300, -10, 10, -10, 10,(4-1*x)/-2, (4-1*x)/-2+6.75),
graph( 400, 300, -10, 10, -10, 10,(4-1*x)/-2, (4-1*x)/-2+7.5),
graph( 400, 300, -10, 10, -10, 10,(4-1*x)/-2, (4-1*x)/-2+8.25),
graph( 400, 300, -10, 10, -10, 10,(4-1*x)/-2, (4-1*x)/-2+9),
graph( 400, 300, -10, 10, -10, 10,(4-1*x)/-2, (4-1*x)/-2+9.75),
graph( 400, 300, -10, 10, -10, 10,10000(x-1000/1000), 10000(x-1000/1000-0)),
graph( 400, 300, -10, 10, -10, 10,10000(x-1000/1000), 10000(x-1000/1000-0.75)),
graph( 400, 300, -10, 10, -10, 10,10000(x-1000/1000), 10000(x-1000/1000-1.5)),
graph( 400, 300, -10, 10, -10, 10,10000(x-1000/1000), 10000(x-1000/1000-2.25)),
graph( 400, 300, -10, 10, -10, 10,10000(x-1000/1000), 10000(x-1000/1000-3)),
graph( 400, 300, -10, 10, -10, 10,10000(x-1000/1000), 10000(x-1000/1000-3.75)),
graph( 400, 300, -10, 10, -10, 10,10000(x-1000/1000), 10000(x-1000/1000-4.5)),
graph( 400, 300, -10, 10, -10, 10,10000(x-1000/1000), 10000(x-1000/1000-5.25)),
graph( 400, 300, -10, 10, -10, 10,10000(x-1000/1000), 10000(x-1000/1000-6)),
graph( 400, 300, -10, 10, -10, 10,10000(x-1000/1000), 10000(x-1000/1000-6.75)),
graph( 400, 300, -10, 10, -10, 10,10000(x-1000/1000), 10000(x-1000/1000-7.5)),
graph( 400, 300, -10, 10, -10, 10,10000(x-1000/1000), 10000(x-1000/1000-8.25)),
graph( 400, 300, -10, 10, -10, 10,10000(x-1000/1000), 10000(x-1000/1000-9)),
graph( 400, 300, -10, 10, -10, 10,10000(x-1000/1000), 10000(x-1000/1000-9.75)))}}}

Here is a cleaner look at the intersection of regions


{{{drawing( 400, 300, -10, 10, -10, 10,
  graph( 400, 300, -10, 10, -10, 10,(4-1*x)/-2,(1000-1000*x)/1),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(1,2,0.05),
 circle(1,2,0.08),circle(1,2.5,0.05),
 circle(1,2.5,0.08),circle(1,3,0.05),
 circle(1,3,0.08),circle(1,3.5,0.05),
 circle(1,3.5,0.08),circle(1,4,0.05),
 circle(1,4,0.08),circle(1,4.5,0.05),
 circle(1,4.5,0.08),circle(1,5,0.05),
 circle(1,5,0.08),circle(1,5.5,0.05),
 circle(1,5.5,0.08),circle(1,6,0.05),
 circle(1,6,0.08),circle(1,6.5,0.05),
 circle(1,6.5,0.08),circle(1,7,0.05),
 circle(1,7,0.08),circle(1,7.5,0.05),
 circle(1,7.5,0.08),circle(1,8,0.05),
 circle(1,8,0.08),circle(1,8.5,0.05),
 circle(1,8.5,0.08),circle(1,9,0.05),
 circle(1,9,0.08),circle(1,9.5,0.05),
 circle(1,9.5,0.08),circle(1.5,-1,0.05),
 circle(1.5,-1,0.08),circle(1.5,-0.5,0.05),
 circle(1.5,-0.5,0.08),circle(1.5,0,0.05),
 circle(1.5,0,0.08),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(1.5,1.5,0.05),
 circle(1.5,1.5,0.08),circle(1.5,2,0.05),
