SOLUTION: Solve the system of inequalities by graphing the solution region: { x+2y<_ 3 2x-3y<_6 }

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Question 151526: Solve the system of inequalities by graphing the solution region:
{ x+2y<_ 3
2x-3y<_6 }
Answer by jim_thompson5910(21685) About Me  (Show Source):
You can put this solution on YOUR website!


Jump to Inequality #1
Jump to Inequality #2
Jump to Solution






Inequality #1


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x%2B2y%3C=3 Start with the 1st inequality.


2y%3C=3-x Subtract x from both sides.


y%3C=%283-x%29%2F%282%29 Divide both sides by 2 to isolate y.


y%3C=-%281%2F2%29x%2B3%2F2 Simplify.


So in order to plot x%2B2y%3C=3 or y%3C=-%281%2F2%29x%2B3%2F2 (which is the same thing), we need to plot the equation y=-%281%2F2%29x%2B3%2F2 first.
(note: if you need help with graphing, check out this solver)


+graph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C-%281%2F2%29x%2B3%2F2%29+ Graph of y=-%281%2F2%29x%2B3%2F2


Now plug in a test point (0,0) into the inequality y%3C=-%281%2F2%29x%2B3%2F2


y%3C=-%281%2F2%29x%2B3%2F2 Start with the given inequality.


0%3C=-%281%2F2%29%2A%280%29%2B3%2F2 Plug in x=0 and y=0


0%3C=3%2F2 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 entire region that is below the line.


drawing%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C%0D%0A++++graph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C-%281%2F2%29x%2B3%2F2%29%2C%0D%0Agraph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C-%281%2F2%29x%2B3%2F2%2C-%281%2F2%29x%2B3%2F2-0%29%2C%0D%0Agraph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C-%281%2F2%29x%2B3%2F2%2C-%281%2F2%29x%2B3%2F2-1%29%2C%0D%0Agraph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C-%281%2F2%29x%2B3%2F2%2C-%281%2F2%29x%2B3%2F2-2%29%2C%0D%0Agraph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C-%281%2F2%29x%2B3%2F2%2C-%281%2F2%29x%2B3%2F2-3%29%2C%0D%0Agraph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C-%281%2F2%29x%2B3%2F2%2C-%281%2F2%29x%2B3%2F2-4%29%2C%0D%0Agraph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C-%281%2F2%29x%2B3%2F2%2C-%281%2F2%29x%2B3%2F2-5%29%2C%0D%0Agraph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C-%281%2F2%29x%2B3%2F2%2C-%281%2F2%29x%2B3%2F2-6%29%2C%0D%0Agraph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C-%281%2F2%29x%2B3%2F2%2C-%281%2F2%29x%2B3%2F2-7%29%2C%0D%0Agraph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C-%281%2F2%29x%2B3%2F2%2C-%281%2F2%29x%2B3%2F2-8%29%2C%0D%0Agraph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C-%281%2F2%29x%2B3%2F2%2C-%281%2F2%29x%2B3%2F2-9%29%2C%0D%0Agraph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C-%281%2F2%29x%2B3%2F2%2C-%281%2F2%29x%2B3%2F2-10%29%2C%0D%0Agraph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C-%281%2F2%29x%2B3%2F2%2C-%281%2F2%29x%2B3%2F2-11%29%2C%0D%0Agraph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C-%281%2F2%29x%2B3%2F2%2C-%281%2F2%29x%2B3%2F2-12%29%2C%0D%0Agraph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C-%281%2F2%29x%2B3%2F2%2C-%281%2F2%29x%2B3%2F2-13%29%2C%0D%0Agraph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C-%281%2F2%29x%2B3%2F2%2C-%281%2F2%29x%2B3%2F2-14%29%2C%0D%0Agraph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C-%281%2F2%29x%2B3%2F2%2C-%281%2F2%29x%2B3%2F2-15%29%2C%0D%0Agraph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C-%281%2F2%29x%2B3%2F2%2C-%281%2F2%29x%2B3%2F2-16%29%0D%0A%0D%0A+%29 Graph of y%3C=-%281%2F2%29x%2B3%2F2 with the shaded region in green

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


Now move onto the next inequality





Inequality #2


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2x-3y%3C=6 Start with the 2nd inequality.


