Question 146396
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<a href="#1">Jump to Inequality #1</a>
<a href="#2">Jump to Inequality #2</a>
<a href="#solution">Jump to Solution</a>




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Inequality #1 



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{{{x<=2}}} Start with the 1st inequality.



{{{x=2}}} Replace the inequality sign with an equals sign.



So in order to graph {{{x<=2}}}, we need to graph {{{x=2}}} first. So simply draw a vertical line through {{{x=2}}}.



{{{ graph( 500, 500, -10, 10, -10, 10,1000(x-2)) }}} Graph of {{{x=2}}}



Since the inequality sign is "<=", this tells us to shade to the left.


 {{{drawing( 500, 500, -10, 10, -10, 10,
    graph( 500, 500, -10, 10, -10, 10,1000(x-2)),
graph( 500, 500, -10, 10, -10, 10,1000(x-2),1000(x-2+0)),
graph( 500, 500, -10, 10, -10, 10,1000(x-2),1000(x-2+1)),
graph( 500, 500, -10, 10, -10, 10,1000(x-2),1000(x-2+2)),
graph( 500, 500, -10, 10, -10, 10,1000(x-2),1000(x-2+3)),
graph( 500, 500, -10, 10, -10, 10,1000(x-2),1000(x-2+4)),
graph( 500, 500, -10, 10, -10, 10,1000(x-2),1000(x-2+5)),
graph( 500, 500, -10, 10, -10, 10,1000(x-2),1000(x-2+6)),
graph( 500, 500, -10, 10, -10, 10,1000(x-2),1000(x-2+7)),
graph( 500, 500, -10, 10, -10, 10,1000(x-2),1000(x-2+8)),
graph( 500, 500, -10, 10, -10, 10,1000(x-2),1000(x-2+9)),
graph( 500, 500, -10, 10, -10, 10,1000(x-2),1000(x-2+10)),
graph( 500, 500, -10, 10, -10, 10,1000(x-2),1000(x-2+11)),
graph( 500, 500, -10, 10, -10, 10,1000(x-2),1000(x-2+12)),
graph( 500, 500, -10, 10, -10, 10,1000(x-2),1000(x-2+13)),
graph( 500, 500, -10, 10, -10, 10,1000(x-2),1000(x-2+14)),
graph( 500, 500, -10, 10, -10, 10,1000(x-2),1000(x-2+15)),
graph( 500, 500, -10, 10, -10, 10,1000(x-2),1000(x-2+16))

 )}}} Graph of {{{x<=2}}} with the shaded region in green

  

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Now move onto the next inequality



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Inequality #2 



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{{{y>=-2x+3}}} Start with the 2nd inequality.



{{{y=-2x+3}}} Replace the inequality sign with an equals sign.



So in order to plot {{{y>=-2x+3}}}, we need to plot the equation {{{y=-2x+3}}} first. 
(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, -10, 10, -10, 10,-2x+3) }}} Graph of {{{y=-2x+3}}}



Now plug in a test point (0,0) into the inequality {{{y>-2x+3}}}



{{{y>=-2x+3}}} Start with the given inequality.



{{{0>=-2*(0)+3}}} Plug in {{{x=0}}} and {{{y=0}}}



{{{0>=3}}} Evaluate and simplify.



Since the inequality is false, this means that we shade the entire region that does <b>NOT</b> contain the point (0,0)



In other words, we simply shade the <b>entire</b> region that is above the line.



 {{{drawing( 500, 500, -10, 10, -10, 10,
    graph( 500, 500, -10, 10, -10, 10,-2x+3),
graph( 500, 500, -10, 10, -10, 10,-2x+3,-2x+3+0),
graph( 500, 500, -10, 10, -10, 10,-2x+3,-2x+3+4),
graph( 500, 500, -10, 10, -10, 10,-2x+3,-2x+3+8),
graph( 500, 500, -10, 10, -10, 10,-2x+3,-2x+3+12),
graph( 500, 500, -10, 10, -10, 10,-2x+3,-2x+3+16),
graph( 500, 500, -10, 10, -10, 10,-2x+3,-2x+3+20),
graph( 500, 500, -10, 10, -10, 10,-2x+3,-2x+3+24),
graph( 500, 500, -10, 10, -10, 10,-2x+3,-2x+3+28),
graph( 500, 500, -10, 10, -10, 10,-2x+3,-2x+3+32),
graph( 500, 500, -10, 10, -10, 10,-2x+3,-2x+3+36),
graph( 500, 500, -10, 10, -10, 10,-2x+3,-2x+3+40),
graph( 500, 500, -10, 10, -10, 10,-2x+3,-2x+3+44),
graph( 500, 500, -10, 10, -10, 10,-2x+3,-2x+3+48),
graph( 500, 500, -10, 10, -10, 10,-2x+3,-2x+3+52),
graph( 500, 500, -10, 10, -10, 10,-2x+3,-2x+3+56),
graph( 500, 500, -10, 10, -10, 10,-2x+3,-2x+3+60),
graph( 500, 500, -10, 10, -10, 10,-2x+3,-2x+3+64)

 )}}} Graph of {{{y>=-2x+3}}} with the shaded region in green

 






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 Now draw all of the inequalities on the same coordinate system.

