IM UNIT 2 Equations and Inequalities Teacher Guide

24.4 Scavenger Hunt

Optional: 15 minutes
This optional activity reinforces the idea that the solutions to a system of inequalities can
be effectively represented by a region on the graphs of the inequalities in the system. The
activity is designed to be completed without the use of graphing technology.

Addressing

• HSA-REI.D.12

Launch
Ask students to put away any devices.

Anticipated Misconceptions
Some students may have trouble interpreting the graph of the fourth system, wondering if
a point in either of the shaded regions on the graph could be where an item is hidden. Ask
them to pick a point on the graph and consider whether it satisfies the first inequality, and
then whether it satisfies the second inequality. Remind them that a solution to a system
needs to satisfy both.

Student Task Statement
Members of a high school math club are doing a scavenger hunt. Three items are
hidden in the park, which is a rectangle that measures 50 meters by 20 meters.

• The clues are written as systems of inequalities. One system has no solutions.
• The locations of the items can be narrowed down by solving the systems. A

coordinate plane can be used to describe the solutions.

Can you find the hidden items? Sketch a graph to show where each item could be
hidden.

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Clue 1: Clue 2:
Clue 3: Clue 4:

Student Response Solution to Clue 2:
Solution to Clue 1:

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Solution to Clue 3: Solution to Clue 4:

Are You Ready for More? .
Two non-negative numbers and satisfy

1. Find a second inequality, also using and values greater than or equal to
zero, to make a system of inequalities with exactly one solution.

2. Find as many ways to answer this question as you can.

Student Response

Sample response: .Other responses include and any inequality of the form

, where is negative.

Activity Synthesis

Invite students to share their graphs and strategies for finding the solution regions. In
particular, discuss how they found out which system had no solutions.

Remind students that a system of linear equations has no solutions if the graphs of the
equations are two parallel lines that never intersect. Explain that a system of linear

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inequalities has no solutions if their regions are bound by two parallel lines and the
solution region of each one is on the "outside" of the parallel lines, as is the case with the
last given system.

Lesson Synthesis

To help students make connections between systems of equations and systems of
inequalities, display the following graphs for all to see.

Ask students:

• "How are the two sets of graphs alike?" (They have the same two lines. They can tell

us about the solutions to individual equations or inequalities, as well as the solutions
to systems.)

• "How are they different?" (The first set of graphs show two regions that overlap,

bounded by dotted lines. The second set shows two intersecting lines and the lines
are solid. One set represents the solutions to a system of linear inequalities.)

• "How can we tell the number of solutions from each set of graphs?" (The graphs

representing a system of equations shows one point of intersection, so there is only
one solution. The graphs representing a system of inequalities show one region of
overlap, but there are many points in that region. This means that there are many
solutions.)

24.5 Oh Good, Another Riddle

Cool Down: 5 minutes
Addressing

• HSA-REI.D.12

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Student Task Statement
Here is another riddle:

• The sum of two numbers is less than 2.
• If we subtract the second number from the first, the difference is greater than

1.
What are the two numbers?

1. The riddle can be represented by a system of inequalities. Write an inequality
for each statement.

2. These graphs represent the inequalities in the system.
Which graph represents which
inequality?

3. Name a possible solution to the riddle. Explain or show how you know.

Student Response
1.

2. The region with solid blue shading represents the first clue. The region with line
shading represents the second clue.

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3. Sample response: . The sum is 1.5, which is less than 2. The difference is 1.5,

which is greater than 1. The point is located in the region where the two

shaded regions overlap, which means it is a solution to both inequalities in the

system.

Student Lesson Summary

In this lesson, we used two linear inequalities in two variables to represent the
constraints in a situation. Each pair of inequalities forms a system of inequalities.

A solution to the system is any pair that makes both inequalities true, or any

pair of values that simultaneously meet both constraints in the situation. The

solution to the system is often best represented by a region on a graph.

Suppose there are two numbers, and , and there are two things we know about
them:

• The value of one number is more than double the value of the other.
• The sum of the two numbers is less than 10.

We can represent these constraints with a system of
inequalities.

There are many possible pairs of numbers that meet the first constraint, for
example: 1 and 3, or 4 and 9.

The same can be said about the second constraint, for example: 1 and 3, or 2.4 and
7.5.

The pair and meets both constraints, so it is a solution to the system.

The pair and meets the first constraint but not the second ( is a
is not true.)
true statement, but

Remember that graphing is a great way to show all the possible solutions to an
inequality, so let’s graph the solution region for each inequality.

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Because we are looking for a pair of numbers that meet both constraints or make
both inequalities true at the same time, we want to find points that are in the
solution regions of both graphs.
To do that, we can graph both inequalities on the same
coordinate plane.
The solution set to the system of inequalities is
represented by the region where the two graphs
overlap.

Glossary
• solutions to a system of inequalities
• system of inequalities

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Lesson 24 Practice Problems

1. Problem 1

Statement

Two inequalities are graphed on the
same coordinate plane.

Which region represents the solution
to the system of the two inequalities?

