IM UNIT 2 Equations and Inequalities Teacher Guide

-value is greater?” Compare the -values with another value of . Then, ask students, “Is
there a value of that would make the two -values equal? What is that -value?”

Student Task Statement . Let's look at another way to find its
Consider the inequality
solutions.

1. Use graphing technology to graph and on the same
coordinate plane.

2. Use your graphs to answer the following questions:

a. Find the values of and when is 1.

b. What value of makes and equal?

c. For what values of is less than ?

d. For what values of is greater than ?

3. What is the solution to the inequality ? Be prepared to
explain how you know.

Student Response
1. Sample graph:

Algebra1 Unit 2 Lesson 19 CC BY 2019 by Illustrative Mathematics 20

2. a. When is 1, the value of is and the value of is 1.

b. When is equal to 2, the value of is equal to that of .

c. When is greater than 2, the value of is less than that of .

d. When is less than 2, the value of is greater than that of .

3.

Activity Synthesis

Focus the discussion on how this way of solving an inequality in one variable is like and
unlike the strategy of solving a related equation, which students used in an earlier activity.
Discuss questions such as:

• “Previously, we saw that we could solve an inequality like this by first solving a related

equation: . Is the method of graphing similar to that process in any

way?” (It is similar in that we can find the value of that makes the two expressions
equal.)

• “How is the graphing method different?” (Instead of comparing the values of the

expressions by calculation, we can graph and

and see where the two graphs intersect. The

intersection tells us the -value that produces the same -value.)

• “Previously, to find the solutions to , we would test -values that

are greater and less than the solution to and see which one would

make the inequality true. How is the graphing method similar and how is it different?”

Algebra1 Unit 2 Lesson 19 CC BY 2019 by Illustrative Mathematics 21

(It is similar in that we are still comparing the values of the two expressions. It is
different in that the graphs allow us to compare visually and see which graph has a
greater -value on either side of their intersection.)

Lesson Synthesis

To help students synthesize the work in this lesson, display the following prompt for all to

see: "How does solving the equation help with solving the inequality

?"

Ask students to explain in their own words and in writing. If time permits, ask students to
share their response with a partner, and then invite a student or two to share with the
class a particularly clear explanation their partner has written.

19.6 Seeking Solutions

Cool Down: 5 minutes
Addressing

• HSA-REI.B.3

Student Task Statement ? Show
Which graph correctly shows the solution to the inequality
or explain your reasoning.

A

B

C

D

Algebra1 Unit 2 Lesson 19 CC BY 2019 by Illustrative Mathematics 22

Student Response
Graph C. Sample reasoning:

• I tested 2, 3, and 4 for the values of , and 3 and 4 make the inequality true.
• I solved a related equation and then tested a couple of values on either side of the

solution.

Student Lesson Summary

The equation is an equation in one variable. Its solution is any value of

that makes the equation true. Only meets that requirement, so 20 is the only
solution.

The inequality is an inequality in one variable. Any value of that makes the

inequality true is a solution. For instance, 30 and 48 are both solutions because

substituting these values for produces true inequalities. is true, as is

. Because the inequality has a range of values that make it true, we

sometimes refer to all the solutions as the solution set.

One way to find the solutions to an inequality is by reasoning. For example, to find

the solution to , we can reason that if 2 times a value is less than 8, then that

value must be less than 4. So a solution to is any value of that is less than

4.

Another way to find the solutions to is to solve the related equation .

In this case, dividing each side of the equation by 2 gives . This point, where

is 4, is the boundary of the solution to the inequality.

To find out the range of values that make the inequality true, we can try values less
than and greater than 4 in our inequality and see which ones make a true
statement.

Algebra1 Unit 2 Lesson 19 CC BY 2019 by Illustrative Mathematics 23

Let's try some values less than 4: Let's try values greater than 4:

• If , the inequality is or • If , the inequality is or
, which is false.
, which is true.

• If , the inequality is • If , the inequality is
, which is also true. , which is also false.
or or

In general, the inequality is false when is greater than or equal to 4 and true when
is less than 4.

We can represent the solution set to an inequality by writing an inequality, , or

by graphing on a number line. The ray pointing to the left represents all values less

than 4.

Algebra1 Unit 2 Lesson 19 CC BY 2019 by Illustrative Mathematics 24

Lesson 19 Practice Problems

1. Problem 1
Statement

Here is an inequality:

Select all of the values that are a solution to the inequality.
A.
B.
C.
D.
E.
F.
G.

Solution

["A", "B"]

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

2. Problem 2
Statement

Find the solution set to this inequality:

A.
B.
C.
D.

Solution

B

3. Problem 3
Statement

Algebra1 Unit 2 Lesson 19 CC BY 2019 by Illustrative Mathematics 26

Here is an inequality:
What value of will produce equality (or make the two sides equal)?

Solution

4. Problem 4

Statement . First, he solves the equation

Noah is solving the inequality .
and gets

How does the solution to the equation help Noah solve the

inequality ? Explain your reasoning.

Solution

Sample response: It helps Noah solve the inequality because is he knows that
makes the 2 sides of the inequality equal, so he knows that the solution must be
either all the values greater than 6 or all the values less than 6. He can just substitute
a 5 or a 7 into the inequality to see what the correct solution is.

Algebra1 Unit 2 Lesson 19 CC BY 2019 by Illustrative Mathematics 27

5. Problem 5 ?
Statement

Which graph represents the solution to
A.
B.
C.
D.

