Question 87987

Start with the given system of inequalities

{{{3x-y<6}}}

{{{x>1}}}

{{{y<3}}}


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



First lets graph the first inequality {{{3x-y<6}}}

In order to graph {{{3x-y<6}}}, we need to graph the <b>equation</b> {{{3x-y=6}}} (just replace the inequality sign with an equal sign).
So lets graph the line {{{3x-y=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( 500, 500, -20, 20, -20, 20, 3x-6) }}} graph of {{{3x-y=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 {{{3x-y<6}}}


Substitute (0,0) into the inequality

{{{3(0)-(0)<6}}} Plug in {{{x=0}}} and {{{y=0}}}

{{{0<6}}} Simplify

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

{{{drawing( 500, 500, -20, 20, -20, 20,
graph(  500, 500, -20, 20, -20, 20,3x-6,3x-6+1),
graph(  500, 500, -20, 20, -20, 20,3x-6,3x-6+2),
graph(  500, 500, -20, 20, -20, 20,3x-6,3x-6+3),
graph(  500, 500, -20, 20, -20, 20,3x-6,3x-6+4),
graph(  500, 500, -20, 20, -20, 20,3x-6,3x-6+5),
graph(  500, 500, -20, 20, -20, 20,3x-6,3x-6+6),
graph(  500, 500, -20, 20, -20, 20,3x-6,3x-6+7),
graph(  500, 500, -20, 20, -20, 20,3x-6,3x-6+8),
graph(  500, 500, -20, 20, -20, 20,3x-6,3x-6+9),
graph(  500, 500, -20, 20, -20, 20,3x-6,3x-6+10),
graph(  500, 500, -20, 20, -20, 20,3x-6,3x-6+11),
graph(  500, 500, -20, 20, -20, 20,3x-6,3x-6+12),
graph(  500, 500, -20, 20, -20, 20,3x-6,3x-6+13),
graph(  500, 500, -20, 20, -20, 20,3x-6,3x-6+14),
graph(  500, 500, -20, 20, -20, 20,3x-6,3x-6+15),
graph(  500, 500, -20, 20, -20, 20,3x-6,3x-6+16),
graph(  500, 500, -20, 20, -20, 20,3x-6,3x-6+17),
graph(  500, 500, -20, 20, -20, 20,3x-6,3x-6+18),
graph(  500, 500, -20, 20, -20, 20,3x-6,3x-6+19))}}} Graph of {{{3x-y<6}}} with the boundary (which is the line {{{3x-y=6}}} in red) and the shaded region (in green) 
(note: since the inequality contains a less-than sign, this means the boundary is excluded. This means the solid red line is really a dashed line)

---------------------------------------------------------------



Now lets graph the second inequality {{{x>1}}}

In order to graph {{{x>1}}}, we need to graph the <b>equation</b> {{{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( 500, 500, -20, 20, -20, 20, 1000(x-1)) }}} 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}}}


Substitute (0,0) into the inequality

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

{{{0>1}}} Simplify

Since this inequality is <b>not</b> true, we do <b>not</b> shade the entire region that contains (0,0). So this means we shade the region that is on the opposite side of the line

