IM UNIT 2 Equations and Inequalities Teacher Guide

3. Problem 3
Statement

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

Solution

Infinitely many solutions. Sample explanation: Both equations can be written as
, so their graphs are exactly the same.

4. Problem 4
Statement

Algebra1 Unit 2 Lesson 17 CC BY 2019 by Illustrative Mathematics 23

Select all systems of equations that have no solutions.
A.
B.
C.
D.
E.

Solution

["A", "D"]

5. Problem 5
Statement

Solve each system of equations without graphing.
a.

b.

Algebra1 Unit 2 Lesson 17 CC BY 2019 by Illustrative Mathematics 24

Solution

a.
b.

(From Unit 2, Lesson 16.)

6. Problem 6

Statement

Algebra1 Unit 2 Lesson 17 CC BY 2019 by Illustrative Mathematics 25

Select all the dot plots that appear to contain outliers.

A.

B.

C.
D.

E.

Solution

["A", "C", "D"]

(From Unit 1, Lesson 14.)

7. Problem 7
Statement

Algebra1 Unit 2 Lesson 17 CC BY 2019 by Illustrative Mathematics 26

Here is a system of equations:

Would you rather use subtraction or addition to solve the system? Explain
your reasoning.

Solution

Sample responses:

◦ I would rather use addition because this eliminates and allows us to solve for

.

◦ I would rather use subtraction because this eliminates and allows us to solve

for .

(From Unit 2, Lesson 14.)

Algebra1 Unit 2 Lesson 17 CC BY 2019 by Illustrative Mathematics 27

8. Problem 8

Statement

Here is a system of linear equations:

Select all the steps that would help to eliminate a variable and enable solving.
A. Multiply the first equation by 2, then subtract the second equation from
the result.
B. Multiply the first equation by 4 and the second equation by 6, then
subtract the resulting equations.
C. Multiply the first equation by 2, then add the result to the second
equation.
D. Divide the second equation by 2, then add the result to the first equation.
E. Multiply the second equation by 6, then subtract the result from the first
equation.

Solution

["B", "C", "D"]

(From Unit 2, Lesson 16.)

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9. Problem 9

Algebra1 Unit 2 Lesson 17 CC BY 2019 by Illustrative Mathematics 29

Statement

Consider this system of equations, which has one solution:

Here are some equivalent systems. Each one is a step in solving the original
system.

Step 1: Step 2: Step 3:

a. Look at the original system and the system in Step 1.
i. What was done to the original system to get the system in Step 1?

ii. Explain why the system in Step 1 shares a solution with the original
system.

b. Look at the system in Step 1 and the system in Step 2.
i. What was done to the system in Step 1 to get the system in Step 2?

ii. Explain why the system in Step 2 shares a solution with that in Step
1.

c. What is the solution to the original system?

Algebra1 Unit 2 Lesson 17 CC BY 2019 by Illustrative Mathematics 30

Solution

a. Sample response:
i. The first equation in the original system is multiplied by 3.5. The second
equation stays the same.

ii. Multiplying both sides of the equation by the same number

(3.5) keeps the two sides equal and gives an equivalent equation. The

pair that is a solution to is still a solution to ,

so it is also a solution to the system in Step 1.

b. Sample response:

i. Subtracting the second equation in Step 1 from the first equation gives the

equation in Step 2. The second equation is unchanged.

ii. Because is equal to 78, subtracting from the left side of

and 78 from the right side means subtracting equal

amounts from both sides, which keeps the two sides equal. The pair

that is a solution to the system in Step 1 is also a solution to the resulting

equation.

c. and or

(From Unit 2, Lesson 16.)

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Lesson 18: Representing Situations with
Inequalities

Goals
• Interpret and write inequalities that represent the constraints in a situation.

Learning Targets
• I can write inequalities that represent the constraints in a situation.

Lesson Narrative

Prior to this point, students have worked primarily with equations in one and two
variables. In this lesson, they shift their attention to inequalities. Here and in the next two
lessons, students interpret, write, and find solutions to inequalities in one variable. The
activities here are grounded in contexts, and the ideas build on the work students have
done in grade 7.

The focus of this lesson is on interpreting and writing inequalities that represent the
constraints in various situations. As students consider key quantities in a situation, select
variables to represent them, and decide whether multiple inequalities (and which
ones) are needed, they engage in aspects of mathematical modeling (MP4).

In future lessons, students will use the insights from this lesson to find and interpret the
solutions to inequalities that model situations.

Alignments
Building On

• 6.EE.B.5: Understand solving an equation or inequality as a process of answering a

question: which values from a specified set, if any, make the equation or inequality
true? Use substitution to determine whether a given number in a specified set makes
an equation or inequality true.

• 7.EE.B.4: Use variables to represent quantities in a real-world or mathematical

problem, and construct simple equations and inequalities to solve problems by
reasoning about the quantities.

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

Algebra1 Unit 2 Lesson 18 CC BY 2019 by Illustrative Mathematics 1

in a modeling context. For example, represent inequalities describing nutritional and
cost constraints on combinations of different foods.

Building Towards

• 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.

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

Instructional Routines

• Aspects of Mathematical Modeling
• MLR2: Collect and Display
• MLR6: Three Reads
• Think Pair Share

Student Learning Goals

• Let’s use inequalities to represent constraints in situations.

