Question 151526

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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+2y<=3}}} Start with the 1st inequality.



{{{2y<=3-x}}} Subtract {{{x}}} from both sides.



{{{y<=(3-x)/(2)}}} Divide both sides by {{{2}}} to isolate {{{y}}}.



{{{y<=-(1/2)x+3/2}}} Simplify.



So in order to plot {{{x+2y<=3}}} or {{{y<=-(1/2)x+3/2}}} (which is the same thing), we need to plot the equation {{{y=-(1/2)x+3/2}}} 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,-(1/2)x+3/2) }}} Graph of {{{y=-(1/2)x+3/2}}}



Now plug in a test point (0,0) into the inequality {{{y<=-(1/2)x+3/2}}}



{{{y<=-(1/2)x+3/2}}} Start with the given inequality.



{{{0<=-(1/2)*(0)+3/2}}} Plug in {{{x=0}}} and {{{y=0}}}



{{{0<=3/2}}} Evaluate and simplify.



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



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



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

 )}}} Graph of {{{y<=-(1/2)x+3/2}}} with the shaded region in green

 

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



Now move onto the next inequality



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Inequality #2 



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{{{2x-3y<=6}}} Start with the 2nd inequality.



{{{-3y<=6-2x}}} Subtract {{{2x}}} from both sides.



{{{y>=(6-2x)/(-3)}}} Divide both sides by {{{-3}}} to isolate {{{y}}}. This will flip the inequality sign.



{{{y>=(2/3)x-2}}} Simplify.



So in order to plot {{{2x-3y<=6}}} or {{{y>=(2/3)x-2}}} (which is the same thing), we need to plot the equation {{{y=(2/3)x-2}}} 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,(2/3)x-2) }}} Graph of {{{y=(2/3)x-2}}}



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



{{{y>=(2/3)x-2}}} Start with the given inequality.



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



{{{0>=-2}}} Evaluate and simplify.



Since the inequality is true, this means that we shade the entire region that contains 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,(2/3)x-2),
graph( 500, 500, -10, 10, -10, 10,(2/3)x-2,(2/3)x-2+0),
graph( 500, 500, -10, 10, -10, 10,(2/3)x-2,(2/3)x-2+1.33333333333333),
graph( 500, 500, -10, 10, -10, 10,(2/3)x-2,(2/3)x-2+2.66666666666667),
graph( 500, 500, -10, 10, -10, 10,(2/3)x-2,(2/3)x-2+4),
graph( 500, 500, -10, 10, -10, 10,(2/3)x-2,(2/3)x-2+5.33333333333333),
graph( 500, 500, -10, 10, -10, 10,(2/3)x-2,(2/3)x-2+6.66666666666667),
graph( 500, 500, -10, 10, -10, 10,(2/3)x-2,(2/3)x-2+8),
graph( 500, 500, -10, 10, -10, 10,(2/3)x-2,(2/3)x-2+9.33333333333333),
graph( 500, 500, -10, 10, -10, 10,(2/3)x-2,(2/3)x-2+10.6666666666667),
graph( 500, 500, -10, 10, -10, 10,(2/3)x-2,(2/3)x-2+12),
graph( 500, 500, -10, 10, -10, 10,(2/3)x-2,(2/3)x-2+13.3333333333333),
graph( 500, 500, -10, 10, -10, 10,(2/3)x-2,(2/3)x-2+14.6666666666667),
graph( 500, 500, -10, 10, -10, 10,(2/3)x-2,(2/3)x-2+16),
graph( 500, 500, -10, 10, -10, 10,(2/3)x-2,(2/3)x-2+17.3333333333333),
graph( 500, 500, -10, 10, -10, 10,(2/3)x-2,(2/3)x-2+18.6666666666667),
graph( 500, 500, -10, 10, -10, 10,(2/3)x-2,(2/3)x-2+20),
graph( 500, 500, -10, 10, -10, 10,(2/3)x-2,(2/3)x-2+21.3333333333333),
graph( 500, 500, -10, 10, -10, 10,(2/3)x-2,(2/3)x-2+22.6666666666667)