{{{drawing( 500, 500, -20, 20, -20, 20,
graph( 500, 500, -20, 20, -20, 20,1000(x-1),1000(x-1+-1)),
graph( 500, 500, -20, 20, -20, 20,1000(x-1),1000(x-1+-2)),
graph( 500, 500, -20, 20, -20, 20,1000(x-1),1000(x-1+-3)),
graph( 500, 500, -20, 20, -20, 20,1000(x-1),1000(x-1+-4)),
graph( 500, 500, -20, 20, -20, 20,1000(x-1),1000(x-1+-5)),
graph( 500, 500, -20, 20, -20, 20,1000(x-1),1000(x-1+-6)),
graph( 500, 500, -20, 20, -20, 20,1000(x-1),1000(x-1+-7)),
graph( 500, 500, -20, 20, -20, 20,1000(x-1),1000(x-1+-8)),
graph( 500, 500, -20, 20, -20, 20,1000(x-1),1000(x-1+-9)),
graph( 500, 500, -20, 20, -20, 20,1000(x-1),1000(x-1+-10)),
graph( 500, 500, -20, 20, -20, 20,1000(x-1),1000(x-1+-11)),
graph( 500, 500, -20, 20, -20, 20,1000(x-1),1000(x-1+-12)),
graph( 500, 500, -20, 20, -20, 20,1000(x-1),1000(x-1+-13)),
graph( 500, 500, -20, 20, -20, 20,1000(x-1),1000(x-1+-14)),
graph( 500, 500, -20, 20, -20, 20,1000(x-1),1000(x-1+-15)),
graph( 500, 500, -20, 20, -20, 20,1000(x-1),1000(x-1+-16)),
graph( 500, 500, -20, 20, -20, 20,1000(x-1),1000(x-1+-17)),
graph( 500, 500, -20, 20, -20, 20,1000(x-1),1000(x-1+-18)),
graph( 500, 500, -20, 20, -20, 20,1000(x-1),1000(x-1+-19)))}}} Graph of {{{x>1}}} with the boundary (which is the line {{{x=1}}} in red) and the shaded region (in green) 
(note: since the inequality contains a greater-than sign, this means the boundary is excluded. This means the solid red line is really a dashed line)

---------------------------------------------------------------



Now lets graph the third inequality {{{y<3}}}

In order to graph {{{y<3}}}, we need to graph the <b>equation</b> {{{y=3}}} (just replace the inequality sign with an equal sign).
So lets graph the line {{{y=3}}} (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( 500, 500, -20, 20, -20, 20, 3) }}} graph of {{{y=3}}} 

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 {{{y<3}}}


Substitute (0,0) into the inequality

{{{(0)<3}}} Plug in {{{x=0}}} and {{{y=0}}}

{{{0<3}}} Simplify

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

{{{drawing( 500, 500, -20, 20, -20, 20,
graph(  500, 500, -20, 20, -20, 20,3,3+-1),
graph(  500, 500, -20, 20, -20, 20,3,3+-2),
graph(  500, 500, -20, 20, -20, 20,3,3+-3),
graph(  500, 500, -20, 20, -20, 20,3,3+-4),
graph(  500, 500, -20, 20, -20, 20,3,3+-5),
graph(  500, 500, -20, 20, -20, 20,3,3+-6),
graph(  500, 500, -20, 20, -20, 20,3,3+-7),
graph(  500, 500, -20, 20, -20, 20,3,3+-8),
graph(  500, 500, -20, 20, -20, 20,3,3+-9),
graph(  500, 500, -20, 20, -20, 20,3,3+-10),
graph(  500, 500, -20, 20, -20, 20,3,3+-11),
graph(  500, 500, -20, 20, -20, 20,3,3+-12),
graph(  500, 500, -20, 20, -20, 20,3,3+-13),
graph(  500, 500, -20, 20, -20, 20,3,3+-14),
graph(  500, 500, -20, 20, -20, 20,3,3+-15),
graph(  500, 500, -20, 20, -20, 20,3,3+-16),
graph(  500, 500, -20, 20, -20, 20,3,3+-17),
graph(  500, 500, -20, 20, -20, 20,3,3+-18),
graph(  500, 500, -20, 20, -20, 20,3,3+-19))}}} Graph of {{{y<3}}} with the boundary (which is the line {{{y=3}}} in red) and the shaded region (in green) 
(note: since the inequality contains a less-than sign, this means the boundary is excluded. This means the solid red line is really a dashed line)

---------------------------------------------------------------



So we essentially have these 3 regions:


