6.6 Systems of Linear
Inequalities Word Problems
Use a system of linear inequalities
to model a real-life situation.
Standard Form of Word Problems
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Define variables
Write out the system
Graph the inequalities
Write 2 possible solutions
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
Can I work negative hours?
Hours:
x + y ≤ 41
Be sure to include x ≥ 0
Money: 5x + 8y ≥ 254 and 𝑦 ≥ 0
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:
Gallons of x:
Gallons of y:
2x + 3y ≤ 18
x ≥0
y≥0
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
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)
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:
Pounds of ham:
Pounds of chicken:
x+y≤6
x≥0
y≥0
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
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
Word Problem #4
Graphing…
x + y ≤ 13
4x + 7y ≥ 65
𝑥≥0
𝑦≥0
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
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
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
y≤7
20x + 10y > 125
He can do at most 9
mowing jobs.
He can do at most 7
raking jobs.
He wants to earn more
than $125.
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
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