Question 87681
Start with the given system of inequalities

{{{y>=x}}}

{{{x<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 {{{y>=x}}}

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

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>=x}}}


Substitute (0,0) into the inequality

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

{{{1>=0}}} Simplify

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

{{{drawing( 500, 500, -20, 20, -20, 20,
graph(  500, 500, -20, 20, -20, 20,x,x+2),
graph(  500, 500, -20, 20, -20, 20,x,x+4),
graph(  500, 500, -20, 20, -20, 20,x,x+6),
graph(  500, 500, -20, 20, -20, 20,x,x+8),
graph(  500, 500, -20, 20, -20, 20,x,x+10),
graph(  500, 500, -20, 20, -20, 20,x,x+12),
graph(  500, 500, -20, 20, -20, 20,x,x+14),
graph(  500, 500, -20, 20, -20, 20,x,x+16),
graph(  500, 500, -20, 20, -20, 20,x,x+18),
graph(  500, 500, -20, 20, -20, 20,x,x+20),
graph(  500, 500, -20, 20, -20, 20,x,x+22),
graph(  500, 500, -20, 20, -20, 20,x,x+24),
graph(  500, 500, -20, 20, -20, 20,x,x+26),
graph(  500, 500, -20, 20, -20, 20,x,x+28),
graph(  500, 500, -20, 20, -20, 20,x,x+30),
graph(  500, 500, -20, 20, -20, 20,x,x+32),
graph(  500, 500, -20, 20, -20, 20,x,x+34),
graph(  500, 500, -20, 20, -20, 20,x,x+36),
graph(  500, 500, -20, 20, -20, 20,x,x+38))}}} Graph of {{{y>=x}}} with the boundary (which is the line {{{y=x}}} in red) and the shaded region (in green) 


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



Now lets graph the second inequality {{{x<3}}}

In order to graph {{{x<3}}}, we need to graph the <b>equation</b> {{{x=3}}} (just replace the inequality sign with an equal sign).
So lets graph the line {{{x=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, 1000(x-3)) }}} graph of {{{x=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 {{{x<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, 20,1000(x-3),1000(x-3+2)),
graph( 500, 500, -20, 20, -20, 20,1000(x-3),1000(x-3+4)),
graph( 500, 500, -20, 20, -20, 20,1000(x-3),1000(x-3+6)),
graph( 500, 500, -20, 20, -20, 20,1000(x-3),1000(x-3+8)),
graph( 500, 500, -20, 20, -20, 20,1000(x-3),1000(x-3+10)),
graph( 500, 500, -20, 20, -20, 20,1000(x-3),1000(x-3+12)),
graph( 500, 500, -20, 20, -20, 20,1000(x-3),1000(x-3+14)),
graph( 500, 500, -20, 20, -20, 20,1000(x-3),1000(x-3+16)),
graph( 500, 500, -20, 20, -20, 20,1000(x-3),1000(x-3+18)),
graph( 500, 500, -20, 20, -20, 20,1000(x-3),1000(x-3+20)),
graph( 500, 500, -20, 20, -20, 20,1000(x-3),1000(x-3+22)),
graph( 500, 500, -20, 20, -20, 20,1000(x-3),1000(x-3+24)),
graph( 500, 500, -20, 20, -20, 20,1000(x-3),1000(x-3+26)),
graph( 500, 500, -20, 20, -20, 20,1000(x-3),1000(x-3+28)),
graph( 500, 500, -20, 20, -20, 20,1000(x-3),1000(x-3+30)),
graph( 500, 500, -20, 20, -20, 20,1000(x-3),1000(x-3+32)),
graph( 500, 500, -20, 20, -20, 20,1000(x-3),1000(x-3+34)),
graph( 500, 500, -20, 20, -20, 20,1000(x-3),1000(x-3+36)),
graph( 500, 500, -20, 20, -20, 20,1000(x-3),1000(x-3+38)))}}} Graph of {{{x<3}}} with the boundary (which is the line {{{x=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 2 regions:


Region #1
{{{drawing( 500, 500, -20, 20, -20, 20,
graph( 500, 500, -20, 20, -20, 20,x),
graph( 500, 500, -20, 20, -20, 20,x,x+2),
graph( 500, 500, -20, 20, -20, 20,x,x+4),
graph( 500, 500, -20, 20, -20, 20,x,x+6),
graph( 500, 500, -20, 20, -20, 20,x,x+8),
graph( 500, 500, -20, 20, -20, 20,x,x+10),
graph( 500, 500, -20, 20, -20, 20,x,x+12),
graph( 500, 500, -20, 20, -20, 20,x,x+14),
graph( 500, 500, -20, 20, -20, 20,x,x+16),
graph( 500, 500, -20, 20, -20, 20,x,x+18),
graph( 500, 500, -20, 20, -20, 20,x,x+20),
graph( 500, 500, -20, 20, -20, 20,x,x+22),
graph( 500, 500, -20, 20, -20, 20,x,x+24),
graph( 500, 500, -20, 20, -20, 20,x,x+26),
graph( 500, 500, -20, 20, -20, 20,x,x+28),
graph( 500, 500, -20, 20, -20, 20,x,x+30),
graph( 500, 500, -20, 20, -20, 20,x,x+32),
graph( 500, 500, -20, 20, -20, 20,x,x+34),
graph( 500, 500, -20, 20, -20, 20,x,x+36),
graph( 500, 500, -20, 20, -20, 20,x,x+38))}}} Graph of {{{y>=x}}}



Region #2
{{{drawing( 500, 500, -20, 20, -20, 20,
graph( 500, 500, -20, 20, -20, 20,1000(x-3)),
graph( 500, 500, -20, 20, -20, 20,1000(x-3),1000(x-3+2)),
graph( 500, 500, -20, 20, -20, 20,1000(x-3),1000(x-3+4)),
graph( 500, 500, -20, 20, -20, 20,1000(x-3),1000(x-3+6)),
graph( 500, 500, -20, 20, -20, 20,1000(x-3),1000(x-3+8)),
graph( 500, 500, -20, 20, -20, 20,1000(x-3),1000(x-3+10)),
graph( 500, 500, -20, 20, -20, 20,1000(x-3),1000(x-3+12)),
graph( 500, 500, -20, 20, -20, 20,1000(x-3),1000(x-3+14)),
graph( 500, 500, -20, 20, -20, 20,1000(x-3),1000(x-3+16)),
graph( 500, 500, -20, 20, -20, 20,1000(x-3),1000(x-3+18)),
graph( 500, 500, -20, 20, -20, 20,1000(x-3),1000(x-3+20)),
graph( 500, 500, -20, 20, -20, 20,1000(x-3),1000(x-3+22)),
graph( 500, 500, -20, 20, -20, 20,1000(x-3),1000(x-3+24)),
graph( 500, 500, -20, 20, -20, 20,1000(x-3),1000(x-3+26)),
graph( 500, 500, -20, 20, -20, 20,1000(x-3),1000(x-3+28)),
graph( 500, 500, -20, 20, -20, 20,1000(x-3),1000(x-3+30)),
graph( 500, 500, -20, 20, -20, 20,1000(x-3),1000(x-3+32)),
graph( 500, 500, -20, 20, -20, 20,1000(x-3),1000(x-3+34)),
graph( 500, 500, -20, 20, -20, 20,1000(x-3),1000(x-3+36)),