 circle(1.5,2,0.08),circle(1.5,2.5,0.05),
 circle(1.5,2.5,0.08),circle(1.5,3,0.05),
 circle(1.5,3,0.08),circle(1.5,3.5,0.05),
 circle(1.5,3.5,0.08),circle(1.5,4,0.05),
 circle(1.5,4,0.08),circle(1.5,4.5,0.05),
 circle(1.5,4.5,0.08),circle(1.5,5,0.05),
 circle(1.5,5,0.08),circle(1.5,5.5,0.05),
 circle(1.5,5.5,0.08),circle(1.5,6,0.05),
 circle(1.5,6,0.08),circle(1.5,6.5,0.05),
 circle(1.5,6.5,0.08),circle(1.5,7,0.05),
 circle(1.5,7,0.08),circle(1.5,7.5,0.05),
 circle(1.5,7.5,0.08),circle(1.5,8,0.05),
 circle(1.5,8,0.08),circle(1.5,8.5,0.05),
 circle(1.5,8.5,0.08),circle(1.5,9,0.05),
 circle(1.5,9,0.08),circle(1.5,9.5,0.05),
 circle(1.5,9.5,0.08),circle(2,-0.5,0.05),
 circle(2,-0.5,0.08),circle(2,0,0.05),
 circle(2,0,0.08),circle(2,0.5,0.05),
 circle(2,0.5,0.08),circle(2,1,0.05),
 circle(2,1,0.08),circle(2,1.5,0.05),
 circle(2,1.5,0.08),circle(2,2,0.05),
 circle(2,2,0.08),circle(2,2.5,0.05),
 circle(2,2.5,0.08),circle(2,3,0.05),
 circle(2,3,0.08),circle(2,3.5,0.05),
 circle(2,3.5,0.08),circle(2,4,0.05),
 circle(2,4,0.08),circle(2,4.5,0.05),
 circle(2,4.5,0.08),circle(2,5,0.05),
 circle(2,5,0.08),circle(2,5.5,0.05),
 circle(2,5.5,0.08),circle(2,6,0.05),
 circle(2,6,0.08),circle(2,6.5,0.05),
 circle(2,6.5,0.08),circle(2,7,0.05),
 circle(2,7,0.08),circle(2,7.5,0.05),
 circle(2,7.5,0.08),circle(2,8,0.05),
 circle(2,8,0.08),circle(2,8.5,0.05),
 circle(2,8.5,0.08),circle(2,9,0.05),
 circle(2,9,0.08),circle(2,9.5,0.05),
 circle(2,9.5,0.08),circle(2.5,-0.5,0.05),
 circle(2.5,-0.5,0.08),circle(2.5,0,0.05),
 circle(2.5,0,0.08),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(2.5,1.5,0.05),
 circle(2.5,1.5,0.08),circle(2.5,2,0.05),
 circle(2.5,2,0.08),circle(2.5,2.5,0.05),
 circle(2.5,2.5,0.08),circle(2.5,3,0.05),
 circle(2.5,3,0.08),circle(2.5,3.5,0.05),
 circle(2.5,3.5,0.08),circle(2.5,4,0.05),
 circle(2.5,4,0.08),circle(2.5,4.5,0.05),
 circle(2.5,4.5,0.08),circle(2.5,5,0.05),
 circle(2.5,5,0.08),circle(2.5,5.5,0.05),
 circle(2.5,5.5,0.08),circle(2.5,6,0.05),
 circle(2.5,6,0.08),circle(2.5,6.5,0.05),
 circle(2.5,6.5,0.08),circle(2.5,7,0.05),
 circle(2.5,7,0.08),circle(2.5,7.5,0.05),
 circle(2.5,7.5,0.08),circle(2.5,8,0.05),
 circle(2.5,8,0.08),circle(2.5,8.5,0.05),
 circle(2.5,8.5,0.08),circle(2.5,9,0.05),
 circle(2.5,9,0.08),circle(2.5,9.5,0.05),
 circle(2.5,9.5,0.08),circle(3,0,0.05)
  )}}} Here is the intersection of the 2 regions represented by the dots (note: this region extends to infinity)