Region #1
{{{drawing( 500, 500, -20, 20, -20, 20,
graph( 500, 500, -20, 20, -20, 20,3x-6),
graph( 500, 500, -20, 20, -20, 20,3x-6,3x-6+1),
graph( 500, 500, -20, 20, -20, 20,3x-6,3x-6+2),
graph( 500, 500, -20, 20, -20, 20,3x-6,3x-6+3),
graph( 500, 500, -20, 20, -20, 20,3x-6,3x-6+4),
graph( 500, 500, -20, 20, -20, 20,3x-6,3x-6+5),
graph( 500, 500, -20, 20, -20, 20,3x-6,3x-6+6),
graph( 500, 500, -20, 20, -20, 20,3x-6,3x-6+7),
graph( 500, 500, -20, 20, -20, 20,3x-6,3x-6+8),
graph( 500, 500, -20, 20, -20, 20,3x-6,3x-6+9),
graph( 500, 500, -20, 20, -20, 20,3x-6,3x-6+10),
graph( 500, 500, -20, 20, -20, 20,3x-6,3x-6+11),
graph( 500, 500, -20, 20, -20, 20,3x-6,3x-6+12),
graph( 500, 500, -20, 20, -20, 20,3x-6,3x-6+13),
graph( 500, 500, -20, 20, -20, 20,3x-6,3x-6+14),
graph( 500, 500, -20, 20, -20, 20,3x-6,3x-6+15),
graph( 500, 500, -20, 20, -20, 20,3x-6,3x-6+16),
graph( 500, 500, -20, 20, -20, 20,3x-6,3x-6+17),
graph( 500, 500, -20, 20, -20, 20,3x-6,3x-6+18),
graph( 500, 500, -20, 20, -20, 20,3x-6,3x-6+19))}}} Graph of {{{3x-y<6}}}



Region #2
{{{drawing( 500, 500, -20, 20, -20, 20,
graph( 500, 500, -20, 20, -20, 20,1000(x-1)),
graph( 500, 500, -20, 20, -20, 20,1000(x-1),1000(x-1+-1)),
graph( 500, 500, -20, 20, -20, 20,1000(x-1),1000(x-1+-2)),
graph( 500, 500, -20, 20, -20, 20,1000(x-1),1000(x-1+-3)),
graph( 500, 500, -20, 20, -20, 20,1000(x-1),1000(x-1+-4)),
graph( 500, 500, -20, 20, -20, 20,1000(x-1),1000(x-1+-5)),
graph( 500, 500, -20, 20, -20, 20,1000(x-1),1000(x-1+-6)),
graph( 500, 500, -20, 20, -20, 20,1000(x-1),1000(x-1+-7)),
graph( 500, 500, -20, 20, -20, 20,1000(x-1),1000(x-1+-8)),
graph( 500, 500, -20, 20, -20, 20,1000(x-1),1000(x-1+-9)),
graph( 500, 500, -20, 20, -20, 20,1000(x-1),1000(x-1+-10)),
graph( 500, 500, -20, 20, -20, 20,1000(x-1),1000(x-1+-11)),
graph( 500, 500, -20, 20, -20, 20,1000(x-1),1000(x-1+-12)),
graph( 500, 500, -20, 20, -20, 20,1000(x-1),1000(x-1+-13)),
graph( 500, 500, -20, 20, -20, 20,1000(x-1),1000(x-1+-14)),
graph( 500, 500, -20, 20, -20, 20,1000(x-1),1000(x-1+-15)),
graph( 500, 500, -20, 20, -20, 20,1000(x-1),1000(x-1+-16)),
graph( 500, 500, -20, 20, -20, 20,1000(x-1),1000(x-1+-17)),
graph( 500, 500, -20, 20, -20, 20,1000(x-1),1000(x-1+-18)),
graph( 500, 500, -20, 20, -20, 20,1000(x-1),1000(x-1+-19)))}}} Graph of {{{x>1}}}