-3y%3C=6-2x Subtract 2x from both sides.


y%3E=%286-2x%29%2F%28-3%29 Divide both sides by -3 to isolate y. This will flip the inequality sign.


y%3E=%282%2F3%29x-2 Simplify.


So in order to plot 2x-3y%3C=6 or y%3E=%282%2F3%29x-2 (which is the same thing), we need to plot the equation y=%282%2F3%29x-2 first.
(note: if you need help with graphing, check out this solver)


+graph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C%282%2F3%29x-2%29+ Graph of y=%282%2F3%29x-2


Now plug in a test point (0,0) into the inequality y%3E=%282%2F3%29x-2


y%3E=%282%2F3%29x-2 Start with the given inequality.


0%3E=%282%2F3%29%2A%280%29-2 Plug in x=0 and y=0


0%3E=-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 entire region that is above the line.


drawing%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C%0D%0A++++graph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C%282%2F3%29x-2%29%2C%0D%0Agraph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C%282%2F3%29x-2%2C%282%2F3%29x-2%2B0%29%2C%0D%0Agraph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C%282%2F3%29x-2%2C%282%2F3%29x-2%2B1.33333333333333%29%2C%0D%0Agraph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C%282%2F3%29x-2%2C%282%2F3%29x-2%2B2.66666666666667%29%2C%0D%0Agraph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C%282%2F3%29x-2%2C%282%2F3%29x-2%2B4%29%2C%0D%0Agraph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C%282%2F3%29x-2%2C%282%2F3%29x-2%2B5.33333333333333%29%2C%0D%0Agraph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C%282%2F3%29x-2%2C%282%2F3%29x-2%2B6.66666666666667%29%2C%0D%0Agraph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C%282%2F3%29x-2%2C%282%2F3%29x-2%2B8%29%2C%0D%0Agraph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C%282%2F3%29x-2%2C%282%2F3%29x-2%2B9.33333333333333%29%2C%0D%0Agraph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C%282%2F3%29x-2%2C%282%2F3%29x-2%2B10.6666666666667%29%2C%0D%0Agraph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C%282%2F3%29x-2%2C%282%2F3%29x-2%2B12%29%2C%0D%0Agraph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C%282%2F3%29x-2%2C%282%2F3%29x-2%2B13.3333333333333%29%2C%0D%0Agraph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C%282%2F3%29x-2%2C%282%2F3%29x-2%2B14.6666666666667%29%2C%0D%0Agraph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C%282%2F3%29x-2%2C%282%2F3%29x-2%2B16%29%2C%0D%0Agraph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C%282%2F3%29x-2%2C%282%2F3%29x-2%2B17.3333333333333%29%2C%0D%0Agraph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C%282%2F3%29x-2%2C%282%2F3%29x-2%2B18.6666666666667%29%2C%0D%0Agraph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C%282%2F3%29x-2%2C%282%2F3%29x-2%2B20%29%2C%0D%0Agraph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C%282%2F3%29x-2%2C%282%2F3%29x-2%2B21.3333333333333%29%2C%0D%0Agraph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C%282%2F3%29x-2%2C%282%2F3%29x-2%2B22.6666666666667%29%0D%0A%0D%0A+%29 Graph of y%3E=%282%2F3%29x-2 with the shaded region in green