 )}}} Graph of {{{y>=(2/3)x-2}}} 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,-(1/2)x+3/2,-(1/2)x+3/2-0),
graph( 500, 500, -10, 10, -10, 10,-(1/2)x+3/2,-(1/2)x+3/2-1),
graph( 500, 500, -10, 10, -10, 10,-(1/2)x+3/2,-(1/2)x+3/2-2),
graph( 500, 500, -10, 10, -10, 10,-(1/2)x+3/2,-(1/2)x+3/2-3),
graph( 500, 500, -10, 10, -10, 10,-(1/2)x+3/2,-(1/2)x+3/2-4),
graph( 500, 500, -10, 10, -10, 10,-(1/2)x+3/2,-(1/2)x+3/2-5),
graph( 500, 500, -10, 10, -10, 10,-(1/2)x+3/2,-(1/2)x+3/2-6),
graph( 500, 500, -10, 10, -10, 10,-(1/2)x+3/2,-(1/2)x+3/2-7),
graph( 500, 500, -10, 10, -10, 10,-(1/2)x+3/2,-(1/2)x+3/2-8),
graph( 500, 500, -10, 10, -10, 10,-(1/2)x+3/2,-(1/2)x+3/2-9),
graph( 500, 500, -10, 10, -10, 10,-(1/2)x+3/2,-(1/2)x+3/2-10),
graph( 500, 500, -10, 10, -10, 10,-(1/2)x+3/2,-(1/2)x+3/2-11),
graph( 500, 500, -10, 10, -10, 10,-(1/2)x+3/2,-(1/2)x+3/2-12),
graph( 500, 500, -10, 10, -10, 10,-(1/2)x+3/2,-(1/2)x+3/2-13),
graph( 500, 500, -10, 10, -10, 10,-(1/2)x+3/2,-(1/2)x+3/2-14),
graph( 500, 500, -10, 10, -10, 10,-(1/2)x+3/2,-(1/2)x+3/2-15),
graph( 500, 500, -10, 10, -10, 10,-(1/2)x+3/2,-(1/2)x+3/2-16),
graph( 500, 500, -10, 10, -10, 10,(2/3)x-2,(2/3)x-2+0),
graph( 500, 500, -10, 10, -10, 10,(2/3)x-2,(2/3)x-2+1.33333333333333),
graph( 500, 500, -10, 10, -10, 10,(2/3)x-2,(2/3)x-2+2.66666666666667),
graph( 500, 500, -10, 10, -10, 10,(2/3)x-2,(2/3)x-2+4),
graph( 500, 500, -10, 10, -10, 10,(2/3)x-2,(2/3)x-2+5.33333333333333),
graph( 500, 500, -10, 10, -10, 10,(2/3)x-2,(2/3)x-2+6.66666666666667),
graph( 500, 500, -10, 10, -10, 10,(2/3)x-2,(2/3)x-2+8),
graph( 500, 500, -10, 10, -10, 10,(2/3)x-2,(2/3)x-2+9.33333333333333),
graph( 500, 500, -10, 10, -10, 10,(2/3)x-2,(2/3)x-2+10.6666666666667),
graph( 500, 500, -10, 10, -10, 10,(2/3)x-2,(2/3)x-2+12),
graph( 500, 500, -10, 10, -10, 10,(2/3)x-2,(2/3)x-2+13.3333333333333),
graph( 500, 500, -10, 10, -10, 10,(2/3)x-2,(2/3)x-2+14.6666666666667),
graph( 500, 500, -10, 10, -10, 10,(2/3)x-2,(2/3)x-2+16),
graph( 500, 500, -10, 10, -10, 10,(2/3)x-2,(2/3)x-2+17.3333333333333),
graph( 500, 500, -10, 10, -10, 10,(2/3)x-2,(2/3)x-2+18.6666666666667),
graph( 500, 500, -10, 10, -10, 10,(2/3)x-2,(2/3)x-2+20),
graph( 500, 500, -10, 10, -10, 10,(2/3)x-2,(2/3)x-2+21.3333333333333),
graph( 500, 500, -10, 10, -10, 10,(2/3)x-2,(2/3)x-2+22.6666666666667)
 )}}} Graph of the 2 inequalities and their shaded regions.