Solution

D

6. Problem 6
Statement

Algebra1 Unit 2 Lesson 19 CC BY 2019 by Illustrative Mathematics 28

Solve this system of linear equations without graphing:

Solution

(From Unit 2, Lesson 15.)

7. Problem 7
Statement

Kiran has 27 nickels and quarters in his pocket, worth a total of $2.75.
a. Write a system of equations to represent the relationships between the
number of nickels , the number of dimes , and the dollar amount in
this situation.

b. How many nickels and quarters are in Kiran’s pocket? Show your
reasoning.

Algebra1 Unit 2 Lesson 19 CC BY 2019 by Illustrative Mathematics 29

Solution

a.

b. 20 nickels and 7 quarters. Sample reasoning: The graphs of the two equations

intersect at .

(From Unit 2, Lesson 12.)

8. Problem 8

Statement

How many solutions does this system of equations have? Explain how you
know.

Solution

The system of equations has infinitely many solutions. Sample explanations:

◦ Rewriting both equations into slope-intercept form gives the same equation:

. This means the two equations are equivalent.

◦ Multiplying the first equation by 3 gives . Rearranging the second
. The two equations are
equation into standard form also gives

equivalent.

◦ The graphs of the two equations are the exact same line.

(From Unit 2, Lesson 17.)

9. Problem 9

Statement

Algebra1 Unit 2 Lesson 19 CC BY 2019 by Illustrative Mathematics 30

The principal of a school is hosting a small luncheon for her staff. She plans to
prepare two sandwiches for each person. Some staff members offer to
bring salads and beverages.

The principal has a budget of $225 and expects at least 16 people to attend.
Sandwiches cost $3 each.

Select all of the equations and inequalities that could represent the
constraints in the situation, where is number of people attending and is
number of sandwiches.

A.

B.

C.

D.

E.

F.

Solution

["A", "D", "F"]

(From Unit 2, Lesson 18.)

10. Problem 10

Statement

Students at the college are allowed to work on campus no more than 20 hours
per week. The jobs that are available pay different rates, starting from $8.75
an hour. Students can earn a maximum of $320 per week.

Write at least two inequalities that could represent the constraints in this
situation. Be sure to specify what your variables represent.

Algebra1 Unit 2 Lesson 19 CC BY 2019 by Illustrative Mathematics 31

Solution

Sample responses:

◦ , where represents number of hours worked
◦ , where is the hourly rate in dollars
◦

(From Unit 2, Lesson 18.)

Algebra1 Unit 2 Lesson 19 CC BY 2019 by Illustrative Mathematics 32

Lesson 20: Writing and Solving Inequalities in
One Variable

Goals
• Analyze and use the structure in inequalities to determine whether the solution is

greater or less than the solution to a related equation.

• Write and solve inequalities in one variable to represent the constraints in situations

and to solve problems.

Learning Targets
• I can analyze the structure of an inequality in one variable to help determine if the

solution is greater or less than the solution to the related equation.

• I can write and solve inequalities to answer questions about a situation.

Lesson Narrative

This lesson serves two main goals. The first is to prompt students to write and solve
inequalities to answer questions about a situation. They consider not only whether the
inequalities appropriately model the situations, but also whether there are assumptions
that need to be stated and whether the solution sets make sense in context. Along the
way, they practice reasoning quantitatively and abstractly (MP2) and engage in aspects of
mathematical modeling (MP4).

The second goal is to practice finding the solution set to an inequality by reasoning about

its composition and parts. Take for example. We can reason that for 0.5 times

a number to be greater than 10 times the same number, the number must be negative, so

the solution is . Likewise, we can see that all values of are solutions to

because 5 more than a number will always be greater than that number. Students practice

looking for and making use of structure (MP7) as they reason about solutions this way.

The lesson includes an optional activity for practice solving inequalities without a context.

Alignments

Addressing

• HSA-CED.A.1: Create equations and inequalities in one variable and use them to solve

problems. Include equations arising from linear and quadratic functions, and simple
rational and exponential functions.

Algebra1 Unit 2 Lesson 20 CC BY 2019 by Illustrative Mathematics 1

• 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.B.3: Solve linear equations and inequalities in one variable, including

equations with coefficients represented by letters.

Instructional Routines

• Anticipate, Monitor, Select, Sequence, Connect
• Aspects of Mathematical Modeling
• MLR5: Co-Craft Questions
• MLR7: Compare and Connect
• Think Pair Share

Student Learning Goals

• Let’s solve problems by writing and solving inequalities in one variable.

20.1 Dinner for Drama Club

Warm Up: 5 minutes
In this warm-up, students practice writing an inequality to represent a constraint,
reasoning about its solutions, and interpreting the solutions. The work here engages
students in aspects of mathematical modeling (MP4).

To write an inequality, students need to attend carefully to verbal clues so they can
appropriately model the situation. The word "budget," for instance, implies that the exact
amount given or any amount less than it meets a certain constraint, without explicitly
stating this. When thinking about how many pounds of food Kiran can buy, students
should also recognize that the answer involves a range, rather than a single value.

Addressing

• HSA-CED.A.1
• HSA-CED.A.3

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

Instructional Routines

• Aspects of Mathematical Modeling

Student Task Statement
Kiran is getting dinner for his drama club on the evening of their final rehearsal. The
budget for dinner is $60.

Kiran plans to buy some prepared dishes from a supermarket. The prepared dishes
are sold by the pound, at $5.29 a pound. He also plans to buy two large bottles of
sparkling water at $2.49 each.

