1. 3.7. Inequality Word Problems
College Algebra
3.7. Inequality Word Problems

2. Do Now: PLUS In your homework packet #19/20
Translate the following statements into inequalities:
X is at most 14
Y is larger than -2
Y is positive
Two times a number is less than or equal to zero.

3. Homework
Questions?
Comments?
Confusions?
Concerns?
ASK ASK ASK!

4. Today:
We have looked at system of equation word problems– now let’s talk system of INEQUALITY word problems!

5. Generally:
You have two types of inequality equations that show up in word problems
1. Two Variable Inequalities
2. One Variable Inequalities

6. 2 Variable Inequalities:
Generally deal with one of three topics:
1. Time
2. Money
3. A certain amount of materials/Items

7. Example One: Time
2 Variable Time Examples. Write the inequalities represented in each statement and identify the variables used.
It takes 15 minutes to make a batch of chocolate chip cookies and 24 minutes to make a batch of oatmeal raison. You only have three hours of free time total. *Watch the change in units!*
Sally wants at least 5 hours of recorded time between 80 minute CDs and 100 minute tapes.

8. Example Two: Money
2 Variable Money Examples. Write the inequalities represented in each statement and identify the variables used.
It costs $7.00 to buy a scarf and $12.00 to buy a shirt. You only brought $ to the mall with you.
You sell lemonade and ice tea at a stand. The lemonade is sold for $3.00 a glass and the ice tea is sold for $4.00 a glass. You want to make at least $ in all.

9. Example Three: Amount of Material/Items
2 Variable Material/Item Examples. Write the inequalities represented in each statement and identify the variables used.
You have 24 ounces of cocoa butter. Lavender Lotion uses 3 ounces of cocoa butter and Lemon Lotion uses 4 ounces of cocoa butter.
You use glitter and ribbons to make a special ornament. A star ornament uses 1 ounce of glitter and a inch of ribbon and a tree ornament uses 2 ounces of glitter and inches of ribbon. You have 20 inches of ribbon and 16 ounces of glitter total.

10. You Try!
Write the inequalities represented in each statement and identify the variables used.
Ms. Stonebraker has 28 desks in her classroom for all her boy and girl students
You sell $12.00 adult tickets and $7.00 student tickets and you want to make at least $ total but the auditorium you are using only seats 500 people. *Note– there are two inequalities here!*
You sell necklaces and bracelets. Your boss wants you to sell at least 500 items.

11. So…. When is it One Variable?
You use one variable when you are just talking about one thing!

12. Example Four: One Variable Examples:
You want to buy at least five cookies and twelve oranges.
You want to buy between 10 and 20 apples at the market.
You want to buy at least seven tapes and four CDs, but don’t want to buy more than ten tapes.

13. You Try! One Variable Examples
The soccer coach wants to allow at least ten students onto the team, but does not want to exceed twenty students.
You go to the store and want to buy at least five oranges and three apples.

14. Do Now: Identify variables and create inequalities
You go to the store and want to buy at least ten oranges. You also want to buy between three and five cookies, but you don’t want to buy more than twenty items of food.
You have a gift certificate to the store for $ You can buy t-shirts for $8.00 or button down shirts for $13.00.
A club is selling muffins and cookies. A muffin is sold for $1.50 and a cookie is sold for $1.00. The club wants to make at least $ but can only bake as much as 400 food items.

15. Example Five: Let’s combine it all together into one problem!
A carpenter makes bookcases in two different sizes: small and large. It takes 4 hours to make a large bookcase and 2 hours to make a small one. The carpenter can spend only 32 hours per week making bookcases and he must make at least 2 of the large and at least 4 of the small each week. Determine the inequalities in this situation, and identify the variables

16. Example Six:
A company makes two models of light fixtures, A and B, each of which needs to be assembled and packed. The time required to assemble model A is 12 minutes and model B takes 18 minutes. It takes 2 minutes to package model A and 1 minute to package model B. Each week there are available 240 hours of assembly time and 20 hours of packaging time. Determine the inequalities in this situation and identify the variables

17. Example Seven:
A small town is trying to establish a transportation system of large and small vans. It can spend no more than $100,000 purchasing both sizes of vehicles and no more than $ per month for maintenance. The town can purchase a small van for $10,000 and maintain it for $ per month. Alternatively, they can purchase large vans for $20,000, which can be maintained for $50.00 per month. Determine the inequalities in this situation and identify the variables.

18. You Try!
Determine the inequalities and identify the variables in each of the problems below
1) A ski company makes two types of skis and has a fabrication and a finishing department. A pair of downhill skis requires 6 hours to fabricate and 1 hour to finish. A pair of cross-country skis require 4 hours to fabricate and 1 hour to finish. The fabricating department has 108 hours of labor available per day. The finishing department has 24 hours of labor available per day.
2) You are stenciling wooden boxes to sell at a crafts fair. Stenciling takes 2 hours for a small box and 3 hours for a large box. You have no more than 30 hours available but want at least 12 boxes to sell at the fair.
3) A chemical company uses two types of fuel for heating and processing. At least 3,800 gallons total fuel are used each day. The burning of each gallon of #1 fuel leaves a residue of 0.02 pounds of ash and 0.06 pounds of soot. The burning of one gallon of #2 fuel leaves a residue of 0.05 pounds of ash and 0.01 pounds of soot. The factory needs at least 120 pounds of ash and at least 136 pounds of soot each day.

