SOLUTION: Dry Boat Works wholesales aluminum boats for $200 and fiberglass boats for $150. Northland Marina wants to order at least $600 worth but no more than $1,200 worth of boats from the

Question 1152496: Dry Boat Works wholesales aluminum boats for $200 and fiberglass boats for $150. Northland Marina wants to order at least $600 worth but no more than $1,200 worth of boats from them. Graph a system of inequalities that will show the possible combinations of the number of aluminum boats (x) and the number of fiberglass boats (y) that can be ordered.

Found 2 solutions by jim_thompson5910, Theo:
Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!

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

"Northland Marina wants to order at least $600 worth "
So which leads to after plugging in

Solve for y

Subtract 200x from both sides

Divide both sides by 150

To graph this, we first graph 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

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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 which turns into

Solve for y

Subtract 200x from both sides

Divide both sides by 150

Graphing 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

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

Graph of in blue and in red

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)

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

So instead of an infinitely wide (slightly rotated) rectangle, we have this trapezoid that forms

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)

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 dollars. This total T value satisfies the inequality (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.

Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
here's the graph.

anything in the area that's not shaded is possible.
also anything on the lines that enclose the non shaded area are posible since the inequality includes equal as well (<= not <)(>= not >).
the points that fit the requirements have been noted.
they are shown in (x,y) format.
the inequality is 600 <= 200x + 150y <= 1200
for example, the point (3,4) is possible since 200 * 3 + 150 * 4 = 600 + 600 = 1200 which is the upper limit of what's possible.
likewise, the point (0,4) is possible since 200 * 0 + 150 * 4 = 600 which is the lower limit possible.
likewise, the point (2,4) is possible since 200 * 2 + 150 * 4 = 400 + 600 = 1000 which is above 600 and less than 1200.
the value of x and y have to be integers since only whole boats can be sold.

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