1. Represent the constraints in the situation mathematically. If you use variables,
specify what each one means.

2. How many pounds of prepared dishes can Kiran buy? Explain or show your
reasoning.

Student Response

1. Sample response: or (or equivalent). is the

pounds of prepared foods Kiran could buy without going over budget.

2. Up to 10.4 pounds, or . Sample reasoning: After removing the cost of

sparkling water, Kiran still has $55.02. Dividing that amount by 5.29 gives 10.4.

Activity Synthesis

Make sure students see one or more inequalities that appropriately model the situation.
Then, focus the discussion on the solution set. Discuss questions such as:

• "What strategy did you use to find the number of pounds of dishes Kiran could buy?"

• "Does Kiran have to buy exactly 10.4 pounds of dishes?" (No.) "Can he buy less? Why

or why not?" (Yes. He can buy any amount as long as the cost of the food doesn't
exceed $55.02. This means he can buy up to 10.4 pounds.)

• "What is the minimum amount he could buy?" (He could buy 0 pounds of food, but it

wouldn't make sense if his goal is to feed the club members.)

Algebra1 Unit 2 Lesson 20 CC BY 2019 by Illustrative Mathematics 3

20.2 Gasoline in the Tank

10 minutes
This activity enables students to use inequalities to solve a problem about a
situation whose constraints might be unfamiliar and in which multiple quantities are
unknown.

To reason about the problem, students need to interpret the descriptions carefully and
consider their assumptions about the situation. To make sense of the situation, some
students may define additional variables or use diagrams, tables, or other representations.
Along the way, they engage in aspects of modeling (MP4).

Monitor for these approaches students may use to find the possible number of hours of
mowing:

• Start with an empty tank and think about the hours of mowing if there is 0, 1, 2, 3,...

gallons of gasoline in the tank. (If the tank is empty, Han could mow 0 hours. If it has
1 gallon, he could mow or 2.5 hours. And so on, up to 5 gallons.)

• Write a series of inequalities and equations (for instance, , where is the

capacity of the tank in gallons, , and ) and then solve .

• Start with a full tank and the maximum hours of mowing and work down to an empty

tank. (If the tank has 5 gallons, Han could mow hours. If it has 0 gallons, then Han

could mow 0 hours.)

Many students are likely to say that represents all possible hours of mowing.

Look for students who also specify that must be positive or who also write (or

) for the solution set. Ask them to share their thinking with the class during

discussion.

Addressing

• HSA-CED.A.1

• HSA-CED.A.3

Instructional Routines

• Anticipate, Monitor, Select, Sequence, Connect

• Aspects of Mathematical Modeling

• MLR5: Co-Craft Questions

Algebra1 Unit 2 Lesson 20 CC BY 2019 by Illustrative Mathematics 4

Launch

Support for English Language Learners

Speaking, Reading: MLR5 Co-Craft Questions. Use this routine to increase awareness of
language used to talk about solutions of inequalities. Display only the context of this
problem without revealing the questions that follow. Give students 1–2 minutes to
write their own mathematical questions about the situation, then invite them to share
their questions with a partner. Listen for and amplify any questions that involve
possible constraints. Once students compare their questions, reveal the remainder of
the task.
Design Principle(s): Maximize meta-awareness; Support sense-making

Support for Students with Disabilities

Representation: Internalize Comprehension. Represent the same information through
different modalities by using a diagram. If students are unsure where to begin,
suggest that they draw a diagram of a 5-gallon tank and an unknown quantity of gas.
Encourage students to label the diagram with their own interpretations of the
significant elements. Look for phrases that indicate students are fully internalizing
comprehension of the the diagram and exploring how to apply the information
mathematically. For example, phrases that lend themselves to translation into
mathematical symbols such as “there is at most 5 gallons.”
Supports accessibility for: Conceptual processing; Visual-spatial processing

Anticipated Misconceptions

For students struggling to express the value of as an inequality, suggest they first try
reasoning about the question and finding some possible hours of mowing if Han had, say,
1 gallon or 2 gallons of gasoline.

One likely incorrect answer is , the result of multiplying gallons by 0.4 gallons per

hour. If students make this mistake, ask, “About how long can the mower mow with one

gallon of gas?” Then, ask if it is reasonable that the mower mows for 2 hours with 5 gallons

of gas.

Algebra1 Unit 2 Lesson 20 CC BY 2019 by Illustrative Mathematics 5

Student Task Statement
Han is about to mow some lawns in his
neighborhood. His lawn mower has a
5-gallon fuel tank, but Han is not sure
how much gasoline is in the tank.

He knows, however, that the lawn mower
uses 0.4 gallon of gasoline per hour of
mowing.

What are all the possible values for , the number of hours Han can mow without
refilling the lawn mower?

Write one or more inequalities to represent your response. Be prepared to explain
or show your reasoning.

Student Response

or , and . Sample explanation: If the tank is completely full, Han can

mow for at most 12.5 hours without refilling. If it is all the way empty, he can mow for 0
hours. It is not possible for Han to mow for a negative number of hours.

Activity Synthesis

Select previously identified students to share their responses, in the order as listed in the
Activity Narrative. Also invite students who explicitly stated that must be positive to
explain why they included that lower boundary in their response.

Emphasize that although it is probably understood from the context that the amount of

gasoline in the tank cannot be negative and that the lowest possible number of hours of

mowing is 0, it is not apparent from a mathematical statement like . Writing

(or and ) makes this assumption explicit and leaves less room

for misinterpretation.

20.3 Different Ways of Solving

20 minutes

Algebra1 Unit 2 Lesson 20 CC BY 2019 by Illustrative Mathematics 6

This activity highlights some ways to decide whether the solution set to an inequality is
greater than or less than a particular boundary value (identified by solving a related
equation).

