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5-5 Solving Linear Inequalities
5.5/5.6 Word Problems with
Inequalitites
Essential Q: How do I use
systems of inequalities to solve
real-world problems?
Holt McDougal Algebra 1
5-5 Solving Linear Inequalities
Example 3: Application
Ada has at most 285 beads to make jewelry. A
necklace requires 40 beads, and a bracelet
requires 15 beads.
Let x represent the number of necklaces and y the
number of bracelets.
Write an inequality. Use ≤ for “at most.”
Holt McDougal Algebra 1
5-5 Solving Linear Inequalities
Example 3a Continued
Necklace
beads
40x
plus
bracelet
beads
is at
most
285
beads.
+
15y
≤
285
Solve the inequality for y.
40x + 15y ≤ 285
–40x
–40x
15y ≤ –40x + 285
Subtract 40x from
both sides.
Divide both sides by 15.
Holt McDougal Algebra 1
5-5 Solving Linear Inequalities
Example 3b
b. Graph the solutions.
Step 1 Since Ada cannot make a
negative amount of jewelry, the
system is graphed only in
Quadrant I. Graph the boundary
line
=
for ≤.
Holt McDougal Algebra 1
. Use a solid line
5-5 Solving Linear Inequalities
Example 3b Continued
b. Graph the solutions.
Step 2 Shade below the line. Ada
can only make whole numbers of
jewelry. All points on or below the
line with whole number
coordinates are the different
combinations of bracelets and
necklaces that Ada can make.
Holt McDougal Algebra 1
5-5 Solving Linear Inequalities
Example 3c
c. Give two combinations of necklaces and
bracelets that Ada could make.
Two different combinations of
jewelry that Ada could make
with 285 beads could be 2
necklaces and 8 bracelets or 5
necklaces and 3 bracelets.
(2, 8)


(5, 3)
Holt McDougal Algebra 1
5-5 Solving Linear Inequalities
Check It Out! Example 3
What if…? Dirk is going to bring two types of
olives to the Honor Society induction and
can spend no more than $6. Green olives
cost $2 per pound and black olives cost
$2.50 per pound.
a. Write a linear inequality to describe the
situation.
b. Graph the solutions.
c. Give two combinations of olives that Dirk could
buy.
Holt McDougal Algebra 1
5-5 Solving Linear Inequalities
Check It Out! Example 3 Continued
Let x represent the number of pounds of green
olives and let y represent the number of pounds of
black olives.
Write an inequality. Use ≤ for “no more than.”
Green
olives
plus
black
olives
2x
+
2.50y
Solve the inequality for y.
2x + 2.50y ≤ 6
–2x
–2x
2.50y ≤ –2x + 6
2.50y ≤ –2x + 6
2.50
2.50
Holt McDougal Algebra 1
is no
more
than
≤
total
cost.
6
Subtract 2x from both
sides.
Divide both sides by
2.50.
5-5 Solving Linear Inequalities
Check It Out! Example 3 Continued
y ≤ –0.80x + 2.4
Step 1 Since Dirk cannot
buy negative amounts of
olive, the system is
graphed only in Quadrant
I. Graph the boundary
line for y = –0.80x + 2.4.
Use a solid line for≤.
Black Olives
b. Graph the solutions.
Green Olives
Holt McDougal Algebra 1
5-5 Solving Linear Inequalities
Check It Out! Example 3 Continued
Two different combinations
of olives that Dirk could
purchase with $6 could be
1 pound of green olives and
1 pound of black olives or
0.5 pound of green olives
and 2 pounds of black
olives.
Black Olives
c. Give two combinations of
olives that Dirk could buy.

(0.5, 2)

(1, 1)
Green Olives
Holt McDougal Algebra 1
5-5 Solving Linear Inequalities
Lesson Quiz: Part I
1. You can spend at most $12.00
for drinks at a picnic. Iced tea
costs $1.50 a gallon, and
lemonade costs $2.00 per
gallon. Write an inequality to
describe the situation. Graph
the solutions, describe
reasonable solutions, and then
give two possible
combinations of drinks you
could buy.
1.50x + 2.00y ≤ 12.00
Holt McDougal Algebra 1
5-5 Solving Linear Inequalities
Lesson Quiz: Part I
1.50x + 2.00y ≤ 12.00
Only whole number solutions are
reasonable. Possible answer:
(2 gal tea, 3 gal lemonade) and
(4 gal tea, 1 gal lemonde)
Holt McDougal Algebra 1
5-5 Solving Linear Inequalities
Example 4: Application
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
Holt McDougal Algebra 1
5-5 Solving Linear Inequalities
Example 4 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
Holt McDougal Algebra 1
He can do at most 9
mowing jobs.
He can do at most 7
raking jobs.
He wants to earn
more than $125.
5-5 Solving Linear Inequalities
Example 4 Continued
Step 2 Graph the system.
The graph should be in only the first quadrant
because the number of jobs cannot be negative.
Solutions
Holt McDougal Algebra 1
5-5 Solving Linear Inequalities
Example 4 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
Holt McDougal Algebra 1
5-5 Solving Linear Inequalities
Caution
An ordered pair solution of the system need not
have whole numbers, but answers to many
application problems may be restricted to whole
numbers.
Holt McDougal Algebra 1
5-5 Solving Linear Inequalities
Check It Out! Example 4
At her party, Alice is serving pepper jack cheese
and cheddar cheese. She wants to have at least
2 pounds of each. Alice wants to spend at most
$20 on cheese. Show and describe all possible
combinations of the two cheeses Alice could
buy. List two possible combinations.
Price per Pound ($)
Pepper Jack
4
Cheddar
2
Holt McDougal Algebra 1
5-5 Solving Linear Inequalities
Check It Out! Example 4 Continued
Step 1 Write a system of inequalities.
Let x represent the pounds of pepper jack
and y represent the pounds of cheddar.
x≥2
y≥2
4x + 2y ≤ 20
Holt McDougal Algebra 1
She wants at least 2
pounds of pepper
She
wants at least 2
jack.
pounds of cheddar.
She wants to spend no
more than $20.
5-5 Solving Linear Inequalities
Check It Out! Example 4 Continued
Step 2 Graph the system.
The graph should be in only the first quadrant
because the amount of cheese cannot be
negative.
Solutions
Holt McDougal Algebra 1
5-5 Solving Linear Inequalities
Step 3 Describe all possible combinations.
All possible combinations within the gray region will
meet Alice’s requirement of at most $20 for cheese
and no less than 2 pounds of either type of cheese.
Answers need not be whole numbers as she can buy
fractions of a pound of cheese.
Step 4 Two possible
combinations are (3, 2)
and (2.5, 4). 3 pepper
jack, 2 cheddar or 2.5
pepper jack, 4 cheddar.
Holt McDougal Algebra 1
5-5 Solving Linear Inequalities
Lesson Quiz: Part II
2. Dee has at most $150 to spend on restocking
dolls and trains at her toy store. Dolls cost $7.50
and trains cost $5.00. Dee needs no more than
10 trains and she needs at least 8 dolls. Show
and describe all possible combinations of dolls
and trains that Dee can buy. List two possible
combinations.
Holt McDougal Algebra 1
5-5 Solving Linear Inequalities
Lesson Quiz: Part II Continued
Reasonable answers must
be whole numbers.
Possible answer:
(12 dolls, 6 trains) and
(16 dolls, 4 trains)
Solutions
Holt McDougal Algebra 1