Solution

D

2. Problem 2
Statement

Select all the pairs of and that are solutions to the system of inequalities:

A.
B.
C.
D.
E.

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Solution

["A", "D"]

3. Problem 3

Statement

Jada has $200 to spend on flowers for a school celebration. She decides that
the only flowers that she wants to buy are roses and carnations. Roses cost
$1.45 each and carnations cost $0.65 each. Jada buys enough roses so that
each of the 75 people attending the event can take home at least one rose.

a. Write an inequality to represent the constraint that every person takes
home at least one rose.

b. Write an inequality to represent the cost constraint.

Solution

Sample responses:
a.
b.

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4. Problem 4

Statement

Here are the graphs of the equations
and on the same

coordinate plane.

a. Label each graph with the equation it represents.

b. Identify the region that represents the solution set to . Is the
boundary line a part of the solution? Use a colored pencil or
cross-hatching to shade the region.

c. Identify the region that represents the solution set to . Is the

boundary line a part of the solution? Use a different colored pencil or

cross-hatching to shade the region.

d. Identify a point that is a solution to both and .

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Solution

See graph for solutions to the first three questions.

Sample response on the last question:

5. Problem 5 ?

Statement

Which coordinate pair is a solution to the inequality
A.
B.
C.
D.

Solution

B

(From Unit 2, Lesson 21.)

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6. Problem 6

Statement .

Consider the linear equation

a. The pair is a solution to the equation. Find another pair that is

a solution to the equation.

b. Are and solutions to the inequality ? Explain

how you know.

Solution ,, ,

a. Sample responses:

b. is a solution but is not. Since is a solution to the equation

, it has to be a solution to . is not a solution

because substituting these numbers into the inequality gives , which is

false.

(From Unit 2, Lesson 21.)

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7. Problem 7
Statement

Elena is considering buying bracelets and necklaces as gifts for her friends.
Bracelets cost $3, and necklaces cost $5. She can spend no more than $30 on
the gifts.

a. Write an inequality to represent the number of bracelets, , and the
number of necklaces , she could buy while sticking to her budget.

b. Graph the solutions to the inequality on the coordinate plane.

c. Explain how we could check if the boundary is included or excluded from
the solution set.

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Solution , where represents the number of bracelets and the number of

a.
necklaces.

b. Sample graph:

c. Sample response: The boundary is included because the coordinates of the

points on the line satisfy the inequality. For example, means 0 bracelets

and 6 necklaces, which would cost $30 and is still within Elena's budget.

(From Unit 2, Lesson 22.)

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8. Problem 8

Statement

In physical education class, Mai takes 10 free throws and 10 jump shots. She
earns 1 point for each free throw she makes and 2 points for each jump shot
she makes. The greatest number of points that she can earn is 30.

a. Write an inequality to describe the constraints. Specify what each
variable represents.

b. Name one solution to the inequality and explain what it represents in
that situation.

Solution

Sample responses:

1. , where is the number of free throws Mai makes and is the number
of jump shots that Mai makes.

2. One solution is . That means that Mai makes 5 free throws and 6 jumpshots,

for a total of 17 points.

(From Unit 2, Lesson 23.)

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9. Problem 9

Statement

A rectangle with a width of and a
length of has a perimeter greater
than 100.

Here is a graph that represents this
situation.

a. Write an inequality that represents this situation.

b. Can the rectangle have width of 45 and a length of 10? Explain your
reasoning.

c. Can the rectangle have a width of 30 and a length of 20? Explain your
reasoning.

Solution

Sample responses:

a.

b. Yes, because the point is in the solution region.

c. No, represents a rectangle with a perimeter of exactly 200. This is not a

solution to the inequality because it is on the boundary line, which is not

included in the solution region.

(From Unit 2, Lesson 23.)

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Lesson 25: Solving Problems with Systems of
Linear Inequalities in Two Variables

Goals
• Analyze given information about a situation involving multiple constraints and

determine what additional information is needed to solve problems.

• Given a system of inequalities and their graphs, explain (orally and in writing) how to

tell if a pair of values is a solution to the system.

• Practice writing systems of inequalities in two variables and finding the solution sets

by reasoning or by graphing.

Learning Targets
• I can explain how to tell if a point on the boundary of the graph of the solutions to a

system of inequalities is a solution or not.

Lesson Narrative

In a previous lesson, students learned that the solutions to a system of linear inequalities
can be represented graphically with overlapping regions.

In this lesson, students take a closer look at whether points on the boundary lines of the
system's solution region are included in the solutions. Analyzing graphs and
communicating observations about them require attention to precision (MP6). Students
also apply these insights to solve more challenging contextual problems. This work
involves making sense of the information needed to solve the problems (MP1).

Alignments
Addressing

• HSA-REI.D.12: Graph the solutions to a linear inequality in two variables as a

half-plane (excluding the boundary in the case of a strict inequality), and graph the
solution set to a system of linear inequalities in two variables as the intersection of
the corresponding half-planes.

Building Towards

• HSA-REI.D.12: Graph the solutions to a linear inequality in two variables as a

half-plane (excluding the boundary in the case of a strict inequality), and graph the

Algebra1 Unit 2 Lesson 25 CC BY 2019 by Illustrative Mathematics 1

solution set to a system of linear inequalities in two variables as the intersection of
the corresponding half-planes.