Students are presented with two approaches of solving an inequality. Both characters
(Priya and Andre) took similar steps to solve the related equation, which gave them the
same boundary value. But they took different paths to decide the direction of the
inequality symbol.

One solution path involves testing a value on both sides of the boundary value, and
another involves analyzing the structure of the inequality. The former is more familiar
given previous work. The latter is less familiar and thus encourages students to make
sense of problems and persevere in solving them (MP1). It is also a great opportunity for
students to practice looking for and making use of structure (MP7).

Expect students to need some support in reasoning structurally, as it is something that
takes time and practice to develop.

Addressing

• HSA-REI.B.3

Instructional Routines

• MLR7: Compare and Connect

• Think Pair Share

Launch

Arrange students in groups of 2. Give students a few minutes of quiet time to make sense
of what Andre and Priya have done, and then time to discuss their thinking with
their partner. Pause for a class discussion before students move to the second set of
questions and try to solve inequalities using Andre and Priya's methods.

Select students to share their analyses of Andre and Priya's work. Make sure students can

follow Priya's reasoning and understand how Priya decided on by focusing her

comparison on and .

Algebra1 Unit 2 Lesson 20 CC BY 2019 by Illustrative Mathematics 7

Support for Students with Disabilities

Representation: Develop Language and Symbols. Display or provide copies of Priya’s work
and Andre’s work on separate displays. Write Priya and Andre’s descriptive phrases
(as given on the student-facing materials) in contrasting colors/texts, so students can
more easily sort and process information. Some students may benefit from time to
read and interpret each strategy one at a time.
Supports accessibility for: Conceptual processing; Memory

Anticipated Misconceptions

Students who perform procedural steps on the inequality may find incorrect answers. For

instance, in the third inequality, they may divide each side by -9 and arrive at the incorrect

solution . Encourage these students to check their work by substituting numbers

into the original inequality.

Student Task Statement

Andre and Priya used different strategies to solve the following inequality but
reached the same solution.

1. Make sense of each strategy until you can explain what each student has
done.

Algebra1 Unit 2 Lesson 20 CC BY 2019 by Illustrative Mathematics 8

Andre Priya

Testing to see if is a solution: In , there is on the
left and on the right.

If is a negative number, could

be positive or negative, but will

always be positive.

The inequality is false, so 4 is not a

solution. If a number greater than 3 is For to be true, must

not a solution, the solution must be include negative numbers or must be

less than 3, or . less than 3.

2. Here are four inequalities.

a. .

b.

c.

d.

Work with a partner to decide on at least two inequalities to solve. Solve one
inequality using Andre's strategy (by testing values on either side the given
solution), while your partner uses Priya's strategy (by reasoning about the
parts of the inequality). Switch strategies for the other inequality.

Student Response
1. No response required.

2. a. . Sample reasoning (after solving to get ):

Algebra1 Unit 2 Lesson 20 CC BY 2019 by Illustrative Mathematics 9

▪ Using Andre's strategy: Test a value more than or less than -50, and check

to see if that value gives a true statement when substituting back to the
original inequality.

▪ Using Priya's strategy: If is a negative number that is far away from 0, say

-100, then will be less than, not greater than, -10. This means the

solutions must include positive numbers, so the solutions must be greater
than -50.

b. . Sample reasoning using Priya's strategy (after solving

and getting ):
▪ After dividing each side by 4, the left side of the inequality has while the

right side has . This means the expression on the left side will be less

than that on the right when includes positive numbers, so must be

greater than or equal to -1. (In other words, for most negative values of ,

will be less than , which would make the inequality untrue.)

c. . Sample reasoning using Priya's strategy (after solving to get

): will be smaller than 36, so

▪ If includes large positive numbers, then

must include positive numbers which are greater than -4.

▪ If is less than -4, then it would include negative numbers like -10, which

means the value of would be a positive number that is greater than 36,
which makes the inequality untrue. So the solution must be greater than
-4.

d. . Sample reasoning using Priya's strategy (after solving and

getting ):
▪ The right side of the inequality has
while the left side has . If is a

large positive number, will be less than . If is a negative

number, will be positive and greater than , so the solution must

include negative numbers which are less than 6.

Are You Ready for More?

Using positive integers between 1 and 9 and each positive integer at most once, fill

in values to get two constraints so that is the only integer that will satisfy both

constraints at the same time.

Algebra1 Unit 2 Lesson 20 CC BY 2019 by Illustrative Mathematics 10

Student Response ,
Sample response:

Activity Synthesis

Students should recognize that the solution set for each inequality should be the same
regardless of the reasoning method used.

Select as many students as time permits to share how they used a strategy similar to
Priya's to determine the solution set of each inequality. There may be more than one way
to reason structurally about a solution set. Invite students who reason in different ways to
share their thinking. Record and display their thinking for all to see.

Support for English Language Learners

Representing, Conversing: MLR7 Compare and Connect. Use this routine to prepare
students for the whole-class discussion. At the appropriate time, invite students to
create a visual display showing the two inequalities they selected to solve. Students
should consider how to display their work so that another student can interpret what
they see. Encourage students to add notes or details to their displays to help
communicate their thinking. Begin the whole-class discussion by selecting and
arranging 2–4 displays for all to see. Give students 2–3 minutes of quiet think time to
interpret the displays before inviting the authors to present their work.
Design Principle(s): Optimize output; Cultivate conversation

20.4 Matching Inequalities and Solutions

Optional: 15 minutes
This optional activity gives students additional practice in reasoning about the solutions to
inequalities without a context. Students can match the inequalities and solutions in a
variety of ways—by testing different values, by solving a related equation and then testing
values on either side of that solution, by reasoning about the parts of an inequality or its
structure, or by graphing each side of an inequality as a linear function.

