Question 85713
12.


Start with the given system of inequalities

{{{3x-y<2}}}


{{{x+y>2}}}




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-y<2}}} we need to graph the equation {{{3x-y=2}}} (just replace the inequality sign with an equal sign). So lets graph the line {{{3x-y=2}}} (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,(2-3*x)/-1)}}} graph of {{{3x-y=2}}}

    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-y<2}}}

     {{{3(0)-(0)<2}}} Plug in x=0, y=0


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

 Here is the graph of {{{3x-y<2}}} with the graph of the line({{{3x-y=2}}}) in red and the shaded region in green

(note: The red line should be a dashed line since it is <b>not</b> included in the region. Since the inequalityis a <font size=6><</font> sign, it tells us <b>not</b> to include the boundaries.)





Now lets graph the second inequality


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

    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+y>2}}}

     {{{(0)+(0)>2}}} Plug in x=0, y=0


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

 Here is the graph of {{{x+y>2}}} with the graph of the line({{{x+y=2}}}) in red and the shaded region in green

(note: The red line should be a dashed line since it is <b>not</b> included in the region. Since the inequalityis a <font size=6>></font> sign, it tells us <b>not</b> to include the boundaries.)

So we essentially have these 2 regions

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

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



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,(2-3*x)/-1, (2-3*x)/-1+0),
graph( 400, 300, -10, 10, -10, 10,(2-3*x)/-1, (2-3*x)/-1+0.75),
graph( 400, 300, -10, 10, -10, 10,(2-3*x)/-1, (2-3*x)/-1+1.5),
graph( 400, 300, -10, 10, -10, 10,(2-3*x)/-1, (2-3*x)/-1+2.25),
graph( 400, 300, -10, 10, -10, 10,(2-3*x)/-1, (2-3*x)/-1+3),
graph( 400, 300, -10, 10, -10, 10,(2-3*x)/-1, (2-3*x)/-1+3.75),
graph( 400, 300, -10, 10, -10, 10,(2-3*x)/-1, (2-3*x)/-1+4.5),
graph( 400, 300, -10, 10, -10, 10,(2-3*x)/-1, (2-3*x)/-1+5.25),
graph( 400, 300, -10, 10, -10, 10,(2-3*x)/-1, (2-3*x)/-1+6),
graph( 400, 300, -10, 10, -10, 10,(2-3*x)/-1, (2-3*x)/-1+6.75),
graph( 400, 300, -10, 10, -10, 10,(2-3*x)/-1, (2-3*x)/-1+7.5),
graph( 400, 300, -10, 10, -10, 10,(2-3*x)/-1, (2-3*x)/-1+8.25),
graph( 400, 300, -10, 10, -10, 10,(2-3*x)/-1, (2-3*x)/-1+9),
graph( 400, 300, -10, 10, -10, 10,(2-3*x)/-1, (2-3*x)/-1+9.75),
graph( 400, 300, -10, 10, -10, 10,(2-1*x)/1, (2-1*x)/1+0),
graph( 400, 300, -10, 10, -10, 10,(2-1*x)/1, (2-1*x)/1+0.75),
graph( 400, 300, -10, 10, -10, 10,(2-1*x)/1, (2-1*x)/1+1.5),
graph( 400, 300, -10, 10, -10, 10,(2-1*x)/1, (2-1*x)/1+2.25),
graph( 400, 300, -10, 10, -10, 10,(2-1*x)/1, (2-1*x)/1+3),
graph( 400, 300, -10, 10, -10, 10,(2-1*x)/1, (2-1*x)/1+3.75),
graph( 400, 300, -10, 10, -10, 10,(2-1*x)/1, (2-1*x)/1+4.5),
graph( 400, 300, -10, 10, -10, 10,(2-1*x)/1, (2-1*x)/1+5.25),
graph( 400, 300, -10, 10, -10, 10,(2-1*x)/1, (2-1*x)/1+6),
graph( 400, 300, -10, 10, -10, 10,(2-1*x)/1, (2-1*x)/1+6.75),
graph( 400, 300, -10, 10, -10, 10,(2-1*x)/1, (2-1*x)/1+7.5),
graph( 400, 300, -10, 10, -10, 10,(2-1*x)/1, (2-1*x)/1+8.25),
graph( 400, 300, -10, 10, -10, 10,(2-1*x)/1, (2-1*x)/1+9),
graph( 400, 300, -10, 10, -10, 10,(2-1*x)/1, (2-1*x)/1+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,(2-3*x)/-1,(2-1*x)/1),circle(-6.5,9.25,0.05),
 circle(-6.5,9.25,0.08),circle(-4.75,7.5,0.05),
 circle(-4.75,7.5,0.08),circle(-4.75,9.25,0.05),
 circle(-4.75,9.25,0.08),circle(-3,5.75,0.05),
 circle(-3,5.75,0.08),circle(-3,7.5,0.05),
 circle(-3,7.5,0.08),circle(-3,9.25,0.05),
 circle(-3,9.25,0.08),circle(-1.25,4,0.05),
 circle(-1.25,4,0.08),circle(-1.25,5.75,0.05),
 circle(-1.25,5.75,0.08),circle(-1.25,7.5,0.05),
 circle(-1.25,7.5,0.08),circle(-1.25,9.25,0.05),
 circle(-1.25,9.25,0.08),circle(0.5,2.25,0.05),
 circle(0.5,2.25,0.08),circle(0.5,4,0.05),
 circle(0.5,4,0.08),circle(0.5,5.75,0.05),
 circle(0.5,5.75,0.08),circle(0.5,7.5,0.05),
 circle(0.5,7.5,0.08),circle(0.5,9.25,0.05),
 circle(0.5,9.25,0.08),circle(2.25,5.75,0.05),
 circle(2.25,5.75,0.08),circle(2.25,7.5,0.05),
 circle(2.25,7.5,0.08),circle(2.25,9.25,0.05),
 circle(2.25,9.25,0.08))}}} Here is the intersection of the 2 regions represented by the dots