Instructional Routines

• MLR4: Information Gap Cards
• MLR8: Discussion Supports
• Think Pair Share
• Which One Doesn’t Belong?

Required Materials
Pre-printed slips, cut from copies of the blackline master

Student Learning Goals

• Let’s use systems of inequalities to solve some problems.

25.1 Which One Doesn’t Belong: Graphs of
Solutions

Warm Up: 5 minutes
This warm-up prompts students to carefully analyze and compare graphs that represent
linear equations and inequalities. Making comparisons prompts students to think about
the solutions to the equations, inequalities, or systems that are being represented. It also
gives students a reason to use language precisely (MP6).

Arrange students in groups of 2–4. Display the graphs for all to see. Give students 1
minute of quiet think time and then time to share their thinking with their small group. In
their small groups, ask each student to share their reasoning why a particular item does
not belong, and together find at least one reason each item doesn't belong.

Building Towards

• HSA-REI.D.12

Instructional Routines

• Which One Doesn’t Belong?

Algebra1 Unit 2 Lesson 25 CC BY 2019 by Illustrative Mathematics 2

Launch

Arrange students in groups of 2–4. Display the graphs for all to see. Ask students to
indicate when they have noticed one that does not belong and can explain why. Give
students 1 minute of quiet think time and then time to share their thinking with their small
group. In their small groups, tell each student to share their reasoning why a particular
item does not belong and together find at least one reason each item doesn't belong.

Student Task Statement B
Which one doesn’t belong?

A

CD

Student Response
Sample responses:

• A is the only set of graphs that represent a system of equations. It is the only system

with exactly one solution.

• B is the only set of graphs that represent a system of inequalities with no solutions.

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• C is the only set of graphs in which the boundaries are horizontal and vertical lines. It

is the only one representing a system with an infinite number of solutions.

• D is the only graph that does not represent a system. It only represents one

inequality.

Activity Synthesis
Ask each group to share one reason why a particular item does not belong. Record and
display the responses for all to see. After each response, ask the class if they agree or
disagree. Since there is no single correct answer to the question of which one does not
belong, attend to students’ explanations and ensure the reasons given are correct.

During the discussion, ask students to explain the meaning of any terminology they use,
such as infinitely many solutions or boundary line. Also, press students on
unsubstantiated claims.

25.2 Focusing on the Details

10 minutes
Previously, students have learned that any point that is in the overlapping solution regions
of the graphs of two inequalities is a solution to the system formed by those inequalities.
In this activity, students take a closer look at whether points that are on the boundary lines
are solutions to the system.

Addressing

• HSA-REI.D.12

Instructional Routines

• MLR8: Discussion Supports
• Think Pair Share

Launch
Arrange students in groups of 2. Display the system of inequalities and the graphs for all
to see.

Give students a minute of quiet time to think about which region represents the solutions
to each inequality and be prepared to explain how they know. Then, give students another
minute to discuss their thinking with a partner. Follow with a class discussion.

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Students are likely to identify the inequality that each graph represents by considering the

equation of the boundary line. They may relate the solidly shaded region to because

the dashed line is the graph of . Or they may relate the hashed region to the

solutions of because the boundary line has a negative slope and it intersects

the -axis at .

Other students may test some coordinate pairs in each region and see if they make an

inequality true. For example, they may say that all points above the graph of has an

-value that is less than the -value.

If these strategies for connecting the algebraic and graphical representations are not
mentioned by students, bring them up.

Tell students that they will now think about whether certain points on the coordinate
plane are solutions to the system.

Support for Students with Disabilities

Representation: Access for Perception. Prior to independent work, engage the
whole-class in developing a set of directions that displays criteria for checking if points
are a solution. This can be written as a flow chart or as a list. Recommend students
start with the step of plotting the point on their graph. Support them in articulating
criteria that address evaluating shaded regions and boundary lines. Check for
understanding by inviting students to rephrase directions in their own words.
Consider keeping the display of directions visible throughout the activity.
Supports accessibility for: Language; Memory

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Student Task Statement

Here are the graphs of the inequalities in
this system:

Decide whether each point is a solution to the system. Be prepared to explain how
you know.

1.
2.
3.
4.
5.

Student Response

1. No. is in the shaded region containing the solutions to one of the inequalities

but not the other.

2. Yes. is in the region where the solutions to the two inequalities overlap.

3. Yes. is on the solid boundary of because , and within
.
the shaded region representing solutions to

4. No. While is clearly a solution to , it is on the dashed boundary of

. The statement is not true.

5. No, because is not true. Even though it is on the solid boundary of one, it is

on the dashed boundary of the other.

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Are You Ready for More?
Find a system of inequalities with this triangle as its set of solutions.

Student Response or equivalent
,,

Activity Synthesis

Focus the discussion on the points on the boundary lines and how students determined if
they are or are not solutions to the system.

Highlight explanations that state that a solution to a system of linear inequalities must be
a solution to every inequality in the system. If a point on the boundary line is not included
in the solution set of one inequality (so the graph is a dashed line), then it is also not
included in the solution set of the system.