 {{{drawing( 500, 500, -10, 10, -10, 10,
 graph( 500, 500, -10, 10, -10, 10,1000(x-2),-2x+3+0),
graph( 500, 500, -10, 10, -10, 10,-2x+3,-2x+3+0),
graph( 500, 500, -10, 10, -10, 10,-2x+3,-2x+3+4),
graph( 500, 500, -10, 10, -10, 10,-2x+3,-2x+3+8),
graph( 500, 500, -10, 10, -10, 10,-2x+3,-2x+3+12),
graph( 500, 500, -10, 10, -10, 10,-2x+3,-2x+3+16),
graph( 500, 500, -10, 10, -10, 10,-2x+3,-2x+3+20),
graph( 500, 500, -10, 10, -10, 10,-2x+3,-2x+3+24),
graph( 500, 500, -10, 10, -10, 10,-2x+3,-2x+3+28),
graph( 500, 500, -10, 10, -10, 10,-2x+3,-2x+3+32),
graph( 500, 500, -10, 10, -10, 10,-2x+3,-2x+3+36),
graph( 500, 500, -10, 10, -10, 10,-2x+3,-2x+3+40),
graph( 500, 500, -10, 10, -10, 10,-2x+3,-2x+3+44),
graph( 500, 500, -10, 10, -10, 10,-2x+3,-2x+3+48),
graph( 500, 500, -10, 10, -10, 10,-2x+3,-2x+3+52),
graph( 500, 500, -10, 10, -10, 10,-2x+3,-2x+3+56),
graph( 500, 500, -10, 10, -10, 10,-2x+3,-2x+3+60),
graph( 500, 500, -10, 10, -10, 10,-2x+3,-2x+3+64),
graph( 500, 500, -10, 10, -10, 10,1000(x-2),1000(x-2+0)),
graph( 500, 500, -10, 10, -10, 10,1000(x-2),1000(x-2+1)),
graph( 500, 500, -10, 10, -10, 10,1000(x-2),1000(x-2+2)),
graph( 500, 500, -10, 10, -10, 10,1000(x-2),1000(x-2+3)),
graph( 500, 500, -10, 10, -10, 10,1000(x-2),1000(x-2+4)),
graph( 500, 500, -10, 10, -10, 10,1000(x-2),1000(x-2+5)),
graph( 500, 500, -10, 10, -10, 10,1000(x-2),1000(x-2+6)),
graph( 500, 500, -10, 10, -10, 10,1000(x-2),1000(x-2+7)),
graph( 500, 500, -10, 10, -10, 10,1000(x-2),1000(x-2+8)),
graph( 500, 500, -10, 10, -10, 10,1000(x-2),1000(x-2+9)),
graph( 500, 500, -10, 10, -10, 10,1000(x-2),1000(x-2+10)),
graph( 500, 500, -10, 10, -10, 10,1000(x-2),1000(x-2+11)),
graph( 500, 500, -10, 10, -10, 10,1000(x-2),1000(x-2+12)),
graph( 500, 500, -10, 10, -10, 10,1000(x-2),1000(x-2+13)),
graph( 500, 500, -10, 10, -10, 10,1000(x-2),1000(x-2+14)),
graph( 500, 500, -10, 10, -10, 10,1000(x-2),1000(x-2+15)),
graph( 500, 500, -10, 10, -10, 10,1000(x-2),1000(x-2+16))
 )}}} Graph of the 2 inequalities and their shaded regions.



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  The solution region will be the intersecting region of the inequalities.



  ----------------------------  <font size="4" color="red"><b>Solution</b></font>  --------------------------


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 So the solution looks like:


 {{{drawing( 500, 500, -10, 10, -10, 10,
  graph( 500, 500, -10, 10, -10, 10,1000(x-2),-2x+3),

   circle(-2,8,0.08),
circle(-2,8,0.1),
circle(-2,9,0.08),
circle(-2,9,0.1),
circle(-1,6,0.08),
circle(-1,6,0.1),
circle(-1,7,0.08),
circle(-1,7,0.1),
circle(-1,8,0.08),
circle(-1,8,0.1),
circle(-1,9,0.08),
circle(-1,9,0.1),
circle(0,4,0.08),
circle(0,4,0.1),
circle(0,5,0.08),
circle(0,5,0.1),
circle(0,6,0.08),
circle(0,6,0.1),
circle(0,7,0.08),
circle(0,7,0.1),
circle(0,8,0.08),
circle(0,8,0.1),
circle(0,9,0.08),
circle(0,9,0.1),
circle(1,2,0.08),
circle(1,2,0.1),
circle(1,3,0.08),
circle(1,3,0.1),
circle(1,4,0.08),
circle(1,4,0.1),
circle(1,5,0.08),
circle(1,5,0.1),
circle(1,6,0.08),
circle(1,6,0.1),
circle(1,7,0.08),
circle(1,7,0.1),
circle(1,8,0.08),
circle(1,8,0.1),
circle(1,9,0.08),
circle(1,9,0.1)
  )}}}  Graph of the 2 lines and the final solution region represented by a series of dots.