Algebra1 Unit 2 Lesson 20 CC BY 2019 by Illustrative Mathematics 11

For all of the inequalities, once students find the boundary value by solving a related
equation, the range of the solutions can be determined by analyzing the structure of the
inequality. Monitor for students who do so and ask them to share their reasoning during
class discussion later.
As students work, also make note of any common challenges or errors so they could be
addressed.
Addressing

• HSA-REI.B.3

Student Task Statement
Match each inequality to a graph that represents its solutions. Be prepared to
explain or show your reasoning.

1. A

2. B

C

3.

D

4.

E

5.

F

6.

Student Response

1. Graph F (for )

Algebra1 Unit 2 Lesson 20 CC BY 2019 by Illustrative Mathematics 12

2. Graph E (for )
3. Graph B (for )

4. Graph D (for )

5. Graph A (for )

6. Graph C (for )

Activity Synthesis

Select students who used different strategies—especially those who made use of the
structure of the inequalities—to share their thinking. If no students found solution sets by
thinking about the features of the inequalities, demonstrate the reasoning process with
one or two examples. For instance:

• : After finding as the solution to , we can reason that for to

be less than , must include negative numbers, so the solution must be .

• : After finding as the solution to the related equation

, we can reason that as gets smaller, is also going to get

smaller. For that expression to be greater than -1, will have be to greater than .

If time permits, ask students to choose a different inequality and try reasoning this way
about its solution.

Lesson Synthesis

Display the following inequalities. Remind students that they are examples of inequalities
written and solved in the lesson.

and

Next, tell students that they will see some statements about writing and solving
inequalities. Their job is to decide whether they agree or disagree wtih each statement and
be prepared to defend their response. (One way to collect their responses is by asking
them to give a discrete hand signal.)

Display the following statements—one at a time—for all to see. After students indicate
their agreement or disagreement, select a student from each camp to explain their

Algebra1 Unit 2 Lesson 20 CC BY 2019 by Illustrative Mathematics 13

reasoning. Then invite others who are not convinced by the reasoning to offer a
counterexample or an alternative view.

• We can usually represent the constraints in a situation with a single inequality.

(Disagree. Sometimes multiple inequalities are needed, depending on what's
happening in the situation.)

• The only way to check the solutions to an inequality is to see if they make sense in a

situation. (Disagree. We can also check by substituting some values in the solution set
back into the inequality to see if they make the inequality a true statement.)

• The only way to find the solutions to an inequality in one variable is by testing

different values for the variable and seeing which ones work. (Disagree. We can also
reason about the solutions in context, solve a related equation and test a higher and
lower value, or use the structure of the inequality.)

• To express the solutions to an inequality in one variable, we always use the same

inequality symbol as in the original inequality. (Disagree. There are different ways to
express a solution set. The symbol we use would depend on how we reason about
the solutions. For example, or both represent the solution set
to .)

Make sure students see that there are reasons for disagreeing with each of these
statements and can articulate some of the reasons.

20.5 How Many Hours of Work?

Cool Down: 5 minutes
Addressing

• HSA-CED.A.1

• HSA-CED.A.3

• HSA-REI.B.3

Student Task Statement
Lin’s job pays $8.25 an hour plus $10 of transportation allowance each week. She
has to work at least 5 hours a week to keep the job, and can earn up to $175 per
week (including the allowance).

Algebra1 Unit 2 Lesson 20 CC BY 2019 by Illustrative Mathematics 14

1. Represent this situation mathematically. If you use variables, specify what
each one means.

2. How many hours per week can Lin work? Explain or show your reasoning.

Student Response and , where represents the number of

1. Sample response:
hours Lin works in a week.

2. At least 5 hours and at most 20 hours (or ). Sample reasoning: The

maximum amount she could earn, not including the transportation allowance, is

$165. That amount is equal to 20 hours of work ( ).

Student Lesson Summary

Writing and solving inequalities can help us make sense of the constraints in a
situation and solve problems. Let's look at an example.

Clare would like to buy a video game that costs $130. She has saved $48 so far and
plans on saving $5 of her allowance each week. How many weeks, , will it be until
she has enough money to buy the game? To represent the constraints, we can write

. Let’s reason about the solutions:

• Because Clare has $48 already and needs to have at least $130 to afford the

game, she needs to save at least $82 more.

Algebra1 Unit 2 Lesson 20 CC BY 2019 by Illustrative Mathematics 15

• If she saves $5 each week, it will take at least weeks to reach $82.

• is 16.4. Any time shorter than 16.4 weeks won't allow her to save enough.

• Assuming she saves $5 at the end of each week (instead of saving smaller

amounts throughout a week), it will be at least 17 weeks before she can

afford the game.

We can also solve by writing and solving a related equation to find the boundary
value for , and then determine whether the solutions are less than or greater than
that value.

• Substituting 16.4 for in the original inequality gives a

true statement. (When , we get .)

• Substituting a value greater than 16.4 for also gives

a true statement. (When , we get .)

• Substituting a value less than 16.4 for gives a false

statement. (When , we get .)

• The solution set is therefore .

Sometimes the structure of an inequality can help us see whether the solutions are

less than or greater than a boundary value. For example, to find the solutions to

, we can solve the equation , which gives us . Then, instead of

testing values on either side of 0, we could reason as follows about the inequality:

• If is a positive value, then would be less than .

