SOLUTION: Please help me solve this word problem: You can spend at most $21 on fruit. Blueberries cost $4 per pound and strawberries cost $3 per pound. You need at least 3 pounds to make

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Question 1059142: Please help me solve this word problem:
You can spend at most $21 on fruit. Blueberries cost $4 per pound and strawberries cost $3 per pound. You need at least 3 pounds to make muffins.
a. Define the variables.
b. Write a system of linear inequalities that represents this situation.
c. Graph the system of linear inequalities.
d. Is it possible to buy 4 pounds of blueberries and 1 pound of straberries in this situation? Justify your answer.
Answer by Theo(13342) About Me  (Show Source):
You can put this solution on YOUR website!
let x equal the pounds of blueberries.
let y = the pounds of strawberries.

blueberries cost 4 dollars a pound.
strawberries cost 3 dollars a pound.
total cost has to be less than 21 dollars.

equation for that is:
4x + 3y <= 21

you need at least 3 pounds of fruit to make muffins.

equation for that is:
x + y >= 3

is it possible to buy 4 pounds of blueberries and 1 pound of strawberries in this scenario?

4 pounds of blueberries costs 4 * 4 = 16 dollars.
1 pound of strawberries costs 3 * 1 = 3 dollars.
total cost is 19 dollars which is less than the maximum amount of money than can be spent.

5 pounds of fruit is greater than the minimum amount of fruit required.

answer is yes.
all the requirements are met.
pounds of fruit need to be greater than 3.
dollars of cost need to be less than 21.

following graph shows this relationship.

$$$

in this graph, the area that is NOT shaded represents the region of feasibility.

since 4x + 3y <= 21 represents the area of the graph that satisfies the requirement, then 4x + 3y >= 21 represents the area of the graph that doesn't satisfy the requirement.
that is the area in the graph that is shaded.
from 4x + 3y >= 21, solve for y to get y >= (21 - 4x) / 3.

since x + y >= 3 represents the area of the graph that satisfies the requirement, then x + y <= 3 represents the area of the graph that doesn't satisfy the requirement.
that is the area in the graph that is shaded.
from x + y <= 3, solve for y to get y <= 3 - x.

two implied requirements are that x and y both have to be greater than or equal to 0.

therefore, x and y <= 0 do not satisfy that requirement and are therefore shaded as well.

the region of feasibility on the graph is represented by the area of the graph that is NOT shaded.

why do i do it this way?
because it's a lot easier to see the region of feasibility when it is not shaded than when it is shaded when i am using software to generate the graph.

if i was generating the graph manually, then i would have just shaded the region of feasibility instead.

in that case, the part that you see not shaded would have been the only part of the graph that was shaded.