18.1 What Do Those Symbols Mean?

Warm Up: 5 minutes
In this activity, students recall the meaning of inequality symbols ( , , , and ) and the
meaning of “solutions to an inequality." They are reminded that an inequality in one
variable can have a range of values that make the statement true. Students also pay
attention to the value that is at the boundary of an inequality and consider whether it is or
isn't a solution to an inequality.

Building On

• 6.EE.B.5

Building Towards

• HSA-CED.A.3

Launch
Give students 1–2 minutes of quiet work time. Follow with a whole-class discussion.

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

Student Task Statement
1. Match each inequality to the meaning of a symbol within it.

a. ◦ less than or equal to
b. ◦ greater than
c. ◦ greater than or equal to

2. Is 25 a solution to any of the inequalities? Which one(s)?

3. Is 40 a solution to any of the inequalities? Which one(s)?

4. Is 30 a solution to any of the inequalities? Which one(s)?

Algebra1 Unit 2 Lesson 18 CC BY 2019 by Illustrative Mathematics 3

Student Response

1. a. : “greater than”

b. : “less than or equal to”

c. : “greater than or equal to”

2. 25 is a solution to because 30 is greater than or equal to 25.

3. 40 is not a solution to any of the inequalities because 40 does not make any of the
inequalities true when substituted in for .

4. 30 is a solution to because, while is not a true statement, is a
true statement.

Activity Synthesis

Draw students' attention to the last inequality ( ). Make sure students see that, even

though the symbol is read "greater than or equal to," it doesn't mean that we're looking

for values that are greater than or equal to 30. The statement reads "30 is greater than or

equal to ," which means must be less than or equal to 30.

Next, ask students how they know whether each of those numbers (the 50, 20, and 30, or
the boundary values) is a solution to the inequality. Emphasize that we can test those
boundary values the same way we test other values—by checking if they make the
statement true.

Display these equations in one variable for all to see: , , and . Discuss
with students how these equations are different from the inequalities in one variable
(aside from the fact that the symbols are different). Highlight the idea that there is only
one value that could make each equation true, but there is a range of values that can
make each inequality true.

18.2 Planning the Senior Ball

15 minutes
This activity prompts students to interpret several inequalities that represent
the constraints in a situation. To explain what the letters in the inequalities mean in the
given context, students cannot simply match the numbers in the verbal descriptions and
those in the inequalities. They must attend carefully to the symbols and any operations
(MP6), and reason both quantitatively and abstractly (MP2).

Algebra1 Unit 2 Lesson 18 CC BY 2019 by Illustrative Mathematics 4

The work here also engages students in an aspect of mathematical modeling (MP4).
Although students do not choose the variables to represent the essential features of a
situation, they think carefully about and explain how the given models do represent the
key features of the given situation.

As students discuss with their partners, listen for those who could interpret the
inequalities clearly and accurately. Ask them to share their interpretations later.

Building On

• 7.EE.B.4

Building Towards

• HSA-CED.A.3
• HSN-Q.A.2

Instructional Routines

• Aspects of Mathematical Modeling
• MLR6: Three Reads
• Think Pair Share

Launch
Arrange students in groups of 2. Give them a few minutes of quiet think time, followed by
some time to share their thinking with their partner.

Algebra1 Unit 2 Lesson 18 CC BY 2019 by Illustrative Mathematics 5

Support for English Language Learners

Reading, Listening, Conversing: MLR6 Three Reads. Use this routine to support reading
comprehension of this problem. Use the first read to orient students to the situation.
Ask students to describe what the situation is about without using numbers (students
planning a budget for an upcoming special event). Take time to discuss the meaning
of unfamiliar terms at this time (Senior Ball, budget, chaperone, profit, etc.) Use the
second read to identify quantities and relationships. Ask students what can be
counted or measured without focusing on the values (number of people who
attended in the past, maximum number of people expected this year, ratio of
chaperones to students, maximum ticket price, minimum amount of profit needed,
etc.) After the third read, ask students to brainstorm possible strategies to answer the
questions. This will help students make sense of language used to describe situations
involving inequalities.
Design Principle(s): Support sense-making

Support for Students with Disabilities

Representation: Internalize Comprehension. Provide appropriate reading
accommodations and supports to ensure student access to written directions, word
problems, and other text-based content.
Supports accessibility for: Language; Conceptual processing

Anticipated Misconceptions

Some students may relate an inequality to a written description simply based on the letter
chosen for the variable (for example, “people” begins with “p”). Push these students to
explain how the inequalities express the quantities and constraints in the written
descriptions.

For students unfamiliar with the notation , explain that this is a way of

stating and .

Algebra1 Unit 2 Lesson 18 CC BY 2019 by Illustrative Mathematics 6

Student Task Statement
Seniors in a student council of a high school are trying to come up with a budget for
the Senior Ball. Here is some information they have gathered:

• Last year, 120 people attended. It was a success and is expected to be even

bigger this year. Anywhere up to 200 people might attend.

• There needs to be at least 1 chaperone for every 20 students.
• The ticket price can not exceed $20 per person.
• The revenue from ticket sales needs to cover the cost of the meals and

entertainment, and also make a profit of at least $200 to be contributed to the
school.

Here are some inequalities the seniors wrote about the situation. Each letter stands
for one quantity in the situation. Determine what is meant by each letter.

•

•

•

•

Student Response

Sample response (an interpretation of the meaning of each letter and an interpretation of
the mathematical statement are included):

• means that the ticket price, , must be less than or equal to $20.