Region #3
{{{drawing( 500, 500, -20, 20, -20, 20,
graph( 500, 500, -20, 20, -20, 20,3),
graph( 500, 500, -20, 20, -20, 20,3,3+-1),
graph( 500, 500, -20, 20, -20, 20,3,3+-2),
graph( 500, 500, -20, 20, -20, 20,3,3+-3),
graph( 500, 500, -20, 20, -20, 20,3,3+-4),
graph( 500, 500, -20, 20, -20, 20,3,3+-5),
graph( 500, 500, -20, 20, -20, 20,3,3+-6),
graph( 500, 500, -20, 20, -20, 20,3,3+-7),
graph( 500, 500, -20, 20, -20, 20,3,3+-8),
graph( 500, 500, -20, 20, -20, 20,3,3+-9),
graph( 500, 500, -20, 20, -20, 20,3,3+-10),
graph( 500, 500, -20, 20, -20, 20,3,3+-11),
graph( 500, 500, -20, 20, -20, 20,3,3+-12),
graph( 500, 500, -20, 20, -20, 20,3,3+-13),
graph( 500, 500, -20, 20, -20, 20,3,3+-14),
graph( 500, 500, -20, 20, -20, 20,3,3+-15),
graph( 500, 500, -20, 20, -20, 20,3,3+-16),
graph( 500, 500, -20, 20, -20, 20,3,3+-17),
graph( 500, 500, -20, 20, -20, 20,3,3+-18),
graph( 500, 500, -20, 20, -20, 20,3,3+-19))}}} Graph of {{{y<3}}}





When these inequalities are graphed on the same coordinate system, the 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( 500, 500, -20, 20, -20, 20,
graph( 500, 500, -20, 20, -20, 20,3x-6,3x-6+1),
graph( 500, 500, -20, 20, -20, 20,3x-6,3x-6+2),
graph( 500, 500, -20, 20, -20, 20,3x-6,3x-6+3),
graph( 500, 500, -20, 20, -20, 20,3x-6,3x-6+4),
graph( 500, 500, -20, 20, -20, 20,3x-6,3x-6+5),
graph( 500, 500, -20, 20, -20, 20,3x-6,3x-6+6),
graph( 500, 500, -20, 20, -20, 20,3x-6,3x-6+7),
graph( 500, 500, -20, 20, -20, 20,3x-6,3x-6+8),
graph( 500, 500, -20, 20, -20, 20,3x-6,3x-6+9),
graph( 500, 500, -20, 20, -20, 20,3x-6,3x-6+10),
graph( 500, 500, -20, 20, -20, 20,3x-6,3x-6+11),
graph( 500, 500, -20, 20, -20, 20,3x-6,3x-6+12),
graph( 500, 500, -20, 20, -20, 20,3x-6,3x-6+13),
graph( 500, 500, -20, 20, -20, 20,3x-6,3x-6+14),
graph( 500, 500, -20, 20, -20, 20,3x-6,3x-6+15),
graph( 500, 500, -20, 20, -20, 20,3x-6,3x-6+16),
graph( 500, 500, -20, 20, -20, 20,3x-6,3x-6+17),
graph( 500, 500, -20, 20, -20, 20,3x-6,3x-6+18),
graph( 500, 500, -20, 20, -20, 20,3x-6,3x-6+19),
graph( 500, 500, -20, 20, -20, 20,1000(x-1),1000(x-1+-1)),
graph( 500, 500, -20, 20, -20, 20,1000(x-1),1000(x-1+-2)),
graph( 500, 500, -20, 20, -20, 20,1000(x-1),1000(x-1+-3)),
graph( 500, 500, -20, 20, -20, 20,1000(x-1),1000(x-1+-4)),
graph( 500, 500, -20, 20, -20, 20,1000(x-1),1000(x-1+-5)),
graph( 500, 500, -20, 20, -20, 20,1000(x-1),1000(x-1+-6)),
graph( 500, 500, -20, 20, -20, 20,1000(x-1),1000(x-1+-7)),
graph( 500, 500, -20, 20, -20, 20,1000(x-1),1000(x-1+-8)),
graph( 500, 500, -20, 20, -20, 20,1000(x-1),1000(x-1+-9)),
graph( 500, 500, -20, 20, -20, 20,1000(x-1),1000(x-1+-10)),
graph( 500, 500, -20, 20, -20, 20,1000(x-1),1000(x-1+-11)),
graph( 500, 500, -20, 20, -20, 20,1000(x-1),1000(x-1+-12)),
graph( 500, 500, -20, 20, -20, 20,1000(x-1),1000(x-1+-13)),
graph( 500, 500, -20, 20, -20, 20,1000(x-1),1000(x-1+-14)),
graph( 500, 500, -20, 20, -20, 20,1000(x-1),1000(x-1+-15)),
graph( 500, 500, -20, 20, -20, 20,1000(x-1),1000(x-1+-16)),
graph( 500, 500, -20, 20, -20, 20,1000(x-1),1000(x-1+-17)),
graph( 500, 500, -20, 20, -20, 20,1000(x-1),1000(x-1+-18)),
graph( 500, 500, -20, 20, -20, 20,1000(x-1),1000(x-1+-19)),
graph( 500, 500, -20, 20, -20, 20,3,3+-1),
graph( 500, 500, -20, 20, -20, 20,3,3+-2),
graph( 500, 500, -20, 20, -20, 20,3,3+-3),
graph( 500, 500, -20, 20, -20, 20,3,3+-4),
graph( 500, 500, -20, 20, -20, 20,3,3+-5),
graph( 500, 500, -20, 20, -20, 20,3,3+-6),
graph( 500, 500, -20, 20, -20, 20,3,3+-7),
graph( 500, 500, -20, 20, -20, 20,3,3+-8),
graph( 500, 500, -20, 20, -20, 20,3,3+-9),
graph( 500, 500, -20, 20, -20, 20,3,3+-10),
graph( 500, 500, -20, 20, -20, 20,3,3+-11),
graph( 500, 500, -20, 20, -20, 20,3,3+-12),
graph( 500, 500, -20, 20, -20, 20,3,3+-13),
graph( 500, 500, -20, 20, -20, 20,3,3+-14),
graph( 500, 500, -20, 20, -20, 20,3,3+-15),
graph( 500, 500, -20, 20, -20, 20,3,3+-16),
graph( 500, 500, -20, 20, -20, 20,3,3+-17),
graph( 500, 500, -20, 20, -20, 20,3,3+-18),
graph( 500, 500, -20, 20, -20, 20,3,3+-19))}}}




