Question 172799
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<a href="#1">Graphing Inequality #1</a>
<a href="#2">Graphing Inequality #2</a>
<a href="#3">Graphing Inequality #3</a>
<a href="#combo">Graphing all three inequalities together</a>
<a href="#solution">Solution</a>




<a name="1">



Inequality #1 



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{{{2y+x>=-2}}} Start with the 1st inequality.



{{{2y>=-2-x}}} Subtract {{{x}}} from both sides.



{{{y>=(-2-x)/(2)}}} Divide both sides by {{{2}}} to isolate {{{y}}}.



{{{y>=-(2)/(2)-(x)/(2)}}} Break up the fraction.



{{{y>=-1-(1/2)x}}} Reduce



{{{y>=-(1/2)x-1}}} Rearrange the terms.



So in order to graph {{{y>=-(1/2)x-1}}}, we need to graph the equation {{{y=-(1/2)x-1}}}



{{{ drawing(500, 500, -10, 10, -10, 10,
 graph( 500, 500, -10, 10, -10, 10,-(1/2)x-1)

)}}} Graph {{{y=-(1/2)x-1}}}



Now since the inequality sign of {{{y>=-(1/2)x-1}}} is a "greater-than or equal to" sign, this tells us to shade the region above the graph (since points in that region have larger y values)



So the shading looks like this:



{{{drawing( 500, 500, -20, 20, -20, 20,
graph(  500, 500, -20, 20, -20, 20,-(1/2)x-1,-(1/2)x-1+2),
graph(  500, 500, -20, 20, -20, 20,-(1/2)x-1,-(1/2)x-1+4),
graph(  500, 500, -20, 20, -20, 20,-(1/2)x-1,-(1/2)x-1+6),
graph(  500, 500, -20, 20, -20, 20,-(1/2)x-1,-(1/2)x-1+8),
graph(  500, 500, -20, 20, -20, 20,-(1/2)x-1,-(1/2)x-1+10),
graph(  500, 500, -20, 20, -20, 20,-(1/2)x-1,-(1/2)x-1+12),
graph(  500, 500, -20, 20, -20, 20,-(1/2)x-1,-(1/2)x-1+14),
graph(  500, 500, -20, 20, -20, 20,-(1/2)x-1,-(1/2)x-1+16),
graph(  500, 500, -20, 20, -20, 20,-(1/2)x-1,-(1/2)x-1+18),
graph(  500, 500, -20, 20, -20, 20,-(1/2)x-1,-(1/2)x-1+20),
graph(  500, 500, -20, 20, -20, 20,-(1/2)x-1,-(1/2)x-1+22),
graph(  500, 500, -20, 20, -20, 20,-(1/2)x-1,-(1/2)x-1+24),
graph(  500, 500, -20, 20, -20, 20,-(1/2)x-1,-(1/2)x-1+26),
graph(  500, 500, -20, 20, -20, 20,-(1/2)x-1,-(1/2)x-1+28),
graph(  500, 500, -20, 20, -20, 20,-(1/2)x-1,-(1/2)x-1+30),
graph(  500, 500, -20, 20, -20, 20,-(1/2)x-1,-(1/2)x-1+32))}}} Graph of {{{y>=-(1/2)x-1}}} with the shaded region in green



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




<a name="2">



Inequality #2 



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{{{y+2x<4}}} Start with the 2nd inequality.



{{{y<4-2x}}} Subtract {{{2x}}} from both sides.



{{{y<-2x+4}}} Rearrange the terms.



So in order to graph {{{y<-2x+4}}}, we need to graph the equation {{{y=-2x+4}}}



{{{ drawing(500, 500, -10, 10, -10, 10,
 graph( 500, 500, -10, 10, -10, 10,-2x+4)

)}}} Graph {{{y=-2x+4}}}



Now since the inequality sign of {{{y<-2x+4}}} is a "less-than" sign, this tells us to shade the region below the graph (since points in that region have smaller y values)



So the shading looks like this:



{{{drawing( 500, 500, -20, 20, -20, 20,
graph(  500, 500, -20, 20, -20, 20,-2x+4,-2x+4+-4),
graph(  500, 500, -20, 20, -20, 20,-2x+4,-2x+4+-8),
graph(  500, 500, -20, 20, -20, 20,-2x+4,-2x+4+-12),
graph(  500, 500, -20, 20, -20, 20,-2x+4,-2x+4+-16),
graph(  500, 500, -20, 20, -20, 20,-2x+4,-2x+4+-20),
graph(  500, 500, -20, 20, -20, 20,-2x+4,-2x+4+-24),
graph(  500, 500, -20, 20, -20, 20,-2x+4,-2x+4+-28),
graph(  500, 500, -20, 20, -20, 20,-2x+4,-2x+4+-32),
graph(  500, 500, -20, 20, -20, 20,-2x+4,-2x+4+-36),
graph(  500, 500, -20, 20, -20, 20,-2x+4,-2x+4+-40),
graph(  500, 500, -20, 20, -20, 20,-2x+4,-2x+4+-44),
graph(  500, 500, -20, 20, -20, 20,-2x+4,-2x+4+-48),
graph(  500, 500, -20, 20, -20, 20,-2x+4,-2x+4+-52),
graph(  500, 500, -20, 20, -20, 20,-2x+4,-2x+4+-56),
graph(  500, 500, -20, 20, -20, 20,-2x+4,-2x+4+-60),
graph(  500, 500, -20, 20, -20, 20,-2x+4,-2x+4+-64))}}} Graph of {{{y<-2x+4}}} with the shaded region in green



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




<a name="3">



Inequality #3 



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Finally, to graph {{{y>3}}}, we need to graph {{{y=3}}} (which is a horizontal line at y=3) like this


{{{ drawing(500, 500, -10, 10, -10, 10,
 graph( 500, 500, -10, 10, -10, 10,3)

)}}} Graph {{{y=3}}}




Now shade the region above the line (since the sign is a "greater-than" sign) to get



{{{drawing( 500, 500, -20, 20, -20, 20,
graph(  500, 500, -20, 20, -20, 20,3,3+2),
graph(  500, 500, -20, 20, -20, 20,3,3+4),
graph(  500, 500, -20, 20, -20, 20,3,3+6),
graph(  500, 500, -20, 20, -20, 20,3,3+8),
graph(  500, 500, -20, 20, -20, 20,3,3+10),
graph(  500, 500, -20, 20, -20, 20,3,3+12),
graph(  500, 500, -20, 20, -20, 20,3,3+14),
graph(  500, 500, -20, 20, -20, 20,3,3+16),
graph(  500, 500, -20, 20, -20, 20,3,3+18),
graph(  500, 500, -20, 20, -20, 20,3,3+20),
graph(  500, 500, -20, 20, -20, 20,3,3+22),
graph(  500, 500, -20, 20, -20, 20,3,3+24),
graph(  500, 500, -20, 20, -20, 20,3,3+26),
graph(  500, 500, -20, 20, -20, 20,3,3+28),
graph(  500, 500, -20, 20, -20, 20,3,3+30),
graph(  500, 500, -20, 20, -20, 20,3,3+32))}}} Graph of {{{y>3}}} with the shaded region in green


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<a name="combo">



Graphing the inequalities together on the same coordinate systems



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Now simply graph the inequalities together. To do this, simply graph each line together and their corresponding region to get (note: use different colors to represent the shaded regions)