Support for English Language Learners

Speaking: MLR8 Discussion Supports. Use this routine to support whole-class discussion.
For each observation that is shared, ask students to restate what they heard using
precise mathematical language. Consider providing students time to restate what
they hear to a partner before selecting one or two students to share with the class.
Ask the original speaker if their peer was accurately able to restate their thinking. Call
students’ attention to any words or phrases that helped clarify the original statement.
This provides more students with an opportunity to produce language as they
interpret the reasoning of others.
Design Principle(s): Support sense-making

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25.3 Info Gap: Terms of A Team

20 minutes
This info gap activity gives students an opportunity to use their understandings of systems
of linear inequalities to solve problems that involve satisfying multiple constraints
simultaneously.

The info gap structure requires students to make sense of problems and persevere in
solving them (MP1). Students need determine what information is necessary to solve a
problem, ask for the information, and then explain their request. This may take several
rounds of discussion if their first requests do not yield the information they need. It also
allows them to refine the language they use and ask increasingly more precise questions
until they get the information they need (MP6).

Students may approach the problems in different ways, including by guessing and
checking, but the problems can be efficiently solved by writing and solving systems of
linear inequalities.

Every clue given in a data card can be written as an inequality. If is the number of

children on a team and the number of adults, the clue "a team with only adults is not

allowed" can be expressed as , and "a team without adults is allowed" can be

expressed as . But because other membership rules are more restrictive, and

because thinking only about positive values of and is natural in this situation, it is not

essential for these constraints to be represented. (The blank coordinate planes provided

also show only the first quadrant, implying positive solutions.)

In an upcoming lesson, students will look more closely at the inequalities and .

Here is the text of the cards for reference and planning:

Algebra1 Unit 2 Lesson 25 CC BY 2019 by Illustrative Mathematics 8

Addressing

• HSA-REI.D.12

Instructional Routines

• MLR4: Information Gap Cards

Launch
Explain the info gap structure, and consider demonstrating the protocol if students are
unfamiliar with it.

Arrange students in groups of 2. In each group, distribute a problem card to one student
and a data card to the other student. After reviewing their work on the first problem, give
them the cards for a second problem and instruct them to switch roles.

Because this activity was designed to be completed without technology, ask students to
put away any devices.

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Support for English Language Learners

Conversing: This activity uses MLR4 Information Gap to give students a purpose for
discussing information necessary to solve problems that involve satisfying multiple
constraints simultaneously. Display questions or question starters for students who
need a starting point such as: “Can you tell me . . . (specific piece of information)”, and
“Why do you need to know . . . (that piece of information)?"
Design Principle(s): Cultivate Conversation

Support for Students with Disabilities

Engagement: Develop Effort and Persistence. Display or provide students with a physical
copy of the written directions. Check for understanding by inviting students to
rephrase directions in their own words. Keep the display of directions visible
throughout the activity.
Supports accessibility for: Memory; Organization

Anticipated Misconceptions
Students in both roles may wonder if all clues on the data card constitute a “rule.” Those
holding a data card may not know how to respond if or when asked, “What is the first
rule?” or "What is one of the rules?" Those who are asking for information may not know if
what is given counts as a rule. Clarify that a rule should be general enough to include
multiple possibilities, rather than just one specific case. For example, "If there are 3 adults,
there must be at least 6 children" is a specific case, rather than a general constraint.

Student Task Statement
Your teacher will give you either a problem card or a data card. Do not show or
read your card to your partner.

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If your teacher gives you the data card: If your teacher gives you the problem

card:

1. Silently read the information on

your card. 1. Silently read your card and think

2. Ask your partner “What specific about what information you need to
information do you need?” and wait answer the question.

for your partner to ask for 2. Ask your partner for the specific

information. Only give information information that you need.

that is on your card. (Do not figure 3. Explain to your partner how you are
out anything for your partner!) using the information to solve the

3. Before telling your partner the problem.

information, ask “Why do you need 4. When you have enough information,

to know (that piece of information)?” share the problem card with your

4. Read the problem card, and solve partner, and solve the problem

the problem independently. independently.

5. Share the data card, and discuss 5. Read the data card, and discuss
your reasoning. your reasoning.

Pause here so your teacher can review your work. Ask your teacher for a new set of
cards and repeat the activity, trading roles with your partner.

The blank coordinate planes are provided here in case they are useful.

Student Response
Card 1:

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1. No. The combination meets the rule about the total number of members, but does

not meet the rule about the number adults compared to the number of students.

Sample reasoning:
◦ The point
is not in the shaded region of the graph representing the rule

about children-to-adult ratio.

◦ If 6 is substituted into the inequality representing the rule about , is a false

children-to-adult ratio and then evaluated, the inequality,

statement.

2. The maximum number of adults is 5.

Card 2:

1. Each team can have at least 8 people: 6 children and 2 adults. Sample reasoning: The
region where all three graphs overlap represent combinations that meet all three
rules. The lowest numbers of children and adults in the region are 6 and 2.