• For to be greater than , must include negative values.

• For the solutions to include negative values, they must be less than 0, so the

solution set would be .

Algebra1 Unit 2 Lesson 20 CC BY 2019 by Illustrative Mathematics 16

Lesson 20 Practice Problems

1. Problem 1

Statement

Solve . Explain how to find the solution set.

Solution

. Sample explanation. I divided both sides of by 2 then checked in

the original inequality. Since is true I knew that the solution is and

not .

2. Problem 2

Statement

LIn is solving the inequality . She knows the solution to the
or is the solution to the
equation is

How can Lin determine whether
inequality?

Solution

Sample response: Lin can substitute a number greater than 1 into the inequality and

see if it is true. If it is true, then is the solution, if it is not true, is the

solution.

Algebra1 Unit 2 Lesson 20 CC BY 2019 by Illustrative Mathematics 17

3. Problem 3

Statement

A cell phone company offers two texting plans. People who use plan A pay 10
cents for each text sent or received. People who use plan B pay 12 dollars per
month, and then pay an additional 2 cents for each text sent or received.

a. Write an inequality to represent the fact that it is cheaper for someone to
use plan A than plan B. Use to represent the number of texts they send.

b. Solve the inequality.

Solution or equivalent

a.
b.

4. Problem 4

Statement .

Clare made an error when solving

Describe the error that she made.

Algebra1 Unit 2 Lesson 20 CC BY 2019 by Illustrative Mathematics 18

Solution

Clare's error is in the last step. Sample explanation: She correctly found the value of

at which the left side of the inequality is equal to the right side of the inequality. That

value is -5. However, she identified the wrong side of -5 for the solution set. When is

less than -5, for instance, when , substituting the value into the inequality

makes the statement false: is 24, which is greater than 20, not less than 20.

5. Problem 5

Statement

Diego’s goal is to walk more than 70,000 steps this week. The mean number of
steps that Diego walked during the first 4 days of this week is 8,019.

a. Write an inequality that expresses the mean number of steps that Diego
needs to walk during the last 3 days of this week to walk more than
70,000 steps. Remember to define any variables that you use.

b. If the mean number of steps Diego walks during the last 3 days of the
week is 12,642, will Diego reach his goal of walking more that 70,000
steps this week?

Solution

Sample responses.

a. , where is the mean number of steps that Diego walks
during the last 3 days of the week.

b. Yes, because which is more than 70,000.

Algebra1 Unit 2 Lesson 20 CC BY 2019 by Illustrative Mathematics 19

6. Problem 6

Statement

Here are statistics for the length of some frog jumps in inches:

◦ the mean is 41 inches
◦ the median is 39 inches
◦ the standard deviation is about 9.6 inches
◦ the IQR is 5.5 inches

How does each statistic change if the length of the jumps are measured in feet
instead of inches?

Solution

Because all the data is the same, the statistics for center and variability remain the
same, except that now they will be given in feet instead of inches. There are 12 inches
in a foot or feet in an inch, so we need to divide all the statistics by 12, giving:

◦ the mean is about 3.4 feet ( )

◦ the median is about 3.3 feet ( )

◦ the standard deviation is about 0.8 feet ( )

◦ the IQR is about 0.46 feet ( )

(From Unit 1, Lesson 15.)

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

7. Problem 7
Statement

Solve this system of linear equations without graphing:

Solution

. Sample reasoning: The first equation can be rearranged to .

Adding the second equation to this equation gives or . Substituting -3
or .
for in the first equation gives , so

(From Unit 2, Lesson 15.)

8. Problem 8

Statement

Solve each system of equations without graphing.

a.

b.

Algebra1 Unit 2 Lesson 20 CC BY 2019 by Illustrative Mathematics 21

Solution

a.
b.

(From Unit 2, Lesson 16.)

9. Problem 9

Statement

Noah and Lin are solving this system:

Noah multiplies the first equation by 12 and the second equation by 8, which
gives:

Lin says, “I know you can eliminate by doing that and then subtracting the
second equation from the first, but I can use smaller numbers. Instead of what
you did, try multiplying the first equation by 6 and the second equation by 4."

a. Do you agree with Lin that her approach also works? Explain your
reasoning.

b. What are the smallest whole-number factors by which you can multiply
the equations in order to eliminate ?

Solution

a. Sample response: Yes. With Lin's strategy, the coefficient of in both equations
is 48. Subtracting the two equations eliminates .

b. Multiply the first equation by 3 and the second equation by 2.

(From Unit 2, Lesson 16.)

Algebra1 Unit 2 Lesson 20 CC BY 2019 by Illustrative Mathematics 22

10. Problem 10
Statement

Algebra1 Unit 2 Lesson 20 CC BY 2019 by Illustrative Mathematics 23

What is the solution set of the inequality ?

A.
B.
C.
D.

Solution

B

(From Unit 2, Lesson 19.)

Algebra1 Unit 2 Lesson 20 CC BY 2019 by Illustrative Mathematics 24

Lesson 21: Graphing Linear Inequalities in Two
Variables (Part 1)

Goals
• Given the graph of a related equation, determine the solution region to an inequality

in two variables by testing the points on the line and on either side of the line.

• Understand that the solutions to a linear inequality in two variables are represented

graphically as a half-plane bounded by a line.

Learning Targets
• Given a two-variable inequality and the graph of the related equation, I can

determine which side of the line the solutions to the inequality will fall.

