Question 1152496
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x = number of aluminum boats
y = number of fiberglass boats
x and y cannot be negative, and they must be whole numbers as well


Northland Marina orders x aluminum boats. At $200 each, this will cost them 200x dollars.
Northland Marina orders y fiberglass boats. At $150 each, this will cost them 150y dollars.
Let T be the total amount spent by Northland Marina.
The two subtotals (200x and 150y) would add up to the overall total T
{{{T = 200x+150y}}}


"Northland Marina wants to order at least $600 worth "
So {{{T >= 600}}} which leads to {{{200x+150y >= 600}}} after plugging in {{{T = 200x+150y}}}


Solve for y
{{{200x+150y >= 600}}}


{{{150y >= -200x+600}}} Subtract 200x from both sides


{{{y >= (-200x+600)/150}}} Divide both sides by 150


{{{y >= (-200x)/150+600/150}}}


{{{y >= (-4/3)x+4}}}
To graph this, we first graph {{{y = (-4/3)x+4}}} which passes through the two points (0,4) and (3,0). These are the y and x intercepts respectively. We shade above the boundary line to indicate the solution set. We shade above the boundary line because of the "greater than" portion of the inequality sign. The boundary line is solid (as opposed to dotted or dashed) because we are including it as part of the solution set. This is due to the "or equal to" portion as part of the inequality sign.


Graph of {{{y >= (-4/3)x+4}}}
<img width="35%" src = "https://i.imgur.com/ufpdpQE.png">


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The marina also has a budget of $1200 since it states "Northland Marina wants to order ... no more than $1,200 worth"


This means {{{T <= 1200}}} which turns into {{{200x+150y <= 600}}}


Solve for y
{{{200x+150y <= 1200}}}


{{{150y <= -200x+1200}}} Subtract 200x from both sides


{{{y <= (-200x+1200)/150}}} Divide both sides by 150


{{{y <= (-200x)/150+1200/150}}}


{{{y <= (-4/3)x+8}}}


Graphing {{{y = (-4/3)x+8}}} has a line that passes through (0,8) and (6,0) as the y and x intercepts respectively. We shade below the solid boundary line because the "less than" is part of the inequality sign. The "or equal to" part is still there so this boundary line is solid as well. 


Graph of {{{y <= (-4/3)x+8}}}
<img width="35%" src = "https://i.imgur.com/4GFpWZs.png">


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Graph of {{{y >= (-4/3)x+4}}} in blue and {{{y <= (-4/3)x+8}}} in red
<img width="35%" src = "https://i.imgur.com/HzAG2BC.png">
The red graph overlaps with the blue graph to form the purple region. 


This is an infinitely long rectangle (it has a set height but the width goes on forever in both directions)
<img width="35%" src = "https://i.imgur.com/deB3cGO.png">


Recall that earlier I stated that x and y couldn't be negative. 
This further adds on the restrictions {{{x >= 0}}} and {{{y >= 0}}}


So instead of an infinitely wide (slightly rotated) rectangle, we have this trapezoid that forms
<img width="35%" src = "https://i.imgur.com/2lOwRG7.png">
We only focus on the upper righthand quadrant where x and y are both positive.


In this region is the set of all possible solutions (marked in green)
<img width="35%" src = "https://i.imgur.com/lzvWrGP.png">
 


A point such as (2,3) means that x = 2 aluminum boats are ordered and y = 3 fiberglass boats are ordered. 
That gives a total cost of {{{T = 200x+150y = 200*2+150*3 = 400+450 = 850}}} dollars. This total T value satisfies the inequality {{{600 <= T <= 1200}}} (ie the total cost T = 850 is between 600 and 1200 dollars)


I'll let you check the other points to algebraically confirm they are indeed solutions. I also recommend picking points outside the trapezoid to see how they are not solutions. 
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