Systems of Equations and Inequalities

ALGEBRA 1

Task 1

You are starting a new business in which you have decided to sell two products instead of just one. Determine a business you could start and choose two products that you could sell. How much of your own money are you willing to invest in this business to get started? How much will each item cost you to make? How much will you charge for each item?

a. Explain this business (how much of your own money you're willing to spend on the business, what items you're going to sell, costs for each item, sale price for each item, etc.).

1. What are you going to sell?
2. What does each item cost to make or purchase?
3. How much will you sell each item for?
4. How much will you invest in your business?

b. Write an inequality that represents the fact that while making each item, you can't exceed this limit. Be sure to include the cost per item in this inequality.

• What is the inequality for your business plan?

c. Graph your inequality. Be sure to label your graph and shade the appropriate side of the line.

• Enter the link of your graph fromhttps://www.desmos.com

d. Choose a point that falls in the shaded region. Explain what thex-coordinate andy-coordinate represent and the significance in terms of cost of this point falling in the shaded region.

e. Choose a point that falls directly on the line. Explain what thex-coordinate andy-coordinate represent and the significance in terms of cost of this point falling directly on the line.

f. Choose a point that does not fall in the shaded region. Explain what thex-coordinate andy-coordinate represent and the significance in terms of cost of this point falling outside of the shaded region.

Task 2

Consider the total amount you're willing to spend to start your business. After selling your items, you want your total amount earned to be at least three times the amount you originally spent.

a. How much money are you hoping to earn from selling your products?

b. Determine the price you want to sell each item for. Note: You may need to adjust the original prices that you came up with in Task 1.

c. Write an inequality that represents the fact that you want your total earned to be at leastthree times the amount that you originally spent. Be sure to include the price for each item in this inequality.

d. Graph your inequality. Be sure to label your graph and shade the appropriate side of the line. Enter the link of your graph fromhttps://www.desmos.com

e. Choose a point that falls in the shaded region for both this inequality and the inequality you found in Task 1. Explain what the x-coordinate and y-coordinate represent and the significance in terms of both cost and money earned of this point falling in the shaded regions.

f. What is the ideal number of items you should produce and sell?

g. Why?

Task 3

Pick one of your items. You have been contacted by a customer in Alaska who wants to purchase several of these items and wants you to ship the items to her. You have already established the cost per item and you will only charge the customer $5 to ship these items to Alaska. Suppose another company sells the same item but charges half of your price. However, if the customer buys from this company, she will be charged $20 in shipping costs.

a. Write two equations to represent the customer's total cost based on how many items she buys from each of the two sellersyou and the other company.

• Equation for your company:
• Equation for other company:

b. If the customer in Alaska wants to buy 5 items, from whom should she buy? Explain your answer.

c. If the customer in Alaska wants to buy 50 items, from whom should she buy? Explain your answer.

d. Solve the system of equations from part A.

e. What method did you choose to solve the system?

f. Explain what your solution for part D means in terms of the situation.