graph( 500, 500, -20, 20, -20, 20,1000(x-3),1000(x-3+38)))}}} Graph of {{{x<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,x,x+2),
graph( 500, 500, -20, 20, -20, 20,x,x+4),
graph( 500, 500, -20, 20, -20, 20,x,x+6),
graph( 500, 500, -20, 20, -20, 20,x,x+8),
graph( 500, 500, -20, 20, -20, 20,x,x+10),
graph( 500, 500, -20, 20, -20, 20,x,x+12),
graph( 500, 500, -20, 20, -20, 20,x,x+14),
graph( 500, 500, -20, 20, -20, 20,x,x+16),
graph( 500, 500, -20, 20, -20, 20,x,x+18),
graph( 500, 500, -20, 20, -20, 20,x,x+20),
graph( 500, 500, -20, 20, -20, 20,x,x+22),
graph( 500, 500, -20, 20, -20, 20,x,x+24),
graph( 500, 500, -20, 20, -20, 20,x,x+26),
graph( 500, 500, -20, 20, -20, 20,x,x+28),
graph( 500, 500, -20, 20, -20, 20,x,x+30),
graph( 500, 500, -20, 20, -20, 20,x,x+32),
graph( 500, 500, -20, 20, -20, 20,x,x+34),
graph( 500, 500, -20, 20, -20, 20,x,x+36),
graph( 500, 500, -20, 20, -20, 20,x,x+38),
graph( 500, 500, -20, 20, -20, 20,1000(x-3),1000(x-3+2)),
graph( 500, 500, -20, 20, -20, 20,1000(x-3),1000(x-3+4)),
graph( 500, 500, -20, 20, -20, 20,1000(x-3),1000(x-3+6)),
graph( 500, 500, -20, 20, -20, 20,1000(x-3),1000(x-3+8)),
graph( 500, 500, -20, 20, -20, 20,1000(x-3),1000(x-3+10)),
graph( 500, 500, -20, 20, -20, 20,1000(x-3),1000(x-3+12)),
graph( 500, 500, -20, 20, -20, 20,1000(x-3),1000(x-3+14)),
graph( 500, 500, -20, 20, -20, 20,1000(x-3),1000(x-3+16)),
graph( 500, 500, -20, 20, -20, 20,1000(x-3),1000(x-3+18)),
graph( 500, 500, -20, 20, -20, 20,1000(x-3),1000(x-3+20)),
graph( 500, 500, -20, 20, -20, 20,1000(x-3),1000(x-3+22)),
graph( 500, 500, -20, 20, -20, 20,1000(x-3),1000(x-3+24)),
graph( 500, 500, -20, 20, -20, 20,1000(x-3),1000(x-3+26)),
graph( 500, 500, -20, 20, -20, 20,1000(x-3),1000(x-3+28)),
graph( 500, 500, -20, 20, -20, 20,1000(x-3),1000(x-3+30)),
graph( 500, 500, -20, 20, -20, 20,1000(x-3),1000(x-3+32)),
graph( 500, 500, -20, 20, -20, 20,1000(x-3),1000(x-3+34)),
graph( 500, 500, -20, 20, -20, 20,1000(x-3),1000(x-3+36)),
graph( 500, 500, -20, 20, -20, 20,1000(x-3),1000(x-3+38)))}}}




