6 .6 Systems of Linear Inequalities Word Problems

1. 6.6 Systems of Linear Inequalities Word Problems Use a system of linear inequalities to model a real-life situation.

2. Standard Form of Word Problems • Define variables • Write out the system • Graph the inequalities • Write 2 possible solutions

3. Word Problem #1 You can work a total of no more than 41 hours each week at your two jobs. Housecleaning pays $5 per hour and your sales job pays $8 per hour. You need to earn at least $254 each week to pay your bills. Write a system of inequalities that shows the various numbers of hours you can work at each job. x = housecleaning hours y = sales job hours Hours: x + y ≤ 41 Money: 5x + 8y ≥ 254 Can I work negative hours? Be sure to include and

4. Word Problem #2 Fuel x costs $2 per gallon and fuel y costs $3 per gallon. You have at most $18 to spend on fuel. Write and graph a system of linear inequalities to represent this situation. x = gallons of fuel x y = gallons of fuel y Price: 2x + 3y ≤ 18 Gallons of x: x ≥0 Gallons of y: y ≥ 0

5. Word Problem #2 Graphing… Price: 2x + 3y ≤ 18 Gallons of x: x ≥0 Gallons of y: y ≥ 0 Think about your intercepts when graphing the first inequality

6. Word Problem #2 Graphing… Price: 2x + 3y ≤ 18 Gallons of x: x ≥0 Gallons of y: y ≥ 0 The shaded region are All the possible solutions That satisfy each inequality Possible answers ( 1,1) (2, 4) (0,6)

7. Word Problem #3 A salad contains ham and chicken. There are at most 6 pounds of ham and chicken in the salad. Write and graph a system of inequalities to represent this situation. x = lbs of ham y = lbs of chicken Total Pounds: x + y ≤ 6 Pounds of ham: x ≥ 0 Pounds of chicken: y ≥ 0

8. Word Problem #3 Graphing… Total Pounds: x + y ≤ 6 Pounds of ham: x ≥ 0 Pounds of chicken: y ≥ 0 Remember we cannot have negative lbs!! Possible solutions 1 lb of ham and 1 lb of chicken 3 lbs of ham and 2lbs of chicken

9. Word Problem #4 Mary babysits for $4 per hour. She also works as a tutor for $7 per hour. She is only allowed to work at most 13 hours per week. She wants to make at least $65. Write and graph a system of inequalities to represent this situation. x = hours babysitting y = hours tutoring Hours: x + y ≤ 13 Money: 4x + 7y ≥ 65 Constraints:

10. Word Problem #4 Graphing… x + y ≤ 13 4x + 7y ≥ 65

11. Word Problem #4 Graphing… x + y ≤ 13 4x + 7y ≥ 65 Possible Solutions 2 hours babysitting and 9 hours tutoring 1 hour babysitting and 10 hours tutoring

12. Example 5 In one week, Ed can mow at most 9 times and rake at most 7 times. He charges $20 for mowing and $10 for raking. He needs to make more than $125 in one week. Show and describe all the possible combinations of mowing and raking that Ed can do to meet his goal. List two possible combinations. Earnings per Job ($) Mowing 20 Raking 10

13. Example 5- Continued Step 1 Write a system of inequalities. Let x represent the number of mowing jobs and y represent the number of raking jobs. x ≤ 9 He can do at most 9 mowing jobs. y ≤ 7 He can do at most 7 raking jobs. 20x + 10y > 125 He wants to earn more than $125.

14. Solutions Example 5 Continued Step 2 Graph the system. The graph should be in only the first quadrant because the number of jobs cannot be negative.

15. Example 5 - Continued Step 3 Describe all possible combinations. All possible combinations represented by ordered pairs of whole numbers in the solution region will meet Ed’s requirement of mowing, raking, and earning more than $125 in one week. Answers must be whole numbers because he cannot work a portion of a job. Step 4 List the two possible combinations. Two possible combinations are: 7 mowing and 4 raking jobs 8 mowing and 1 raking jobs