• means that between 120 and 200 people are expected to attend (120

is less than or equal to the number of people; the number of people is less than or
equal to 200). represents number of people attending the ball.

Algebra1 Unit 2 Lesson 18 CC BY 2019 by Illustrative Mathematics 7

• means that the total amount collected from ticket sales in dollars, ,

minus the cost of meals and entertainment in dollars, , must be greater than or

equal to $200.

• means that the number of chaperones, , must be more than of the

number of people attending.

Are You Ready for More? .
Kiran says we should add the constraint

1. What is the reasoning behind this constraint?

2. What other "natural constraint" like this should be added?

Student Response
Sample responses:

1. We can't have a negative ticket price.

2. By the same reasoning we might want to include . Note that the other variables

should also be greater than or equal to 0, but those conditions are already implied by

the existing constraints.

Activity Synthesis

Invite previously identified students to share their responses. Make sure students
understand why the symbols accurately represent the constraints in the situation.

For the last two inequalities, make sure students see how the operations represent the
constraints on profit and on the number of chaperones.

If needed, use numbers to illustrate the relationship between variables. For example, to

help students make sense of , ask: "How many chaperones are needed if there are

120 students?" (at least or 6 chaperones, or ) "180 students?" (at least or 9

chaperones, or ).

Algebra1 Unit 2 Lesson 18 CC BY 2019 by Illustrative Mathematics 8

18.3 Elevator Constraints

15 minutes
Previously, students interpreted given inequalities and made sense of them in terms of a
situation. In this activity, students write inequalities to represent the constraints in a
situation. Students identify key quantities and relationships, and think about ways to
represent them. In doing so, they engage in an aspect of mathematical modeling (MP4).

As students work, look for those who represent the same constraint using different

inequalities or equations. For example, to represent the total weight that the elevator

could carry, some students may write , some may write , and

others may write or . Ask students with contrasting statements to

share their responses later.

Addressing

• HSA-CED.A.3

Building Towards

• HSN-Q.A.2

Instructional Routines

• Aspects of Mathematical Modeling

• MLR2: Collect and Display

Launch
Keep students in groups of 2.

Anticipated Misconceptions

Students who use a variable for the number of adults and another for the number of
children may have trouble accounting for the weight of the gear because it applies to both
groups. Calculating the weight of a specific combination of adults and children (with their
gear) may help.

Student Task Statement

An elevator car in a skyscraper can hold at most 15 people. For safety reasons, each
car can carry a maximum of 1,500 kg. On average, an adult weighs 70 kg and a child
weighs 35 kg. Assume that each person carries 4 kg of gear with them.

Algebra1 Unit 2 Lesson 18 CC BY 2019 by Illustrative Mathematics 9

1. Write as many equations and inequalities as you can think of to represent the
constraints in this situation. Be sure to specify the meaning of any letters that
you use. (Avoid using the letters , , or .)

2. Trade your work with a partner and read each other's equations and
inequalities.

a. Explain to your partner what you think their statements mean, and listen
to their explanation of yours.

b. Make adjustments to your equations and inequalities so that they are
communicated more clearly.

3. Rewrite your equations and inequalities so that they would work for a
different building where:

◦ an elevator car can hold at most people
◦ each car can carry a maximum of kilograms
◦ each person carries kg of gear

Student Response
1. Answers vary. Sample responses:

Algebra1 Unit 2 Lesson 18 CC BY 2019 by Illustrative Mathematics 10

◦ , where represents the number of people in an elevator car.
◦ , where represents the number of adults and represents the

number of children in an elevator car.

◦ , where represents the weight in kilograms that one elevator car

can carry.

◦

2. No response required.
3. Answers vary. Sample responses:

◦

Activity Synthesis

Invite students to share their equations and inequalities, starting with those that are more
concrete (from the first question) and ending with the ones that are more abstract (from
the last question).

Emphasize that the same constraints may be accurately represented by statements of
different forms. Consider reading aloud the different inequalities that represent the same
constraint. For example, if represents the total weight:

• can be read: "The total weight is less than or equal to 1,500 kilograms." or

"The total weight is at most 1,500 kilograms."

• can be read: "Fifteen hundred kilograms is greater than or equal to the

total weight."

• or can be read: "The total weight is less than 1,500 kilograms,

or it is equal to 1,500 kilograms."

For constraints that involve multiple quantities, some students may write, for

instance, , while others may write . Ask

Algebra1 Unit 2 Lesson 18 CC BY 2019 by Illustrative Mathematics 11

students why these expressions are equivalent, encouraging them to use the context in
their explanation.

Point out that although a constraint can be written in different ways, writing it using fewer
terms may be more convenient and may allow us to gain certain insights about the
situation.

Support for English Language Learners

Speaking, Representing: MLR 2 Collect and Display. Use this routine to support
whole-class discussion. As students share their responses, amplify and record any
words and phrases that describe the meaning of each equation or inequality on a
display. Call students’ attention to the different ways the constraints are represented
by language in context and equations or inequalities of different forms.
Design Principle(s): Maximize meta-awareness; Cultivate conversation

Lesson Synthesis

Solicit from students any advantages and disadvantages for representing constraints as
inequalities. Some advantages students might bring up:

• Compared to written words, inequalities are a simpler and quicker way to describe

what is happening in a situation.

• It is easier to see what values a certain quantity could or could not take when the

constraint is written with symbols and numbers.

