Question 146774

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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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{{{y>=-3}}} Start with the 1st inequality.



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



So in order to plot {{{y>=-3}}}, we need to plot the equation {{{y=-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,-3) }}} Graph of {{{y=-3}}}



Now plug in a test point (0,1) into the inequality {{{y>=-3}}}



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



{{{1>=-3}}} Plug in {{{x=0}}} and {{{y=1}}}



{{{1>=-3}}} Evaluate and simplify.



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



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,-3),
graph( 500, 500, -10, 10, -10, 10,-3,-3+0),
graph( 500, 500, -10, 10, -10, 10,-3,-3+2),
graph( 500, 500, -10, 10, -10, 10,-3,-3+4),
graph( 500, 500, -10, 10, -10, 10,-3,-3+6),
graph( 500, 500, -10, 10, -10, 10,-3,-3+8),
graph( 500, 500, -10, 10, -10, 10,-3,-3+10),
graph( 500, 500, -10, 10, -10, 10,-3,-3+12),
graph( 500, 500, -10, 10, -10, 10,-3,-3+14),
graph( 500, 500, -10, 10, -10, 10,-3,-3+16),
graph( 500, 500, -10, 10, -10, 10,-3,-3+18),
graph( 500, 500, -10, 10, -10, 10,-3,-3+20),
graph( 500, 500, -10, 10, -10, 10,-3,-3+22),
graph( 500, 500, -10, 10, -10, 10,-3,-3+24),
graph( 500, 500, -10, 10, -10, 10,-3,-3+26),
graph( 500, 500, -10, 10, -10, 10,-3,-3+28),
graph( 500, 500, -10, 10, -10, 10,-3,-3+30),
graph( 500, 500, -10, 10, -10, 10,-3,-3+32)

 )}}} Graph of {{{y>=-3}}} 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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{{{x<=5}}} Start with the 2nd inequality.



{{{x=5}}} Replace the inequality sign with an equals sign.



So in order to graph {{{x<=5}}}, we need to graph {{{x=5}}} first. So simply draw a vertical line through {{{x=5}}}.



{{{ graph( 500, 500, -10, 10, -10, 10,1000(x-5)) }}} Graph of {{{x=5}}}



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

 )}}} Graph of {{{x<=5}}} 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,-3,1000(x-5)),
 graph( 500, 500, -10, 10, -10, 10,-3,-3+0),
graph( 500, 500, -10, 10, -10, 10,-3,-3+2),
graph( 500, 500, -10, 10, -10, 10,-3,-3+4),
graph( 500, 500, -10, 10, -10, 10,-3,-3+6),
graph( 500, 500, -10, 10, -10, 10,-3,-3+8),
graph( 500, 500, -10, 10, -10, 10,-3,-3+10),
graph( 500, 500, -10, 10, -10, 10,-3,-3+12),
graph( 500, 500, -10, 10, -10, 10,-3,-3+14),
graph( 500, 500, -10, 10, -10, 10,-3,-3+16),
graph( 500, 500, -10, 10, -10, 10,-3,-3+18),
graph( 500, 500, -10, 10, -10, 10,-3,-3+20),
graph( 500, 500, -10, 10, -10, 10,-3,-3+22),
graph( 500, 500, -10, 10, -10, 10,-3,-3+24),
graph( 500, 500, -10, 10, -10, 10,-3,-3+26),
graph( 500, 500, -10, 10, -10, 10,-3,-3+28),
graph( 500, 500, -10, 10, -10, 10,-3,-3+30),
graph( 500, 500, -10, 10, -10, 10,-3,-3+32),
graph( 500, 500, -10, 10, -10, 10,1000(x-5),1000(x-5+0)),
graph( 500, 500, -10, 10, -10, 10,1000(x-5),1000(x-5+1)),
graph( 500, 500, -10, 10, -10, 10,1000(x-5),1000(x-5+2)),
graph( 500, 500, -10, 10, -10, 10,1000(x-5),1000(x-5+3)),
graph( 500, 500, -10, 10, -10, 10,1000(x-5),1000(x-5+4)),
graph( 500, 500, -10, 10, -10, 10,1000(x-5),1000(x-5+5)),
graph( 500, 500, -10, 10, -10, 10,1000(x-5),1000(x-5+6)),
graph( 500, 500, -10, 10, -10, 10,1000(x-5),1000(x-5+7)),
graph( 500, 500, -10, 10, -10, 10,1000(x-5),1000(x-5+8)),
graph( 500, 500, -10, 10, -10, 10,1000(x-5),1000(x-5+9)),
graph( 500, 500, -10, 10, -10, 10,1000(x-5),1000(x-5+10)),
graph( 500, 500, -10, 10, -10, 10,1000(x-5),1000(x-5+11)),
graph( 500, 500, -10, 10, -10, 10,1000(x-5),1000(x-5+12)),
graph( 500, 500, -10, 10, -10, 10,1000(x-5),1000(x-5+13)),
graph( 500, 500, -10, 10, -10, 10,1000(x-5),1000(x-5+14)),
graph( 500, 500, -10, 10, -10, 10,1000(x-5),1000(x-5+15)),
graph( 500, 500, -10, 10, -10, 10,1000(x-5),1000(x-5+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,-3,1000(x-5)),

   circle(-10,-1,0.08),
circle(-10,-1,0.1),
circle(-10,2,0.08),
circle(-10,2,0.1),
circle(-10,5,0.08),
circle(-10,5,0.1),
circle(-10,8,0.08),
circle(-10,8,0.1),
circle(-7,-1,0.08),
circle(-7,-1,0.1),
circle(-7,2,0.08),
circle(-7,2,0.1),
circle(-7,5,0.08),
circle(-7,5,0.1),
circle(-7,8,0.08),
circle(-7,8,0.1),
circle(-4,-1,0.08),
circle(-4,-1,0.1),
circle(-4,2,0.08),
circle(-4,2,0.1),
circle(-4,5,0.08),
circle(-4,5,0.1),
circle(-4,8,0.08),
circle(-4,8,0.1),
circle(-1,-1,0.08),
circle(-1,-1,0.1),
circle(-1,2,0.08),
circle(-1,2,0.1),
circle(-1,5,0.08),
circle(-1,5,0.1),
circle(-1,8,0.08),
circle(-1,8,0.1),
circle(2,-1,0.08),
circle(2,-1,0.1),
circle(2,2,0.08),
circle(2,2,0.1),
circle(2,5,0.08),
circle(2,5,0.1),
circle(2,8,0.08),
circle(2,8,0.1),
circle(5,-1,0.08),
circle(5,-1,0.1),
circle(5,2,0.08),
circle(5,2,0.1),
circle(5,5,0.08),
circle(5,5,0.1),
circle(5,8,0.08),
circle(5,8,0.1)
  )}}}  Graph of the 2 lines and the final solution region represented by a series of dots.