19. Do Now: Determine the system of inequalities:
Charlie goes to the mall with $40.00 in his pocket. He wants to buy at least four items of clothing. Pants cost $10.00 and shirts cost $8.00.
Ms. Stonebraker wants to make at least 12 batches of cookies for her family reunion. Chocolate Chip uses 3 cups of flour and 2 cups of sugar and Peanut Butter uses 5 cups of flour and 3 cups of sugar. Ms. Stonebraker only has 30 cups of flour and 12 cups of sugar.

20. So…..
How do we actually solve these inequality word problems? Well…. To solve inequalities we graph them--- so let’s graph our word problems!

21. Example Eight:
First, let’s get used to graphing a lot of different inequalities on one coordinate plane.

22. Example Nine:
Graph the inequalities on the coordinate plane given.

23. You Try!
Graph the inequalities in the given coordinate planes

24. Do Now: Identify variables, write inequalities
You go the store and want to buy at least five pounds of fruit, but want to buy at least 3 pounds of oranges. Oranges cost $3.00 a pound, apples cost $5.00 a pound and you want to spend less than $30.00 on fruit.
You are making and selling bracelets and necklaces. You can only make 50 items per week. Bracelets use 40 gems and necklaces use 75 gems. You can only get 600 gems per week.

25. Example Ten:
Let’s put it all together: Suppose you want to buy some tapes and CDs. You can afford as many as 10 tapes or 7 CDs. You want at least 4 CDs. Each tape holds 24 minutes of music, and each CD holds an hour of music. You want at least 10 hours of total recorded time between the tapes and CDs.
Write a system of inequalities for this situation, identify the variables, graph the system and then name three possible solutions and what they mean given the context of the problem.

26. Example Ten Graph:

27. Example Eleven:
Write a system of inequalities for this situation, identify the variables, graph the system and then name three possible solutions and what they mean given the context of the problem.
You are going to make and sell bread. A loaf of Irish soda bread is made with 2 cups of flour and ¼ cup of sugar. Cake bread is made with 4 cups of flour and 1 cup of sugar. You have 16 cups of flour and 3 cups of sugar.

28. Example Eleven Graph:

29. Example Twelve:
Write a system of inequalities for this situation, identify the variables, graph the system and then name three possible solutions and what they mean given the context of the problem.
You are making and selling skin lotion. A quart of regular skin lotion contains 2 cups of oil and 1 cup of cocoa butter. A quart of extra rich skin lotion contains 1 cup of oil and 2 cups of cocoa butter. You have 24 cups of oil and 18 cups of cocoa butter.

30. Graph for Example Twelve:

31. You Try #1:
Write a system of inequalities for this situation, identify the variables, graph the system and then name three possible solutions and what they mean given the context of the problem.
A doctor offers 15 minutes for routine office visits and 45 minutes for full physicals. The doctor cannot do more than 10 physicals per day and has 9 hours available for appointments each day.

32. You Try #1 Graph:

33. You Try #2:
Write a system of inequalities for this situation, identify the variables, graph the system and then name three possible solutions and what they mean given the context of the problem.
Mrs. Smith grows peaches and apples. At least 500 peaches and 700 apples must be picked daily to meet minimum demands from the buyers. The workers can pick no more than 1200 apples and 1400 peaches daily. The packaging department can handle a maximum of 2,300 fruit per day.

34. You Try #2 Graph:

35. You Try #3:
Write a system of inequalities for this situation, identify the variables, graph the system and then name three possible solutions and what they mean given the context of the problem.
You are screen-printing T-shirts and sweatshirts to sell at the Polk County Blues Festival and are working with the following constraints.
It takes 10 minutes to make a T-shirt
It takes 20 minutes to make a sweatshirt
You have at most 20 hours to make clothing
Supplies for a T-shirt cost $4.00
Supplies for a sweatshirt cost $20.00.
You can spend no more than $ on supplies.
You want to have at least 50 items to sell at the festival.

36. You Try #3 Graph:

37. Practice Problems Try some on your own/in your table groups
As always don’t hesitate to ask questions if you are confused OR talk to your table-mates. They are your greatest resource!

38. Exit Ticket:
Ms. Stonebraker wants to enroll in graduate school classes. She wants to enroll in education classes and math classes. Math classes meet for a total of 5 hours each week and education classes meet for a total of 4 hours each week. Ms. Stonebraker can only dedicate 20 hours to her classes and she wants to take at most six total classes. She also wants to make sure to take at least two math classes and one education class.