Possible disadvantages:

• Unless we know what the variables stand for, we can't be sure about the meaning of

an inequality.

• If we don't recall what the symbols mean or how to read them, we can't access the

information.

Next, invite students to share some advice for students who might just be learning to write
inequalities to represent constraints. What should they pay attention to? What are some
potential sources of confusion or error they should look out for? Students might mention
that the following are important things to attend to:

• specifying the meaning of each variable

Algebra1 Unit 2 Lesson 18 CC BY 2019 by Illustrative Mathematics 12

• not using the same variable to represent different quantities
• making sure that the correct symbols are used to represent relationships
• using words to read each inequality to make sure that it fully represents a constraint

18.4 Grape Constraints

Cool Down: 5 minutes
Addressing

• HSA-CED.A.3

Student Task Statement
Han has a budget of $25 to buy grapes. Write inequalities to represent the number
of pounds of grapes that Han could buy in each situation:

1. Grapes cost $1.99 per pound.

2. Grapes cost $2.49 per pound.

3. Grapes cost $ per pound.

Student Response
Sample response: Let represent the number of pounds of grapes.

1.
2.
3.

Algebra1 Unit 2 Lesson 18 CC BY 2019 by Illustrative Mathematics 13

Student Lesson Summary

We have used equations and the equal sign to represent relationships and
constraints in various situations. Not all relationships and constraints involve
equality, however.

In some situations, one quantity is, or needs to be, greater than or less than
another. To describe these situations, we can use inequalities and symbols such as

, or .

When working with inequalities, it helps to remember what the symbol means, in
words. For example:

• means “100 is less than .”
• means “ is less than or equal to 55,” or " is not more than 55."
• means “20 is greater than 18.”
• means “ is greater than or equal to 40,” or " is at least 40."

These inequalities are fairly straightforward. Each inequality states the relationship

between two numbers ( ), or they describe the limit or boundary of a

quantity in terms of a number ( ).

Inequalities can also express relationships or constraints that are more complex.
Here are some examples:

• The area of a rectangle, , with a length of 4 meters and a

width meters is no more than 100 square meters.

• To cover all the expenses of a musical production each

week, the number of weekday tickets sold, , and the
number of weekend tickets sold, , must be greater than
4,000.

• Elena would like the number of hours she works in a

week, , to be more than 5 but no more than 20.

Algebra1 Unit 2 Lesson 18 CC BY 2019 by Illustrative Mathematics 14

• The total cost, , of buying adult shirts and child shirts

must be less than 150. Adult shirts are $12 each and
children shirts are $7 each.
In upcoming lessons, we’ll use inequalities to help us solve problems.

Lesson 18 Practice Problems

1. Problem 1

Statement

Tyler goes to the store. His budget is $125. Which inequality represents , the
amount in dollars Tyler can spend at the store?

A.
B.
C.
D.

Solution

A

Algebra1 Unit 2 Lesson 18 CC BY 2019 by Illustrative Mathematics 15

2. Problem 2

Statement

Jada is making lemonade for a get-together with her friends. She expects a
total of 5 to 8 people to be there (including herself). She plans to prepare 2
cups of lemonade for each person.

The lemonade recipe calls for 4 scoops of lemonade powder for each quart of
water. Each quart is equivalent to 4 cups.

Let represent the number of people at the get-together, the number
of cups of water, the number of scoops of lemonade powder.

Select all the mathematical statements that represent the quantities and
constraints in the situation.

A.

B.

C.

D.

E.

F.

Solution

["B", "C", "D", "F"]

Algebra1 Unit 2 Lesson 18 CC BY 2019 by Illustrative Mathematics 16

3. Problem 3
Statement

A doctor sees between 7 and 12 patients each day. On Mondays and
Tuesdays, the appointment times are 15 minutes. On Wednesdays and
Thursdays, they are 30 minutes. On Fridays, they are one hour long. The
doctor works for no more than 8 hours a day.
Here are some inequalities that represent this situation.

a. What does each variable represent?

b. What does the expression in the last inequality mean in this situation?

Solution

a. The variable represents the number of patients the doctor sees each day. The
variable represents the length of each visit, in hours.

b. If the doctor has patients each day and spends hours with each patient, is
the amount of time, in hours, that he spends with patients in one day. That
amount has to be no more than 8 hours.

Algebra1 Unit 2 Lesson 18 CC BY 2019 by Illustrative Mathematics 17

4. Problem 4

Statement

Han wants to build a dog house. He makes a list of the materials needed:

◦ At least 60 square feet of plywood for the surfaces
◦ At least 36 feet of wood planks for the frame of the dog house
◦ Between 1 and 2 quarts of paint

Han's budget is $65. Plywood costs $0.70 per square foot, planks of wood cost
$0.10 per foot, and paint costs $8 per quart.

Write inequalities to represent the material constraints and cost contraints in
this situation. Be sure to specify what your variables represent.

Solution

Sample response: Let represent Han's budget in dollars, the area of plywood in
square feet, the length of wood planks in feet, and the quarts of paint.

Material constraints:

◦
◦
◦

Cost constraints:

◦
◦

5. Problem 5

Statement

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

The equation represents the volume of a cone, where is the

radius of the cone and is the height of the cone.

Which equation is solved for the height of the cone?

A.

B.

C.

D.

Solution

D

(From Unit 2, Lesson 9.)