Here is a cleaner look at the intersection of regions





{{{drawing( 500, 500, -20, 20, -20, 20,
          graph( 500, 500, -20, 20, -20, 20,3x-6,1000(x-1),3),circle(1.5,-1,0.2),
circle(1.5,-0.5,0.2),
circle(1.5,0,0.2),
circle(1.5,0.5,0.2),
circle(1.5,1,0.2),
circle(1.5,1.5,0.2),
circle(1.5,2,0.2),
circle(1.5,2.5,0.2),
circle(2,0.5,0.2),
circle(2,1,0.2),
circle(2,1.5,0.2),
circle(2,2,0.2),
circle(2,2.5,0.2),
circle(2.5,2,0.2),
circle(2.5,2.5,0.2),
circle(1.5,-1,0.2),
circle(1.5,-0.5,0.2),
circle(1.5,0,0.2),
circle(1.5,0.5,0.2),
circle(1.5,1,0.2),
circle(1.5,1.5,0.2),
circle(1.5,2,0.2),
circle(1.5,2.5,0.2),
circle(2,0.5,0.2),
circle(2,1,0.2),
circle(2,1.5,0.2),
circle(2,2,0.2),
circle(2,2.5,0.2),
circle(2.5,2,0.2),
circle(2.5,2.5,0.2),
circle(1.5,-1,0.2),
circle(1.5,-0.5,0.2),
circle(1.5,0,0.2),
circle(1.5,0.5,0.2),
circle(1.5,1,0.2),
circle(1.5,1.5,0.2),
circle(1.5,2,0.2),
circle(1.5,2.5,0.2),
circle(2,0.5,0.2),
circle(2,1,0.2),
circle(2,1.5,0.2),
circle(2,2,0.2),
circle(2,2.5,0.2),
circle(2.5,2,0.2),
circle(2.5,2.5,0.2))}}} Here is the intersection of the 3 regions represented by the series of dots