  <a name="solution">


  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-x)/(2),(6-2x)/(-3)),

   circle(-10,-8,0.08),
circle(-10,-8,0.1),
circle(-10,-7,0.08),
circle(-10,-7,0.1),
circle(-10,-6,0.08),
circle(-10,-6,0.1),
circle(-10,-5,0.08),
circle(-10,-5,0.1),
circle(-10,-4,0.08),
circle(-10,-4,0.1),
circle(-10,-3,0.08),
circle(-10,-3,0.1),
circle(-10,-2,0.08),
circle(-10,-2,0.1),
circle(-10,-1,0.08),
circle(-10,-1,0.1),
circle(-10,0,0.08),
circle(-10,0,0.1),
circle(-10,1,0.08),
circle(-10,1,0.1),
circle(-10,2,0.08),
circle(-10,2,0.1),
circle(-10,3,0.08),
circle(-10,3,0.1),
circle(-10,4,0.08),
circle(-10,4,0.1),
circle(-10,5,0.08),
circle(-10,5,0.1),
circle(-10,6,0.08),
circle(-10,6,0.1),
circle(-9,-8,0.08),
circle(-9,-8,0.1),
circle(-9,-7,0.08),
circle(-9,-7,0.1),
circle(-9,-6,0.08),
circle(-9,-6,0.1),
circle(-9,-5,0.08),
circle(-9,-5,0.1),
circle(-9,-4,0.08),
circle(-9,-4,0.1),
circle(-9,-3,0.08),
circle(-9,-3,0.1),
circle(-9,-2,0.08),
circle(-9,-2,0.1),
circle(-9,-1,0.08),
circle(-9,-1,0.1),
circle(-9,0,0.08),
circle(-9,0,0.1),
circle(-9,1,0.08),
circle(-9,1,0.1),
circle(-9,2,0.08),
circle(-9,2,0.1),
circle(-9,3,0.08),
circle(-9,3,0.1),
circle(-9,4,0.08),
circle(-9,4,0.1),
circle(-9,5,0.08),
circle(-9,5,0.1),
circle(-9,6,0.08),
circle(-9,6,0.1),
circle(-8,-7,0.08),
circle(-8,-7,0.1),
circle(-8,-6,0.08),
circle(-8,-6,0.1),
circle(-8,-5,0.08),
circle(-8,-5,0.1),
circle(-8,-4,0.08),
circle(-8,-4,0.1),
circle(-8,-3,0.08),
circle(-8,-3,0.1),
circle(-8,-2,0.08),
circle(-8,-2,0.1),
circle(-8,-1,0.08),
circle(-8,-1,0.1),
circle(-8,0,0.08),
circle(-8,0,0.1),
circle(-8,1,0.08),
circle(-8,1,0.1),
circle(-8,2,0.08),
circle(-8,2,0.1),
circle(-8,3,0.08),
circle(-8,3,0.1),
circle(-8,4,0.08),
circle(-8,4,0.1),
circle(-8,5,0.08),
circle(-8,5,0.1),
circle(-7,-6,0.08),
circle(-7,-6,0.1),
circle(-7,-5,0.08),
circle(-7,-5,0.1),
circle(-7,-4,0.08),
circle(-7,-4,0.1),
circle(-7,-3,0.08),
circle(-7,-3,0.1),
circle(-7,-2,0.08),
circle(-7,-2,0.1),
circle(-7,-1,0.08),
circle(-7,-1,0.1),
circle(-7,0,0.08),
circle(-7,0,0.1),
circle(-7,1,0.08),
circle(-7,1,0.1),
circle(-7,2,0.08),
circle(-7,2,0.1),
circle(-7,3,0.08),
circle(-7,3,0.1),
circle(-7,4,0.08),
circle(-7,4,0.1),
circle(-7,5,0.08),
circle(-7,5,0.1),