6. Problem 6

Statement

Solve each system of equations without graphing.

a.

b.

Algebra1 Unit 2 Lesson 18 CC BY 2019 by Illustrative Mathematics 19

Solution

a.
b.

(From Unit 2, Lesson 14.)

7. Problem 7
Statement

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

There is a pair of and values that make each equation true in this system of
equations:

Explain why the same pair of values also make true.

Solution

Sample response: Because is equal to 8, I can add to the left side of

the equation and 8 to the right side of the same equation. Adding

equivalent expressions to each side of an equation does not change the solution to

the equation.

(From Unit 2, Lesson 15.)

8. Problem 8

Statement

Which ordered pair is a solution to this system of equations?

A.
B.
C.
D.

Solution

B

(From Unit 2, Lesson 16.)

Algebra1 Unit 2 Lesson 18 CC BY 2019 by Illustrative Mathematics 21

9. Problem 9

Statement

Which equation has exactly one solution in common with the equation
?

A.
B.
C.
D.

Solution

C

(From Unit 2, Lesson 17.)

10. Problem 10

Statement

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

Solution

One solution. Sample explanation: The slope of the first line is -4. The slope of the
second line is . Because the lines have different slopes, they intersect at exactly
one point.

(From Unit 2, Lesson 17.)

Algebra1 Unit 2 Lesson 18 CC BY 2019 by Illustrative Mathematics 22

Lesson 19: Solutions to Inequalities in One
Variable

Goals
• Find the solution to a one-variable inequality by reasoning and by solving a related

equation and testing values greater than and less than that solution.

• Graph the solution to an inequality as a ray on a number line and interpret the

solution in context.

• Understand that the solution to an inequality is a range of values that make the

inequality true.

Learning Targets
• I can graph the solution to an inequality in one variable.

• I can solve one-variable inequalities and interpret the solutions in terms of the

situation.

• I understand that the solution to an inequality is a range of values (such as )

that make the inequality true.

Lesson Narrative

In this lesson, students revisit the meaning of the solutions to an inequality in one variable
and recall that the solution set is a range of values. They also investigate different ways to
find the solution set to an inequality—by reasoning about the quantities and relationships
in context, by guessing some values, substituting them into the inequality, and checking
them to see if they make an inequality true, and by first solving a related equation in one
variable. Along the way, students reason abstractly and quantitatively (MP2).

Two optional activities are included in this lesson. The first is to give students an additional
opportunity to make sense of the solutions to an inequality in terms of a situation. The
second optional activity introduces them to graphing two-variable equations as a way to
find solutions to one-variable inequalities.

Later, students will use the understanding they build here to solve more sophisticated
problems and to find solutions to linear inequalities in two variables.

Algebra1 Unit 2 Lesson 19 CC BY 2019 by Illustrative Mathematics 1

Alignments

Building On

• 7.EE.B.4.b: Solve word problems leading to inequalities of the form or

, where , , and are specific rational numbers. Graph the solution set of

the inequality and interpret it in the context of the problem. For example: As a

salesperson, you are paid $50 per week plus $3 per sale. This week you want your

pay to be at least $100. Write an inequality for the number of sales you need to

make, and describe the solutions.

Addressing

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

equations with coefficients represented by letters.

Building Towards

• 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.

• 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
• Graph It
• MLR5: Co-Craft Questions
• MLR8: Discussion Supports
• Think Pair Share

Required Preparation

Devices that can run Desmos (recommended) or other graphing technology are needed
for the optional activity, More or Less? The digital version with an embedded applet is
recommended for all classes.

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

Student Learning Goals

• Let’s find and interpret solutions to inequalities in one variable.

19.1 Find a Value, Any Value

Warm Up: 5 minutes
This warm-up activates what students know about the solutions to an inequality and ways
to find the solutions. For the first time, students refer to the solutions as the solution
set of the inequality. Throughout their work with one-variable inequalities, students will
use the terms “solutions” and “solution set” interchangeably.

By now, students are likely to have internalized that a solution to an equation in one
variable is a value that makes the equation true. The work here makes it explicit that we
can extend this understanding of solution to inequalities in one variable.

Some students may have been taught to solve inequalities the same way we solve
equations, with the added rule along the lines of "flip the symbol when dividing or
multiplying both sides of an inequality by a negative number." They may or may not have
understood why the rule is the way it is. (If a student shares this method, emphasize that
they will look at different strategies in the lesson and adopt whichever ways that they can
explain or justify.)

If students approach the last question (finding a solution to ) by peforming

operations directly on the inequality but neglect to reverse the inequality symbol, they

would find solutions that result in false statements. Use this opportunity to point out that

we may run into problems with this method. It is not essential to discuss why or to suggest

better approaches at this point. There will be other opportunities in this lesson to reason

about the solutions and to witness the same issue (in the second non-optional

activity—Equality and Inequality).

Building On

• 7.EE.B.4.b

Addressing

• HSA-REI.B.3

Building Towards

• HSA-CED.A.1

Algebra1 Unit 2 Lesson 19 CC BY 2019 by Illustrative Mathematics 3

Launch
Display a number line for all to see that looks like this:

Tell students that, if needed, they could use the number line to help them in reasoning
about the inequalities.