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Now draw all of the inequalities on the same coordinate system.
drawing%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C%0D%0A+graph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C-%281%2F2%29x%2B3%2F2%2C-%281%2F2%29x%2B3%2F2-0%29%2C%0D%0Agraph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C-%281%2F2%29x%2B3%2F2%2C-%281%2F2%29x%2B3%2F2-1%29%2C%0D%0Agraph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C-%281%2F2%29x%2B3%2F2%2C-%281%2F2%29x%2B3%2F2-2%29%2C%0D%0Agraph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C-%281%2F2%29x%2B3%2F2%2C-%281%2F2%29x%2B3%2F2-3%29%2C%0D%0Agraph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C-%281%2F2%29x%2B3%2F2%2C-%281%2F2%29x%2B3%2F2-4%29%2C%0D%0Agraph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C-%281%2F2%29x%2B3%2F2%2C-%281%2F2%29x%2B3%2F2-5%29%2C%0D%0Agraph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C-%281%2F2%29x%2B3%2F2%2C-%281%2F2%29x%2B3%2F2-6%29%2C%0D%0Agraph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C-%281%2F2%29x%2B3%2F2%2C-%281%2F2%29x%2B3%2F2-7%29%2C%0D%0Agraph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C-%281%2F2%29x%2B3%2F2%2C-%281%2F2%29x%2B3%2F2-8%29%2C%0D%0Agraph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C-%281%2F2%29x%2B3%2F2%2C-%281%2F2%29x%2B3%2F2-9%29%2C%0D%0Agraph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C-%281%2F2%29x%2B3%2F2%2C-%281%2F2%29x%2B3%2F2-10%29%2C%0D%0Agraph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C-%281%2F2%29x%2B3%2F2%2C-%281%2F2%29x%2B3%2F2-11%29%2C%0D%0Agraph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C-%281%2F2%29x%2B3%2F2%2C-%281%2F2%29x%2B3%2F2-12%29%2C%0D%0Agraph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C-%281%2F2%29x%2B3%2F2%2C-%281%2F2%29x%2B3%2F2-13%29%2C%0D%0Agraph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C-%281%2F2%29x%2B3%2F2%2C-%281%2F2%29x%2B3%2F2-14%29%2C%0D%0Agraph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C-%281%2F2%29x%2B3%2F2%2C-%281%2F2%29x%2B3%2F2-15%29%2C%0D%0Agraph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C-%281%2F2%29x%2B3%2F2%2C-%281%2F2%29x%2B3%2F2-16%29%2C%0D%0Agraph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C%282%2F3%29x-2%2C%282%2F3%29x-2%2B0%29%2C%0D%0Agraph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C%282%2F3%29x-2%2C%282%2F3%29x-2%2B1.33333333333333%29%2C%0D%0Agraph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C%282%2F3%29x-2%2C%282%2F3%29x-2%2B2.66666666666667%29%2C%0D%0Agraph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C%282%2F3%29x-2%2C%282%2F3%29x-2%2B4%29%2C%0D%0Agraph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C%282%2F3%29x-2%2C%282%2F3%29x-2%2B5.33333333333333%29%2C%0D%0Agraph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C%282%2F3%29x-2%2C%282%2F3%29x-2%2B6.66666666666667%29%2C%0D%0Agraph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C%282%2F3%29x-2%2C%282%2F3%29x-2%2B8%29%2C%0D%0Agraph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C%282%2F3%29x-2%2C%282%2F3%29x-2%2B9.33333333333333%29%2C%0D%0Agraph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C%282%2F3%29x-2%2C%282%2F3%29x-2%2B10.6666666666667%29%2C%0D%0Agraph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C%282%2F3%29x-2%2C%282%2F3%29x-2%2B12%29%2C%0D%0Agraph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C%282%2F3%29x-2%2C%282%2F3%29x-2%2B13.3333333333333%29%2C%0D%0Agraph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C%282%2F3%29x-2%2C%282%2F3%29x-2%2B14.6666666666667%29%2C%0D%0Agraph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C%282%2F3%29x-2%2C%282%2F3%29x-2%2B16%29%2C%0D%0Agraph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C%282%2F3%29x-2%2C%282%2F3%29x-2%2B17.3333333333333%29%2C%0D%0Agraph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C%282%2F3%29x-2%2C%282%2F3%29x-2%2B18.6666666666667%29%2C%0D%0Agraph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C%282%2F3%29x-2%2C%282%2F3%29x-2%2B20%29%2C%0D%0Agraph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C%282%2F3%29x-2%2C%282%2F3%29x-2%2B21.3333333333333%29%2C%0D%0Agraph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C%282%2F3%29x-2%2C%282%2F3%29x-2%2B22.6666666666667%29%0D%0A+%29 Graph of the 2 inequalities and their shaded regions.




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


---------------------------- Solution --------------------------

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So the solution looks like:

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Graph of the 2 lines and the final solution region represented by a series of dots.