Algebraic Solutions to Nonlinear Inequalities in Context

A farmer has a rectangular field and wants to plant two types of crops. Crop A requires at least 3 acres and Crop B requires at least 2 acres. If the total area of the field is 10 acres, graph the system of inequalities and determine the feasible region for planting both crops.

• The feasible region for planting both crops is only (3,2) and (7,3).

• The total area required for both crops is 5 acres, with no feasible region.

• The feasible region is defined by the vertices (0,2), (0,8), (5,5), and (10,0).

• The feasible region for planting both crops is defined by the vertices (3,2), (3,7), (8,2), and (0,10) within the constraints of the inequalities.

A company produces two products, P1 and P2. Each product requires different amounts of resources. P1 requires 2 hours of labor and 3 units of material, while P2 requires 4 hours of labor and 1 unit of material. If the company has a maximum of 20 hours of labor and 12 units of material, graph the inequalities and interpret the feasible production combinations.

• The feasible production combinations are represented by the vertices of the intersection of the inequalities, which can be calculated as (0, 12), (10, 0), (6, 0), and (0, 5).

• The feasible production combinations are (4, 4), (0, 8), (6, 3), and (3, 0).

• The feasible production combinations are (0, 12), (5, 5), (10, 0), and (0, 6).

• The feasible production combinations are (5, 0), (0, 10), (8, 2), and (2, 4).

A school is planning a field trip and has a budget of $500. The cost per student for transportation is $10 and for admission is $15. If the number of students is represented by x, write the inequalities for the budget and graph the system to find the maximum number of students that can attend the trip.

• 25

• 15

• 30

• 20

A local gym offers two types of memberships: Basic and Premium. The Basic membership costs $30 per month and the Premium costs $50 per month. If a customer wants to spend no more than $200 a month, graph the inequalities and determine how many of each type of membership can be purchased.

• Only 2 Premium memberships can be purchased.

• A maximum of 5 Basic memberships or 3 Premium memberships.

• A maximum of 6 Basic memberships or 4 Premium memberships, or any combination that satisfies 30x + 50y ≤ 200.

• A maximum of 8 Basic memberships or 2 Premium memberships.

A restaurant offers two types of meal plans: Plan A and Plan B. Plan A costs $25 and includes 3 meals, while Plan B costs $40 and includes 5 meals. If a customer wants to spend no more than $100, graph the system of inequalities and find the combinations of meal plans that fit the budget.

• (3, 0)

• (0, 2), (1, 1), (2, 0)

• (1, 2)

• (0, 3)

A charity event is selling tickets for two types of seating: VIP and General. VIP tickets cost $100 each and General tickets cost $50 each. If the total revenue goal is $2000, graph the inequalities and interpret the possible combinations of ticket sales.

• The area above the line 100x + 50y = 2000 represents valid combinations.

• VIP tickets can only be sold in pairs.

• The total revenue goal is $1000.

• The possible combinations of ticket sales are represented by the area under the line 100x + 50y = 2000 in the first quadrant, where x ≥ 0 and y ≥ 0.

A tech company is developing two products, X and Y. Product X requires 5 hours of programming and 2 hours of testing, while Product Y requires 3 hours of programming and 4 hours of testing. If the total available hours for programming is 40 and for testing is 30, graph the inequalities and find the feasible production schedule.

• Produce 4 units of Product X and 3 units of Product Y

• Produce 2 units of Product X and 5 units of Product Y

• Produce 6 units of Product X and 0 units of Product Y, or 0 units of Product X and 7 units of Product Y.

• Produce 5 units of Product X and 2 units of Product Y

A clothing store sells two types of shirts: casual and formal. Casual shirts cost $20 each and formal shirts cost $30 each. If the store wants to make at least $600 in sales, graph the inequalities and determine the combinations of shirts that can achieve this goal.

• The inequality is 20x + 30y <= 600.

• The store needs to sell at least 30 shirts.

• The combinations of shirts must total exactly $600.

• The combinations of casual and formal shirts that can achieve at least $600 in sales are represented by the inequality 20x + 30y >= 600.

A concert venue has a maximum capacity of 500 people. Tickets are sold for $20 each for general admission and $50 each for VIP seating. If the venue wants to make at least $15,000 in ticket sales, graph the system of inequalities and interpret the possible ticket sales combinations.

• The possible combinations of ticket sales are represented by the shaded area in the graph of the inequalities.

• VIP tickets are sold for $30 each.

• The venue will make $10,000 in ticket sales.

• The maximum capacity is 300 people.

A bakery sells two types of cakes: chocolate and vanilla. Each chocolate cake requires 2 hours of baking and each vanilla cake requires 1 hour. If the bakery has a total of 10 hours available for baking, graph the inequalities and find the combinations of cakes that can be baked within the time limit.

• (1,9)

• (0,12)

• The combinations of cakes that can be baked are: (0,10), (1,8), (2,6), (3,4), (4,2), (5,0).

• (3,5)