2. The maximum number of adults on a team is 5.

Activity Synthesis

Invite students who solved the problems by graphing a system of inequalities to share
their graphs and their thinking, or display the following graphs.

Discuss questions such as:

• "Which membership rule does each shaded region on the graphs represent?"
• "Can all the clues be written as an inequality? If so, what are they?"
• "Should we graph them all? Why or why not?"
• "How does the first set of graphs help us answer questions about whether 6 adults

and 8 children are acceptable?"

• "How do they tell us about the maximum allowable number of adults on a team?"
• "How does the second set of graphs help us find the minimum number of team

members and the composition of the team?"

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Situation 1: Situation 2:

Lesson Synthesis

Refer to the constraints in the "Terms of a Team" info gap activity. Keep the graphs of the
solution regions displayed for all to see.

Situation 1: Situation 2:

To reiterate the importance of attending carefully to the boundary lines of the regions, ask
questions such as:

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• "In the graphs of the first system, are points on the two lines—say or

—solutions to the system? How do you know?" (Yes. Points on a solid line are

included. Substituting 14 and 2, or 8 and 4, makes each inequality true.)

• "In this situation, are points on the horizontal axis solutions to the system?" (Yes.

Points on the horizontal axis represent no adults. The rules say that a team with only

children and no adults is allowed.)

• "In the graphs of the second system, are points on the boundaries of the overlapping

triangular region solutions to the system?" (Yes)

25.4 Widgets and Zurls

Cool Down: 5 minutes
Addressing

• HSA-REI.D.12

Launch
Graphing technology should not be used in this cool-down.

Student Task Statement

A factory produces widgets and zurls. The combined number of widgets and zurls
made each day cannot be more than 12. The maximum number of widgets the
factory can produce in a day is 4.

Let be the number of widgets and the number of zurls.

1. Select all the inequalities that represent this situation.

a.

b.

c.

d.

e.

2. Here are graphs of and .

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Complete the graphs (by shading
regions and adjusting line types as
needed) to show all the allowable
numbers of widgets and zurls that the
factory can produce in one day.

3. Does each ordered pair represent an allowable combination of widgets and
zurls produced in one day?

Student Response
1. and
2. See graph.
3. Yes, no, no, yes

Student Lesson Summary

A family has at most $25 to spend on activities at Fun Zone. It costs $10 an hour to
use the trampolines and $5 an hour to use the pool. The family can stay less than 4
hours.

What are some combinations of trampoline time and pool time that the family
could choose given their constraints?

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We could find some combinations by trial and error, but writing a system of
inequalities and graphing the solution would allow us to see all the possible
combinations.

Let represent the time, in hours, on the trampolines and represent the time, in
hours, in the pool.

The constraints can be represented with the system of
inequalities:

Here are graphs of the inequalities in the system.

The solution set to the system is represented by the
region where shaded parts of the two graphs overlap.
Any point in that region is a pair of times that meet
both the time and budget constraints.

The graphs give us a complete picture of the possible
solutions.

• Can the family spend 1 hour on the trampolines and 3 hours in the pool?

No. We can reason that it is because those times add up to 4 hours, and the
family wants to spend less than 4 hours. But we can also see that the point

lies on the dashed line of one graph, so it is not a solution.

• Can the family spend 2 hours on the trampolines and 1.5 hours in the pool?

No. We know that these two times add up to less than 4 hours, but to find out

the cost, we need to calculate , which is 27.5 and is more than

the budget.

It may be easier to know that this combination is not an option by noticing

that the point is in the region with line shading, but not in the region

with solid shading. This means it meets one constraint but not the other.

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Lesson 25 Practice Problems

1. Problem 1

Statement

Jada has pennies and nickels that add up to more than 40 cents. She has
fewer than 20 coins altogether.

a. Write a system of inequalities that represents how many pennies and
nickels that Jada could have.

b. Is it possible that Jada has each of the following combinations of coins? If
so, explain or show how you know. If not, state which constraint—the
amount of money or the number of coins—it does not meet.

i. 15 pennies and 5 nickels

ii. 16 pennies and 2 nickels

iii. 10 pennies and 8 nickels

Solution (or equivalent)

a.

b. i. No. This combination doesn't meet either constraint.

ii. No. This combination doesn't add up to more than 40 cents.

iii. Yes. This combination meet both constraints. When substituted into both
inequalities and evaluated, the results are two true statements.

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2. Problem 2

Statement

A triathlon athlete swims at an average rate 2.4 miles per hour, and bikes at an
average rate of 16.1 miles per hour. At the end of one training session, she has
swum and biked more than 20 miles in total.

The inequality and

this graph represent the relationship

between the hours of swimming, , the

hours of biking, , and the total

distance the athlete could have

traveled in miles.

Mai said, "I'm not sure the graph is right. For example, the point is in

the shaded region, but it's not realistic for an athlete to swim for 10 hours and

bike for 3 hours in a training session! I think triathlon athletes generally train

for no more than 2 hours a day."

a. Write an inequality to represent Mai's last statement.

b. Graph the solution set to your inequality.

c. Determine a possible combination of swimming and biking times that
meet both the distance and the time constraints in this situation.