Student Task Statement . Be prepared to explain what

1. Write some solutions to the inequality
makes a value a solution to this inequality.

2. Write one solution to the inequality . Be prepared to explain your
reasoning.

Student Response
Sample responses:

1. 9.2, 8, 4.5, -7

2. 0.5, 0, -1, -6

Activity Synthesis

Invite students to share some solutions to the first inequality and explain what it means

for a value to be a solution to . Be sure to mention some negative numbers that are

solutions. If necessary, show the solution set on a number line. Then, focus the discussion

on the second inequality.

Ask each student to mark the one solution they have for on the class

number line (from the launch). (Students could draw a point or put a dot sticker, if

available, on the number line.) Discuss with students:

• "How did you know that the value you chose is a solution?" (When substituted for in

the inequality, the value makes a true statement.)

• "What do you notice about all the points that are on the line?" (They are all to the left

side of 1.)

Algebra1 Unit 2 Lesson 19 CC BY 2019 by Illustrative Mathematics 4

• "On the number line, we can see that the solutions are values that are less than 1. All

these values form the solution set to the inequality. Is there a way to write the

solution set concisely, without using the number line and without writing out all the

numbers less than 1?" (We can write ).

• "Does the solution set have anything to do with the solution to the equation

?" (The solution to the equation .)

• "Why does the solution set to the inequality involve numbers less than

1?" (The inequality can be taken to say "7 times is greater than 14." For the

inequality to be true, must be greater than 2. For to be greater than 2,

must be less than 1.)

Highlight that we can use a number line to concisely show the solution set to an inequality,
but we can also write another inequality that shows the same information.

19.2 Off to an Orchard

20 minutes
This activity encourages students to interpret an inequality and its solution set in terms of
a situation. A context can help students intuit why the solutions to an inequality form a ray
on the number line.

The activity also prompts students to think about the solutions to an inequality in terms of
a related equation. Here the situation involves choosing between two options. An equation
can be written to represent the two options being equal. The solution to that equation can
be seen as a tipping point, on either side of which one option would be better.

Students may recall this way of solving inequalities from middle school, but they may also
solve by testing different possible values, or by reasoning about the relationship between
quantities in other ways. The work here encourages students to reason quantitatively and
abstractly (MP2), and to make sense of problems and persevere in solving them (MP1).

Monitor the strategies students use to find the solutions to , and
identify students using different approaches. Students may:

• Try different values of until the inequality is no longer true.

• Try different numbers higher than 12 (based on their work on the first question) and

find that, up to 17 students, the cost to go to Orchard B is lower. Beyond 17 students,

the cost for Orchard A is lower.

Algebra1 Unit 2 Lesson 19 CC BY 2019 by Illustrative Mathematics 5

• Solve the equation to find the number of students at which the

costs for both options would be equal. That number is 17. Then, try a higher or lower

number to see which side of the equation has a smaller value.

• Reason about the difference in the cost per student and cost for chaperones. The

cost per student at Orchard A ($9) is $1 lower than at Orchard B ($10). But because 3
chaperones are required at Orchard A ($27 for 3 chaperones) and only 1 at Orchard B
($10 for 1 chaperone), the cost for chaperones is $17 higher at Orchard A than at
Orchard B. So if 17 students go on the trip, the cost would be the same at both
places. If more than 17 students go, Orchard A would be cheaper.

Some students may find the solutions to by manipulating the

inequality to isolate . Depending on the operations performed, they may once again end

up with an incorrect solution set if they forget to reverse the inequality symbol. (For

example, in the final step of solving, they may go from to .) If this

happens, bring the issue to students' attention during activity synthesis.

Building Towards

• HSA-CED.A.1

• HSA-REI.B.3

Instructional Routines

• Anticipate, Monitor, Select, Sequence, Connect

• MLR5: Co-Craft Questions

Launch

Read the first part of the task statement with the class and make sure students
understand the given information.

Arrange students in groups of 2. For the first set of questions, ask one partner to find the
cost of going to Orchard A and the other partner to find the cost of going to Orchard B,
and then compare the costs. Before students move on to the second set of questions,
pause to hear which option works best for 8, 12, and 30 students.

Algebra1 Unit 2 Lesson 19 CC BY 2019 by Illustrative Mathematics 6

Support for English Language Learners

Reading, Writing: MLR 5 Co-Craft Questions. Use this routine to help students consider
the context of this problem and to increase awareness of the language of
mathematical comparisons. Ask students to keep their books or devices closed and
display only the image and the task statement, 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 involving comparing costs. Once students compare their
questions, reveal the remainder of the task.
Design Principles: Maximize Meta-awareness, Support Sense-making

Support for Students with Disabilities

Representation: Internalize Comprehension. Represent the same information through
different modalities by using a table. If students are unsure where to begin, suggest
that they use a table to help organize the information provided. Guide students in
making decisions about what inputs to include in their table. Be sure that they include
17 and some values directly above and below to support their analysis.
Supports accessibility for: Conceptual processing; Visual-spatial processing

Anticipated Misconceptions

If students struggle to interpret the meaning of the equation and of

the inequality , ask them to think about what each side of the

equal sign or the inequality symbol represents.

Student Task Statement
A teacher is choosing between two options for a class field trip to an orchard.

Algebra1 Unit 2 Lesson 19 CC BY 2019 by Illustrative Mathematics 7

• At Orchard A, admission costs $9

per person and 3 chaperones are
required.

• At Orchard B, the cost is $10 per

person, but only 1 chaperone is
required.

• At each orchard, the same price

applies to both chaperones and
students.