circle(-6,-6,0.08),
circle(-6,-6,0.1),
circle(-6,-5,0.08),
circle(-6,-5,0.1),
circle(-6,-4,0.08),
circle(-6,-4,0.1),
circle(-6,-3,0.08),
circle(-6,-3,0.1),
circle(-6,-2,0.08),
circle(-6,-2,0.1),
circle(-6,-1,0.08),
circle(-6,-1,0.1),
circle(-6,0,0.08),
circle(-6,0,0.1),
circle(-6,1,0.08),
circle(-6,1,0.1),
circle(-6,2,0.08),
circle(-6,2,0.1),
circle(-6,3,0.08),
circle(-6,3,0.1),
circle(-6,4,0.08),
circle(-6,4,0.1),
circle(-5,-5,0.08),
circle(-5,-5,0.1),
circle(-5,-4,0.08),
circle(-5,-4,0.1),
circle(-5,-3,0.08),
circle(-5,-3,0.1),
circle(-5,-2,0.08),
circle(-5,-2,0.1),
circle(-5,-1,0.08),
circle(-5,-1,0.1),
circle(-5,0,0.08),
circle(-5,0,0.1),
circle(-5,1,0.08),
circle(-5,1,0.1),
circle(-5,2,0.08),
circle(-5,2,0.1),
circle(-5,3,0.08),
circle(-5,3,0.1),
circle(-5,4,0.08),
circle(-5,4,0.1),
circle(-4,-4,0.08),
circle(-4,-4,0.1),
circle(-4,-3,0.08),
circle(-4,-3,0.1),
circle(-4,-2,0.08),
circle(-4,-2,0.1),
circle(-4,-1,0.08),
circle(-4,-1,0.1),
circle(-4,0,0.08),
circle(-4,0,0.1),
circle(-4,1,0.08),
circle(-4,1,0.1),
circle(-4,2,0.08),
circle(-4,2,0.1),
circle(-4,3,0.08),
circle(-4,3,0.1),
circle(-3,-4,0.08),
circle(-3,-4,0.1),
circle(-3,-3,0.08),
circle(-3,-3,0.1),
circle(-3,-2,0.08),
circle(-3,-2,0.1),
circle(-3,-1,0.08),
circle(-3,-1,0.1),
circle(-3,0,0.08),
circle(-3,0,0.1),
circle(-3,1,0.08),
circle(-3,1,0.1),
circle(-3,2,0.08),
circle(-3,2,0.1),
circle(-3,3,0.08),
circle(-3,3,0.1),
circle(-2,-3,0.08),
circle(-2,-3,0.1),
circle(-2,-2,0.08),
circle(-2,-2,0.1),
circle(-2,-1,0.08),
circle(-2,-1,0.1),
circle(-2,0,0.08),
circle(-2,0,0.1),
circle(-2,1,0.08),
circle(-2,1,0.1),
circle(-2,2,0.08),
circle(-2,2,0.1),
circle(-1,-2,0.08),
circle(-1,-2,0.1),
circle(-1,-1,0.08),
circle(-1,-1,0.1),
circle(-1,0,0.08),
circle(-1,0,0.1),
circle(-1,1,0.08),
circle(-1,1,0.1),
circle(-1,2,0.08),
circle(-1,2,0.1),
circle(0,-2,0.08),
circle(0,-2,0.1),
circle(0,-1,0.08),
circle(0,-1,0.1),
circle(0,0,0.08),
circle(0,0,0.1),
circle(0,1,0.08),
circle(0,1,0.1),
circle(1,-1,0.08),
circle(1,-1,0.1),
circle(1,0,0.08),
circle(1,0,0.1),
circle(1,1,0.08),
circle(1,1,0.1),
circle(2,0,0.08),
circle(2,0,0.1),
circle(3,0,0.08),
circle(3,0,0.1)
  )}}}  Graph of the 2 lines and the final solution region represented by a series of dots.