Algebra1 Unit 2 Lesson 25 CC BY 2019 by Illustrative Mathematics 18

Solution

a.
b. Sample graph:

c. Sample response: 0.5 hour of swimming and 1.5 hours of biking

Algebra1 Unit 2 Lesson 25 CC BY 2019 by Illustrative Mathematics 19

3. Problem 3

Statement

Elena is considering buying bracelets and necklaces as gifts for her friends.
Bracelets cost $3, and necklaces cost $5. She can spend no more than $30 on
the gifts. Elena needs at least 7 gift items.
This graph represents the inequality

, which describes the cost
constraint in this situation.
Let represent the number of
bracelets and the number of
necklaces.

a. Write an inequality that represents the number of gift items that Elena
needs.

b. On the same coordinate plane, graph the solution set to the inequality
you wrote.

c. Use the graphs to find at least two possible combinations of bracelets
and necklaces Elena could buy.

d. Explain how the graphs show that the combination of 2 bracelets and 5
necklaces meet one constraint in the situation but not the other
constraint.

Algebra1 Unit 2 Lesson 25 CC BY 2019 by Illustrative Mathematics 20

Solution

a.

b. See graph.

c. Sample response: 5 bracelets and 3 necklaces, 6
bracelets and 2 necklaces, and 8 necklaces and 1
bracelet

d. Sample response: The point is the solution

region of but not in the solution region of

. The combination meets the

minimum number of items Elena needs, but costs

more than $30.

4. Problem 4
Statement

Algebra1 Unit 2 Lesson 25 CC BY 2019 by Illustrative Mathematics 21

A gardener is buying some topsoil and compost to fill his garden. His budget is
$70. Topsoil costs $1.89 per cubic foot, and compost costs $4.59 per cubic
foot.
Select all statements or representations that correctly describe the
gardener's constraints in this situation. Let represent the cubic feet of topsoil
and the cubic feet of compost.

A. The combination of 7.5 cubic feet of topsoil and 12 cubic feet of compost
is within the gardener's budget.

B. If the line represents the equation
, this graph represents the

solutions to the gardener's budget constraint.

C.
D. The combination of 5 cubic feet of topsoil and 20 cubic feet of compost is

within the gardener's budget.
E.
F. If the line represents the equation

, this graph represents the
solutions to the gardener's budget constraint.

Solution

["A", "E", "F"]

(From Unit 2, Lesson 22.)

Algebra1 Unit 2 Lesson 25 CC BY 2019 by Illustrative Mathematics 22

5. Problem 5

Statement . Write an equation that has:

Priya writes the equation

a. exactly one solution in common with Priya's equation

b. no solutions in common with Priya's equation

c. infinitely many solutions in common with Priya's equation, but looks
different than hers

Solution

a. The equation of any line with a different slope will do. Sample
response:

b. The equation of any line with the same slope but a different -intercept will do.
Sample response:

c. The equation of any line that is a multiple of the original equation or has the

terms rearranged will do. Sample responses: or

(From Unit 2, Lesson 17.)

Algebra1 Unit 2 Lesson 25 CC BY 2019 by Illustrative Mathematics 23

6. Problem 6

Statement

Two inequalities are graphed on the
same coordinate plane.

Which region represents the solution
to the system of the two inequalities?

Solution

C

(From Unit 2, Lesson 24.)

7. Problem 7

Statement

Here is a riddle:

◦ The sum of two numbers is less than 10.
◦ If we subtract the second number from the first, the difference is greater

than 3.
Write a system of inequalities that represents this situation. Let represent
the first number and represent the second number.

Solution

Sample response:

(From Unit 2, Lesson 24.)

Algebra1 Unit 2 Lesson 25 CC BY 2019 by Illustrative Mathematics 24

Lesson 26: Modeling with Systems of
Inequalities in Two Variables

Goals
• Define the constraints in a situation and create a mathematical model to represent

them.

• Interpret a mathematical model, presented as inequalities and graphs, that

represents a situation.

Learning Targets
• I can interpret inequalities and graphs in a mathematical model.

• I know how to choose variables, specify the constraints, and write inequalities to

create a mathematical model.

Lesson Narrative

In this culminating lesson, students integrate the ideas from the unit and engage in
multiple aspects of mathematical modeling (MP4).

In the first activity, they interpret and analyze given models that represent the constraints
and conditions in a situation. In the second activity, they create their own models after
specifying quantities of interest, identifying relevant information, and setting the
constraints.

Alignments
Addressing

• HSA-CED.A.3: Represent constraints by equations or inequalities, and by systems of

equations and/or inequalities, and interpret solutions as viable or nonviable options
in a modeling context. For example, represent inequalities describing nutritional and
cost constraints on combinations of different foods.

• HSA-REI.D.12: Graph the solutions to a linear inequality in two variables as a

half-plane (excluding the boundary in the case of a strict inequality), and graph the
solution set to a system of linear inequalities in two variables as the intersection of
the corresponding half-planes.

• HSN-Q.A.2: Define appropriate quantities for the purpose of descriptive modeling.