1. Which orchard would be cheaper to visit if the class has:

a. 8 students?

b. 12 students?

c. 30 students?

2. To help her compare the cost of her two options, the teacher first writes the

equation , and then she writes the inequality

.

a. What does represent in each statement?

b. In this situation, what does the equation mean?

c. What does the solution to the inequality tell us?

Algebra1 Unit 2 Lesson 19 CC BY 2019 by Illustrative Mathematics 8

d. Graph the solution to the inequality on the number line. Be prepared to
show or explain your reasoning.

Student Response
1. a. Orchard B

b. Orchard B

c. Orchard A

2. a. represents the number of students on the field trip.

b. is the cost of going to Orchard A and is the cost of going to

Orchard B. The equation represents the two options costing the same amount.

c. The solution represents the number of students on the field trip at which it
would be cheaper to visit Orchard A.

d. or

See Activity Narrative for sample reasoning.

Activity Synthesis

Make sure students understand the meaning of the inequality in context and recognize
that there are various ways to find the solutions.

Select previously identified students to share how they found the solution set, in the
sequence shown in the Activity Narrative (starting with guessing and checking, and ending
with reasoning more structurally). It is not necessary to discuss all the listed strategies, but
if the idea of solving a related equation doesn't come up, point it out.

Explain that one way to think about the solutions to the inequality is by thinking about the

solution to a related equation. In this context, the solution to gives us

the number of students at which it would cost the same to go to either orchard. This is a

boundary value for . On one side of the boundary, the cost of Option A would be higher.

On the other, it would be lower. We can test a value of that is higher and one that is lower

than this boundary value to see which one makes the inequality true.

Algebra1 Unit 2 Lesson 19 CC BY 2019 by Illustrative Mathematics 9

If a student brings up “flipping the symbol when multiplying or dividing by a negative
number” as a strategy, invite them to explain why it works. Emphasize that in general it is
more helpful and reliable to use reasoning strategies that we understand and can explain.
If we use a rule without some idea of how it came about or why it works, we might end up
misapplying it (for example, flipping the inequality symbol anytime we see a negative sign,
even if we’re simply adding or subtracting). If we forget or misremember the rule, we
would be stuck or make errors.

19.3 Part-Time Work

Optional: 20 minutes
This optional activity gives students another opportunity to make sense of inequalities and
their solutions in context. The context helps to reinforce why it makes sense for the
solutions to an inequality to be represented by a set of points on one side of a particular
value on the number line. It also reiterates the idea that solving a related equation can
help us find that boundary value. In the context of part-time jobs presented here, that
boundary value is a break-even point at which the two options are equal. On each side of
that point, one job is more lucrative than the other.

Students are given an expression that represents the monthly pay for hours of work at

each job. The expression for the pay at the restaurant is . Some students may

notice that this expression could be rewritten as and use this simpler
equivalent expression throughout the activity. Invite students who do this to share their
rationale during class discussion.

Building On

• 7.EE.B.4.b

Addressing

• HSA-REI.B.3

Building Towards

• HSA-CED.A.1

Instructional Routines

• MLR8: Discussion Supports

• Think Pair Share

Algebra1 Unit 2 Lesson 19 CC BY 2019 by Illustrative Mathematics 10

Launch

Display the situation in the task statement and the equation . To
familiarize students with the quantities in the context, discuss:

• "What does the expression represent?" (the pay at the restaurant for
?" (the pay at the hotel for hours of work)
hours of work) "What about

• "What does the equation tell us about the situation?" (the student earning the same

amount from both jobs for hours of work)

• "What does the solution mean in this situation?" (the number of hours of work that

would enable the student to earn the same amount from either job)

Consider keeping students in groups of 2. Give them a few minutes of quiet work time,
followed by time to discuss their responses with their partner.

Support for Students with Disabilities

Action and Expression: Internalize Executive Functions. Chunk this task into manageable
parts for students who benefit from support with organizational skills in problem
solving. Check in with students after the first 2–3 minutes of work time. Invite 1–2
students to share how they solved the equation and display the number lines they
created. Record their thinking on the display and keep the work visible as students
continue to work.
Supports accessibility for: Organization; Attention

Anticipated Misconceptions
Some students might be stymied by the fact that one of the expressions has a fractional
coefficient for one of its terms. Remind these students that is just a number times .
Rewriting as a decimal may help students to see this concretely.

Student Task Statement

To help pay for his tuition, a college student plans to work in the evenings and on
weekends. He has been offered two part-time jobs: working in the guest-services
department at a hotel and waiting tables at a popular restaurant.

Algebra1 Unit 2 Lesson 19 CC BY 2019 by Illustrative Mathematics 11

• The job at the hotel pays $18 an hour and offers $33 in transportation

allowance per month.

• The job at the restaurant pays $7.50 an hour plus tips. The entire waitstaff

typically collects about $50 in tips each hour. Tips are divided equally among
the 4 waitstaff members who share a shift.

1. The equation represents a possible constraint about
a situation.

a. Solve the equation and check your solution.

b. Here is a graph on a number line.

Put a scale on the number line so that the point marked with a circle
represents the solution to the equation.
2. Does one job pay better if:
a. The student works fewer hours than the solution you found earlier? If so,
which job?

b. The student works more hours than the solution you found earlier? If so,
which job?

Be prepared to explain or show how you know.
3. Here are two inequalities and two graphs that represent the solutions to the

inequalities.