• I can describe the graph that represents the solutions to a linear inequality in two

variables.

Lesson Narrative

In earlier lessons, students wrote and solved linear inequalities in one variable. In this
lesson, they transition to linear inequalities in two variables.

Previously, students learned that the solutions to an equation in two variables are all pairs
of values that make the equation true, and that, when graphed, the solutions are points on
a line. Here, they learn that the solutions to inequalities in two variables also involve pairs
of values. When graphed, the solutions are no longer points on a single line, but comprise
a region that is bounded by a line. This region consists of all points in a plane on one side
of a boundary line. That boundary line is the graph of an equation related to the
inequality.

Students begin by noticing that the plots of solutions and non-solutions occupy different
parts of a coordinate plane. Next, they think about the boundary line between the two
regions and whether it is a part of the solution. Finally, students write some inequalities
given graphs that represent solution regions.

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

Algebra1 Unit 2 Lesson 21 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.

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

Instructional Routines

• Math Talk
• MLR8: Discussion Supports
• Poll the Class

Required Materials
Colored pencils

Required Preparation
The colored pencils are optional. If used in the first activity, each student needs 2 different
colors. As an alternative to two colors, students could just write two different symbols.

Student Learning Goals

• Let’s find out how to use graphs to represent solutions to inequalities in two

variables.

21.1 Math Talk: Less Than, Equal to, or More
Than 12?

Warm Up: 5 minutes

In the first activity of the lesson, students consider whether the expression is

greater than, less than, or equal to 12 for given pairs. This warm-up familiarizes

students with the computation and reasoning that they will need later to determine the

solution region of a linear inequality in two variables.

Students could reason about the answers by considering the signs and relative sizes of the
- and -values, rather than performing full computations. This is an opportunity to notice

and make use of structure (MP7).

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

Building Towards

• HSA-REI.D.12

Instructional Routines

• Math Talk
• MLR8: Discussion Supports

Launch
Display one problem at a time. Give students quiet think time for each problem and ask
them to give a signal when they have an answer and a strategy. Keep all problems
displayed throughout the talk. Follow with a whole-class discussion.

Support for Students with Disabilities

Representation: Internalize Comprehension. To support working memory, provide
students with sticky notes or mini whiteboards.
Supports accessibility for: Memory; Organization

Student Task Statement

Here is an expression: .

Decide if the values in each ordered pair, , make the value of the expression
less than, greater than, or equal to 12.

Algebra1 Unit 2 Lesson 21 CC BY 2019 by Illustrative Mathematics 3

Student Response

• greater than 12
• equal to 12
• less than 12
• greater than 12

Activity Synthesis

Ask students to share their strategies for each problem. Record and display their
responses for all to see. To involve more students in the conversation, consider asking:

• “Who can restate ’s reasoning in a different way?”

• “Did anyone have the same strategy but would explain it differently?”

• “Did anyone solve the problem in a different way?”

• “Does anyone want to add on to ’s strategy?”

• “Do you agree or disagree? Why?”

To help students recall the meaning of a solution to an inequality, ask: "Which pairs, if any,

are solutions to the inequality ?" Make sure students recognize that both

and are solutions because they make the inequality true.

Support for English Language Learners

Speaking: MLR8 Discussion Supports. Display sentence frames to support students when
they explain their strategy. For example, “First, I _____ because….” or “I noticed _____ so
I….” Some students may benefit from the opportunity to rehearse what they will say
with a partner before they share with the whole class.
Design Principle(s): Optimize output (for explanation)

21.2 Solutions and Not Solutions

20 minutes
Earlier in the unit, students saw that a linear inequality in one variable has many solutions,
which are represented by all the points on one side of a number on a number line. They

Algebra1 Unit 2 Lesson 21 CC BY 2019 by Illustrative Mathematics 4

also recalled that a linear equation in two variables has many solutions, which are
represented by all the points on the graph of the equation. The work in this activity builds
on those understandings.

Students see that the solutions to a linear inequality in two variables can be represented
by many points on a coordinate plane, and that the set of points are in a region bounded
by a line. To represent the set of all points that are solutions, we can shade that region.
Students also see that the points that fall on the other side of the boundary line are not
solutions to the inequality.

During the activity, students are not yet expected to recognize that the line is the graph of
an equation related to the inequality (although some students may notice that). That
insight is made explicit in the synthesis and reinforced in the next activity.

Building Towards

• HSA-REI.D.12

Instructional Routines

• MLR8: Discussion Supports

• Poll the Class

Launch for all to see, along with a blank coordinate grid.
Display the inequality

Algebra1 Unit 2 Lesson 21 CC BY 2019 by Illustrative Mathematics 5

Remind students that earlier, in the warm-up, we saw that and are not

solutions to this inequality, but and are solutions. On the coordinate plane,

mark each point that is a solution with an X and each point that is not a solution with an O

(or use different colors to mark the points).

Then, ask each student to identify 3 coordinate pairs that are solutions and 3 pairs that are
not solutions to the inequality. Encourage students to use some negative values of and ,
and to find pairs that are different than those chosen by the people seated around them.

After a few minutes, poll the class to collect all ordered pairs that students identified and
plot them on the blank coordinate grid (again, using different symbols or different colors
for solutions and non-solutions).

When all the points are plotted, the plot might look something like the following. (Blue
points represent solutions and yellow X's represent non-solutions.)

Algebra1 Unit 2 Lesson 21 CC BY 2019 by Illustrative Mathematics 6

Ask students what they notice about the plotted points. Students are not expected to
perfectly articulate the idea of a solution region at this point, but they should notice that
the solutions are separated from non-solutions by what appears to be an invisible line that
slants downwards from left to right.