Algebra1 Unit 2 Lesson 26 CC BY 2019 by Illustrative Mathematics 1

Instructional Routines

• Aspects of Mathematical Modeling
• MLR2: Collect and Display
• MLR7: Compare and Connect
• Notice and Wonder
• Think Pair Share

Required Materials
Graphing technology
Examples of graphing technology are: a handheld graphing calculator, a computer with a
graphing calculator application installed, and an internet-enabled device with access to a
site like desmos.com/calculator or geogebra.org/graphing. For students using the digital
materials, a separate graphing calculator tool isn't necessary; interactive applets are
embedded throughout, and a graphing calculator tool is accessible on the student digital
toolkit page.

Tools for creating a visual display
Any way for students to create work that can be easily displayed to the class. Examples:
chart paper and markers, whiteboard space and markers, shared online drawing tool,
access to a document camera.

Required Preparation
Acquire devices that can run Desmos (recommended) or other graphing technology. It is
ideal if each student has their own device. (Desmos is available under Math Tools.)

Student Learning Goals

• Let’s create mathematical models using systems of inequalities.

26.1 A Solution to Which Inequalities?

Warm Up: 5 minutes
This warm-up gives students a quick exposure to the inequalities , , , and

, so that they are prepared to deal with them later in this lesson. It also reinforces the
idea of thinking carefully about whether the points on the boundary lines of a solution
region are included in the solution set.

Algebra1 Unit 2 Lesson 26 CC BY 2019 by Illustrative Mathematics 2

Addressing

• HSA-REI.D.12

Student Task Statement

Is the ordered pair a solution to all, some, or none of these inequalities? Be

prepared to explain your reasoning.

Student Response

Some. is a solution to , , and . It is not a solution to .

Activity Synthesis

Invite students to share their responses. Then, display a blank four-quadrant coordinate
plane for all to see.

Ask students:

• "If we were to graph the solutions to , what would the region look like?" (We

would shade the right side of -axis.) "Is the -axis included in the solution region?"

(No)

• "What about the graph of the solutions to ?" (We would shade the upper side of

the -axis). "Is the -axis included in the solution region?" (No)

Algebra1 Unit 2 Lesson 26 CC BY 2019 by Illustrative Mathematics 3

• "What about the graph of the solutions to the system and ?" (The solution

region would be the upper-right section of the graph, where the other two regions

overlap.)

Remind students that this upper-right region of the coordinate plane is called the first
quadrant.

26.2 Custom Trail Mix

20 minutes
In this activity, students use their insights from the unit to analyze and interpret a set of
mathematical models and a set of data in context. Each situation involves more than two
constraints, and can therefore be represented with a system with more than two
inequalities.

Interpreting and connecting the inequalities, the graphs, and the data set (which involves
decimals) prompts students to make sense of problems and persevere in solving them
(MP1), and to reason quantitatively and abstractly (MP2).

Addressing

• HSA-REI.D.12

Instructional Routines

• Aspects of Mathematical Modeling
• MLR2: Collect and Display
• Notice and Wonder
• Think Pair Share

Launch

Give students a moment to skim through the task statement and familiarize themselves
with the given information. Ask them to be prepared to share one thing they notice and
one thing they wonder. Invite students to share their observations and questions.

Then, to help students interpret the variables in the given inequalities as representing the

number of grams of each ingredient, ask them to use the table to write an expression to

represent the total amount of fiber if they had grams of almonds and grams of raisins.

Students should see that the expression is .

Algebra1 Unit 2 Lesson 26 CC BY 2019 by Illustrative Mathematics 4

Next, ask for an expression representing the total amount of sugar for the same amounts

of almonds and raisins ( ).

Arrange students in groups of 2. Ask them to analyze and answer the questions about one
student's trail mix (either Tyler's or Jada's). If time permits, the groups could analyze the
other trail mix.

Give students a few minutes of quiet work time and time to share their thinking with their
partner. Follow with a whole-class discussion.

Support for Students with Disabilities

Representation: Internalize Comprehension. Demonstrate and encourage students to use
color coding and annotations to highlight connections between representations in a
problem. For example, highlight values from the table, equations, and graphs in
corresponding colors as they discover the connections. Encourage students to label
regions of the graph and label variables in addition to color coding to reinforce
connections.
Supports accessibility for: Visual-spatial processing

Student Task Statement
Here is the nutrition information for some trail mix ingredients:

Algebra1 Unit 2 Lesson 26 CC BY 2019 by Illustrative Mathematics 5

calories per protein per sugar per fat per fiber per
gram (kcal) gram (g)
gram (g) gram (g) gram (g)

peanuts 5.36 0.21 0.04 0.46 0.07

almonds 5.71 0.18 0.21 0.46 0.07

raisins 3.00 0.03 0.60 0.00 0.05

chocolate 4.76 0.05 0.67 0.19 0.02
pieces

shredded 6.67 0.07 0.07 0.67 0.13
coconut

sunflower 5.50 0.20 0.03 0.47 0.10
seeds

dried 3.25 0.03 0.68 0.00 0.03
cherries

walnuts 6.43 0.14 0.04 0.61 0.07

Tyler and Jada each designed their own custom trail mix using two of these
ingredients. They wrote inequalities and created graphs to represent
their constraints.