{{{drawing( 500, 500, -20, 20, -20, 20,
graph( 500, 500, -20, 20, -20, 20,-(1/2)x-1,-(1/2)x-1+3),
graph( 500, 500, -20, 20, -20, 20,-(1/2)x-1,-(1/2)x-1+6),
graph( 500, 500, -20, 20, -20, 20,-(1/2)x-1,-(1/2)x-1+9),
graph( 500, 500, -20, 20, -20, 20,-(1/2)x-1,-(1/2)x-1+12),
graph( 500, 500, -20, 20, -20, 20,-(1/2)x-1,-(1/2)x-1+15),
graph( 500, 500, -20, 20, -20, 20,-(1/2)x-1,-(1/2)x-1+18),
graph( 500, 500, -20, 20, -20, 20,-(1/2)x-1,-(1/2)x-1+21),
graph( 500, 500, -20, 20, -20, 20,-(1/2)x-1,-(1/2)x-1+24),
graph( 500, 500, -20, 20, -20, 20,-(1/2)x-1,-(1/2)x-1+27),
graph( 500, 500, -20, 20, -20, 20,-(1/2)x-1,-(1/2)x-1+30),
graph( 500, 500, -20, 20, -20, 20,-(1/2)x-1,-(1/2)x-1+33),
graph( 500, 500, -20, 20, -20, 20,-(1/2)x-1,-(1/2)x-1+36),
graph( 500, 500, -20, 20, -20, 20,-(1/2)x-1,-(1/2)x-1+39),
graph( 500, 500, -20, 20, -20, 20,-(1/2)x-1,-(1/2)x-1+42),
graph( 500, 500, -20, 20, -20, 20,-(1/2)x-1,-(1/2)x-1+45),
graph( 500, 500, -20, 20, -20, 20,-(1/2)x-1,-(1/2)x-1+48),
graph( 500, 500, -20, 20, -20, 20,-2x+4,-2x+4+-3),
graph( 500, 500, -20, 20, -20, 20,-2x+4,-2x+4+-6),
graph( 500, 500, -20, 20, -20, 20,-2x+4,-2x+4+-9),
graph( 500, 500, -20, 20, -20, 20,-2x+4,-2x+4+-12),
graph( 500, 500, -20, 20, -20, 20,-2x+4,-2x+4+-15),
graph( 500, 500, -20, 20, -20, 20,-2x+4,-2x+4+-18),
graph( 500, 500, -20, 20, -20, 20,-2x+4,-2x+4+-21),
graph( 500, 500, -20, 20, -20, 20,-2x+4,-2x+4+-24),
graph( 500, 500, -20, 20, -20, 20,-2x+4,-2x+4+-27),
graph( 500, 500, -20, 20, -20, 20,-2x+4,-2x+4+-30),
graph( 500, 500, -20, 20, -20, 20,-2x+4,-2x+4+-33),
graph( 500, 500, -20, 20, -20, 20,-2x+4,-2x+4+-36),
graph( 500, 500, -20, 20, -20, 20,-2x+4,-2x+4+-39),
graph( 500, 500, -20, 20, -20, 20,-2x+4,-2x+4+-42),
graph( 500, 500, -20, 20, -20, 20,-2x+4,-2x+4+-45),
graph( 500, 500, -20, 20, -20, 20,-2x+4,-2x+4+-48),
graph( 500, 500, -20, 20, -20, 20,3,3+3),
graph( 500, 500, -20, 20, -20, 20,3,3+6),
graph( 500, 500, -20, 20, -20, 20,3,3+9),
graph( 500, 500, -20, 20, -20, 20,3,3+12),
graph( 500, 500, -20, 20, -20, 20,3,3+15),
graph( 500, 500, -20, 20, -20, 20,3,3+18),
graph( 500, 500, -20, 20, -20, 20,3,3+21),
graph( 500, 500, -20, 20, -20, 20,3,3+24),
graph( 500, 500, -20, 20, -20, 20,3,3+27),
graph( 500, 500, -20, 20, -20, 20,3,3+30),
graph( 500, 500, -20, 20, -20, 20,3,3+33),
graph( 500, 500, -20, 20, -20, 20,3,3+36),
graph( 500, 500, -20, 20, -20, 20,3,3+39),
graph( 500, 500, -20, 20, -20, 20,3,3+42),
graph( 500, 500, -20, 20, -20, 20,3,3+45),
graph( 500, 500, -20, 20, -20, 20,3,3+48))}}}





Here is a cleaner look at the intersection of regions





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




=============================================================




<a name="solution">


Solution:


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So the solution of the system of inequalities is



{{{drawing( 500, 500, -20, 20, -20, 20,
          graph( 500, 500, -20, 20, -20, 20,-(1/2)x-1,-2x+4,3),circle(-20,10,0.2),
circle(-20,13,0.2),
circle(-20,16,0.2),
circle(-20,19,0.2),
circle(-17,10,0.2),
circle(-17,13,0.2),
circle(-17,16,0.2),
circle(-17,19,0.2),
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(-11,7,0.2),
circle(-11,10,0.2),
circle(-11,13,0.2),
circle(-11,16,0.2),
circle(-11,19,0.2),
circle(-8,4,0.2),
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(-5,4,0.2),
circle(-5,7,0.2),
circle(-5,10,0.2),
circle(-5,13,0.2),
circle(-2,4,0.2),
circle(-2,7,0.2),
circle(-20,10,0.2),
circle(-20,13,0.2),
circle(-20,16,0.2),
circle(-20,19,0.2),
circle(-17,10,0.2),
circle(-17,13,0.2),
circle(-17,16,0.2),
circle(-17,19,0.2),
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(-11,7,0.2),
circle(-11,10,0.2),
circle(-11,13,0.2),
circle(-11,16,0.2),
circle(-11,19,0.2),
circle(-8,4,0.2),
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(-5,4,0.2),
circle(-5,7,0.2),
circle(-5,10,0.2),
circle(-5,13,0.2),
circle(-2,4,0.2),
circle(-2,7,0.2),
circle(-20,10,0.2),
circle(-20,13,0.2),
circle(-20,16,0.2),
circle(-20,19,0.2),
circle(-17,10,0.2),
circle(-17,13,0.2),
circle(-17,16,0.2),
circle(-17,19,0.2),
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(-11,7,0.2),
circle(-11,10,0.2),
circle(-11,13,0.2),
circle(-11,16,0.2),
circle(-11,19,0.2),
circle(-8,4,0.2),
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(-5,4,0.2),
circle(-5,7,0.2),
circle(-5,10,0.2),
circle(-5,13,0.2),
circle(-2,4,0.2),
circle(-2,7,0.2))}}}