Here is a cleaner look at the intersection of regions





{{{drawing( 500, 500, -20, 20, -20, 20,
          graph( 500, 500, -20, 20, -20, 20,x,1000(x-3)),circle(-20,-20,0.2),
circle(-20,-17,0.2),
circle(-20,-14,0.2),
circle(-20,-11,0.2),
circle(-20,-8,0.2),
circle(-20,-5,0.2),
circle(-20,-2,0.2),
circle(-20,1,0.2),
circle(-20,4,0.2),
circle(-20,7,0.2),
circle(-20,10,0.2),
circle(-20,13,0.2),
circle(-20,16,0.2),
circle(-20,19,0.2),
circle(-17,-17,0.2),
circle(-17,-14,0.2),
circle(-17,-11,0.2),
circle(-17,-8,0.2),
circle(-17,-5,0.2),
circle(-17,-2,0.2),
circle(-17,1,0.2),
circle(-17,4,0.2),
circle(-17,7,0.2),
circle(-17,10,0.2),
circle(-17,13,0.2),
circle(-17,16,0.2),
circle(-17,19,0.2),
circle(-14,-14,0.2),
circle(-14,-11,0.2),
circle(-14,-8,0.2),
circle(-14,-5,0.2),
circle(-14,-2,0.2),
circle(-14,1,0.2),
circle(-14,4,0.2),
circle(-14,7,0.2),
circle(-14,10,0.2),
circle(-14,13,0.2),
circle(-14,16,0.2),
circle(-14,19,0.2),
circle(-11,-11,0.2),
circle(-11,-8,0.2),
circle(-11,-5,0.2),
circle(-11,-2,0.2),
circle(-11,1,0.2),
circle(-11,4,0.2),
circle(-11,7,0.2),
circle(-11,10,0.2),
circle(-11,13,0.2),
circle(-11,16,0.2),
circle(-11,19,0.2),
circle(-8,-8,0.2),
circle(-8,-5,0.2),
circle(-8,-2,0.2),
circle(-8,1,0.2),
circle(-8,4,0.2),
circle(-8,7,0.2),
circle(-8,10,0.2),
circle(-8,13,0.2),
circle(-8,16,0.2),
circle(-8,19,0.2),
circle(-5,-5,0.2),
circle(-5,-2,0.2),
circle(-5,1,0.2),
circle(-5,4,0.2),
circle(-5,7,0.2),
circle(-5,10,0.2),
circle(-5,13,0.2),
circle(-5,16,0.2),
circle(-5,19,0.2),
circle(-2,-2,0.2),
circle(-2,1,0.2),
circle(-2,4,0.2),
circle(-2,7,0.2),
circle(-2,10,0.2),
circle(-2,13,0.2),
circle(-2,16,0.2),
circle(-2,19,0.2),
circle(1,1,0.2),
circle(1,4,0.2),
circle(1,7,0.2),
circle(1,10,0.2),
circle(1,13,0.2),
circle(1,16,0.2),
circle(1,19,0.2),
circle(-20,-20,0.2),
circle(-20,-17,0.2),
circle(-20,-14,0.2),
circle(-20,-11,0.2),
circle(-20,-8,0.2),
circle(-20,-5,0.2),
circle(-20,-2,0.2),
circle(-20,1,0.2),
circle(-20,4,0.2),
circle(-20,7,0.2),
circle(-20,10,0.2),
circle(-20,13,0.2),
circle(-20,16,0.2),
circle(-20,19,0.2),
circle(-17,-17,0.2),
circle(-17,-14,0.2),
circle(-17,-11,0.2),
circle(-17,-8,0.2),
circle(-17,-5,0.2),
circle(-17,-2,0.2),
circle(-17,1,0.2),
circle(-17,4,0.2),
circle(-17,7,0.2),
circle(-17,10,0.2),
circle(-17,13,0.2),
circle(-17,16,0.2),
circle(-17,19,0.2),
circle(-14,-14,0.2),
circle(-14,-11,0.2),
circle(-14,-8,0.2),
circle(-14,-5,0.2),
circle(-14,-2,0.2),
circle(-14,1,0.2),
circle(-14,4,0.2),
circle(-14,7,0.2),
circle(-14,10,0.2),
circle(-14,13,0.2),
circle(-14,16,0.2),
circle(-14,19,0.2),
circle(-11,-11,0.2),
circle(-11,-8,0.2),
circle(-11,-5,0.2),
circle(-11,-2,0.2),
circle(-11,1,0.2),
circle(-11,4,0.2),
circle(-11,7,0.2),
circle(-11,10,0.2),
circle(-11,13,0.2),
circle(-11,16,0.2),
circle(-11,19,0.2),
circle(-8,-8,0.2),
circle(-8,-5,0.2),
circle(-8,-2,0.2),
circle(-8,1,0.2),
circle(-8,4,0.2),
circle(-8,7,0.2),
circle(-8,10,0.2),
circle(-8,13,0.2),
circle(-8,16,0.2),
circle(-8,19,0.2),
circle(-5,-5,0.2),
circle(-5,-2,0.2),
circle(-5,1,0.2),
circle(-5,4,0.2),
circle(-5,7,0.2),
circle(-5,10,0.2),
circle(-5,13,0.2),
circle(-5,16,0.2),
circle(-5,19,0.2),
circle(-2,-2,0.2),
circle(-2,1,0.2),
circle(-2,4,0.2),
circle(-2,7,0.2),
circle(-2,10,0.2),
circle(-2,13,0.2),
circle(-2,16,0.2),
circle(-2,19,0.2),
circle(1,1,0.2),
circle(1,4,0.2),
circle(1,7,0.2),
circle(1,10,0.2),
circle(1,13,0.2),
circle(1,16,0.2),
circle(1,19,0.2))}}} Here is the intersection of the 2 regions represented by the series of dots