Algebra1 Unit 2 Lesson 26 CC BY 2019 by Illustrative Mathematics 6

Tyler Jada

••

••

••

••

••

Use the inequalities and graphs to answer these questions about each student's
trail mix. Be prepared to explain your reasoning.

1. Which two ingredients did they choose?

2. What do their variables represent?

3. What does each constraint mean?

4. Which graph represents which constraint?

5. Name one possible combination of ingredients for their trail mix.

Algebra1 Unit 2 Lesson 26 CC BY 2019 by Illustrative Mathematics 7

Student Response
Tyler:

1. Chocolate pieces and shredded coconut

2. represents grams of chocolate pieces and represents grams of coconut.

3. The trail mix contains more than 50 grams of ingredients, a maximum of 400 calories,
and less than 30 grams of sugar. The amounts of chocolate pieces and shredded
coconut are both positive.

4. The region with a dashed boundary and sparser line shading represents .

The region with a solid boundary and denser line shading represents

. The region with a dashed boundary and blue shading

represents .

5. Sample response: 20 grams of chocolate pieces and 40 grams of shredded coconut

Jada:

1. Walnuts and dried cherries

2. represents grams of walnuts and represents grams of dried cherries.

3. The trail mix contains more than 50 grams of ingredients, more than 4 grams of
protein, and no more than 15 grams of fat. The amounts of walnuts and dried
cherries are both positive.

4. The region with a dashed boundary and denser line shading represents .

The region with a dashed boundary and sparser line shading represents

. The region with a solid boundary and blue shading represents

.

5. Sample response: 22 grams of walnuts and 50 grams of dried cherries

Activity Synthesis

Focus the discussion on the connections between the graphs and the inequalities, and on

the inequalities and . Ask questions such as:

• “How did you know which ingredients each person used?” (By matching the

coefficients in two of the inequalities to the nutritional values in the table.)

Algebra1 Unit 2 Lesson 26 CC BY 2019 by Illustrative Mathematics 8

• “The table shows the same values for some nutrients. How can you tell which one

Tyler or Jada chose?” (The coefficients of and in one inequality and those in the

other inequality must be for the same two ingredients.)

• “Why do you think Jada and Tyler both included the inequalities and ?”

(There cannot be only one ingredient, so both and must be greater than 0.)

• “How do those inequalities affect the graph of the solution region?” (They limit the

solution region to the first quadrant.)

• “Jada and Tyler each wrote five inequalities. Could all five form a single system?” (Yes)

What does it mean to have a system with five inequalities?" (There are five
constraints that must be met. The solutions to the system satisfy all five constraints
simultaneously.)

Support for English Language Learners

Conversing: MLR2 Collect and Display. As students discuss their strategies, listen for and
collect the language students use to identify and describe the boundary line or
shading in their inequalities. Write the students’ words and phrases on a visual display
and update it throughout the remainder of the lesson. Remind students to borrow
language from the display as needed. This will help students read and use
mathematical language during their partner and whole-group discussions.
Design Principle(s): Optimize output (for explanation); Maximize meta-awareness

26.3 Design Your Own Trail Mix

This activity is designed to give students opportunities to use their understandings from
this unit to perform mathematical modeling.

The trail mix context is familiar from the previous activity, but students are challenged to
choose quantities, determine how to represent them, interpret and reason about them,
and use the model they create to make choices. It also enables students to reflect on their
model and revise it as needed (MP4).

Students are likely to want to use graphing technology, as the nutritional information
involves decimals and the inequalities written would be inconvenient to graph by hand.
This is an opportunity for students to choose tools strategically (MP5).

Algebra1 Unit 2 Lesson 26 CC BY 2019 by Illustrative Mathematics 9

No time estimate is given here because the time would depend on decisions about the
research students do and on the expectations for collaboration and presentation.

Addressing

• HSA-CED.A.3
• HSA-REI.D.12
• HSN-Q.A.2

Instructional Routines

• Aspects of Mathematical Modeling
• MLR7: Compare and Connect

Launch
Arrange students in groups of 2–4. Provide access to Desmos or other graphing
technology.

Explain the expectations for researching nutritional values, for collaboration with group
members, and for presentation of student work. (If each group is presenting one
response, provide each group with tools for creating a visual display. If each student is
presenting a response, give each student tools for creating a visual display.)

Support for Students with Disabilities

Engagement: Develop Effort and Persistence. Provide prompts, reminders, guides, rubrics,
or checklists that focus on increasing the length of on-task orientation in the face of
distractions. For each question, allot a specific amount of work time and display
expectations. Display a countdown timer, along with a bulleted list of what teams or
individuals are to produce to complete a given step. For instance, during question 1,
display a 5 minute countdown timer and the list: “You’re finished if you. . . (1) Have
your two ingredients (2) Have the nutrition information ready.”
Supports accessibility for: Attention; Social-emotional skills

Student Task Statement
It's time to design your own trail mix!

Algebra1 Unit 2 Lesson 26 CC BY 2019 by Illustrative Mathematics 10