When choosing their coordinate pairs, some students may have started with the

equation and plotted some solutions for that equation. If they suggest that

the invisible line might be the graph of , that's great, but it is unnecessary to

point this out otherwise.

Arrange students in groups of 3–4. Tell students that their job in this activity is to plot
some points that do and do not represent solutions to a few inequalities. If time is limited,
consider assigning 1–2 inequalities to each group.

Algebra1 Unit 2 Lesson 21 CC BY 2019 by Illustrative Mathematics 7

Support for Students with Disabilities

Action and Expression: Provide Access for Physical Action. Eliminate computational

barriers so that students may focus on more efficiently plotting points and noticing

patterns. Provide access to tools and assistive technologies such as a calculator.

Specify that calculators in this activity are only to be used for checking input/output

calculations for a specific pair, and not for graphing lines.

Supports accessibility for: Visual-spatial processing; Conceptual processing; Organization

Anticipated Misconceptions

If students don’t know how to begin finding points that are solutions and points that are

non-solutions, suggest that they pick any pair and then determine if it makes the

inequality true. This way, they don’t have to worry about picking an “incorrect” point.

Student Task Statement

Here are four inequalities. Study each inequality assigned to your group and work
with your group to:

• Find some coordinate pairs that represent solutions to the inequality and

some coordinate pairs that do not represent solutions.

• Plot both sets of points. Either use two different colors or two different

symbols like X and O.

• Plot enough points until you start to see the region that contains solutions and

the region that contains non-solutions. Look for a pattern describing the
region where solutions are plotted.

Algebra1 Unit 2 Lesson 21 CC BY 2019 by Illustrative Mathematics 8

Algebra1 Unit 2 Lesson 21 CC BY 2019 by Illustrative Mathematics 9

Student Response
Sample graphs (blue points are solutions and yellow points are not solutions):

Activity Synthesis
Display four graphs that are representative of students' work for the four inequalities. (A
document camera would be very helpful, if available.) In particular, look for examples
where a student decided to “shade” all the points on one side of the boundary line.

Invite students to make some observations about the graphs. Discuss questions such as:

• "What do the two sets of points represent?" (Solutions and non-solutions to the

inequalities)

Algebra1 Unit 2 Lesson 21 CC BY 2019 by Illustrative Mathematics 10

• "If we plot a new point somewhere on the coordinate plane, how can we tell if it is or

is not a solution?" (There appears to be a line that separates the solutions from
non-solutions. If it's on the same side of the line as other solutions, then it is a
solution.)

• "What might be a good way to show all the possible solutions to a linear inequality in

two variables? Is there a better way other than plotting individual points?" (We can
shade the region that contains the points that are solutions.)

Ask students: "How can we tell where exactly the solution region stops and non-solution
region starts?" Solicit some ideas from students. If no one predicts that the line is the
graph of an equation related to the inequality, remind students that when we solved
inequalities in one variable, we used a related equation to help us identify a boundary
value (a point on a number line). We can do the same here.

Take the first inequality, , as an example. Explain that:

• The solutions to are all coordinate pairs where and are equal. We can graph

it as line that goes through , and so on.

• The solutions to are all coordinate pairs where equals ; (such as the points

previously noted), and where is greater than (such as ). The

latter are all located below the line.

• We can shade the region below the line to represent the solutions to .

In the next activity, students will take a closer look at whether the boundary line itself is
part of the solution region. For now, it is sufficient that students see that the graph of an
equation that is related to each inequality delineates the solution and non-solution
regions.

Algebra1 Unit 2 Lesson 21 CC BY 2019 by Illustrative Mathematics 11

Support for English Language Learners

Listening, 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 to 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

21.3 Sketching Solutions to Inequalities

10 minutes
In the preceding activity, students looked at the regions that represent solutions and
non-solutions to inequalities. They recognized that the boundary between the two regions
is the graph of an equation that is related to the inequality. Students did not, however,
look closely at whether the boundary line itself is a part of the solution. That investigation
is the focus of this activity.

Students reason with algebraic and graphical representations of inequalities in two
directions. They first graph the solutions to given inequalities, and later write inequalities
whose solutions could be represented by given graphs.

If many students get stuck on graphing or writing inequalities, consider moving fairly
quickly through the activity and using the discussion questions in the Lesson Synthesis to
help students gain clarity and focus.

Addressing

• HSA-REI.D.12

Launch
Display the inequalities and for all to see. Then, ask students to consider
whether the following coordinate pairs are solutions to each inequality.

1.

Algebra1 Unit 2 Lesson 21 CC BY 2019 by Illustrative Mathematics 12

2.

3.

Make sure students understand why all three coordinate pairs are solutions to , but

only and are solutions to . Display two graphs, each representing one of

these inequalities.

Ask students to predict which graph represents which inequality. Consider polling the class
on their predictions.

Explain that the solid line is a way to say that all the points on that line ( ) are
solutions, and the dashed line is a way to say otherwise. (This is similar to how we use
solid and open circles to represent the boundary values of a one-variable inequality on a
number line.)

Because the solutions to do not include coordinates where and are equal, the

graph of is drawn with a dashed line. The solutions to do include coordinates

where and are equal, so the graph of is drawn with a solid line.

Tell students they will now sketch the solutions of some other inequalities and think about
whether or not the boundary line is included in the solutions.

Algebra1 Unit 2 Lesson 21 CC BY 2019 by Illustrative Mathematics 13