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 x = housecleaning hours y = sales job hours
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 x≥0
and 𝑦≥0

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… Think about your intercepts
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… The shaded region are
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: 𝑥≥0 𝑎𝑛𝑑 𝑦≥0

10. Word Problem #4
Graphing… x + y ≤ 13 4x + 7y ≥ 65 𝑥≥0 𝑦≥0

11. Word Problem #4 Graphing… x + y ≤ 13 4x + 7y ≥ 65 𝑥≥0 𝑦≥0
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. 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.
Solutions

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