◦ Inequality 1:

Algebra1 Unit 2 Lesson 19 CC BY 2019 by Illustrative Mathematics 12

◦ Inequality 2:

A

B

a. Put the same scale on each number line so that the circle represents the
number of hours that you found earlier.

b. Match each inequality with a graph that shows its solution. Be prepared
to explain or show how you know.

Student Response
1. a.
b. Sample response:

2. a. Yes. The restaurant job pays better if he works fewer than 16.5 hours a month.
Sample reasoning: I found the pay for both jobs using numbers that are less
than 16.5 and found that pay at the restaurant is always higher.

b. Yes. The hotel job pays better if he works more than 16.5 hours a month.
Sample reasoning: I found the pay for both jobs using numbers that are greater
than 16.5 and found that pay at the hotel is always higher.

3. a. Sample response:

A

B

b. Graph A shows the solution to Inequality 1, and Graph B shows the solution to
Inequality 2. Sample explanations:

Algebra1 Unit 2 Lesson 19 CC BY 2019 by Illustrative Mathematics 13

▪ Inequality 1 shows the pay at the hotel ( ) being greater than the

pay at the restaurant ( ). Earlier, we found out that the hotel

job is better when the student works more than 16.5 hours. Graph A

shows . This means Graph A shows the solution to Inequality 1.

▪ If we substitute a value greater than 16.5 for , we have a true statement

for Inequality 1 and a false statement for Inequality 2. This means

is a solution to Inequality 1 but not to Inequality 2.

▪ If we substitute a value less than 16.5 for , we have a true statement for

Inequality 2 but not for Inequality 1. So is a solution to Inequality

2.

Activity Synthesis

Invite students to share how they determined which job pays better for different hours of
work. Display the number lines that show the solution set to each inequality.

If time permits, solicit some ideas from students as to why one job—the restaurant
job—continues to be the more lucrative option after 16.5 hours.

One way to make sense of this is to compare the hourly rates at the two jobs.

• The pay at the restaurant, could be rewritten as , which tells us the

rate of change is $20 per hour.

• At the hotel, the rate is $18 per hour. The $33 transportation allowance makes the

total pay for this job higher up to a point, but then the higher rate of the restaurant

job starts to overtake this one.

If we graph the monthly pay for each job ( and ), we can see that the
is has a lower -value to the left
two graphs intersect when is 16.5. The graph of

of that point but higher value to the right of it.

Algebra1 Unit 2 Lesson 19 CC BY 2019 by Illustrative Mathematics 14

Students could explore this way of solving inequalities in one variable (by graphing) in an
optional activity later in the lesson.

Support for English Language Learners

Conversing: MLR8 Discussion Supports. Use this routine to help students prepare for the
whole-class discussion. Arrange students in groups of 2. Invite Partner A to begin with
this sentence frame: “Graph _____ matches with the inequality _____ , because _____ .”
Invite the listener, Partner B, to press for additional details referring to specific
features of the inequalities, such as solution set, equivalent expression, and boundary
value. Students should switch roles so that Partner B can explain the remaining
match.
Design Principle(s): Support sense-making; Cultivate conversation

19.4 Equality and Inequality

10 minutes
The purpose of this activity is to further develop the idea that we can solve an inequality by
first solving a related equation. Previously, students reasoned about the solutions to an
inequality in context. Here, they transition to solving an inequality without a context.

This activity offers another opportunity to point out the trouble with isolating the

variable directly on the inequality statement, or with assuming that the solution set can be

expressed using the same symbol as in the original inequality. (For example, if in the final

step of solving students go from to , they would end up with the wrong

solution set.)

Algebra1 Unit 2 Lesson 19 CC BY 2019 by Illustrative Mathematics 15

Addressing

• HSA-REI.B.3

Launch

Support for Students with Disabilities

Engagement: Internalize Self Regulation. Chunk this task into more manageable parts to
differentiate the degree of difficulty or complexity. Hand out numbers to test using
sticky notes. Assign each student only a single value to test for, being sure to assign
half the class values greater than two, and the other half values less than two. Label
two areas on a display with “Solution” and “Not a Solution.” Once students have
calculated whether their number is a solution, invite them to place their sticky note
under the appropriate grouping. Invite students to complete the remaining questions
using the display of collected data.
Supports accessibility for: Organization; Attention

Student Task Statement .
1. Solve this equation and check your solution:

Algebra1 Unit 2 Lesson 19 CC BY 2019 by Illustrative Mathematics 16

2. Consider the inequality: .

a. Choose a couple of values less than 2 for . Are they solutions to the
inequality?

b. Choose a couple of values greater than 2 for . Are they solutions to the
inequality?

c. Choose 2 for . Is it a solution?

d. Graph the solution to the inequality on the number line.

Student Response
1.
2. Values chosen vary.
a. Values less than 2 are not solutions.
b. Values greater than 2 are solutions.
c. Yes, 2 is a solution.

d.

Algebra1 Unit 2 Lesson 19 CC BY 2019 by Illustrative Mathematics 17

Are You Ready for More? .
Here is a different type of inequality:

1. Is 1 a solution to the inequality? Is 3 a solution? How about -3?

2. Describe all solutions to this inequality. (If you like, you can graph the solutions
on a number line.)

3. Describe all solutions to the inequality . Test several numbers to make
sure your answer is correct.

Student Response
1. Yes, no, no.

2.

3. or .

Activity Synthesis
Display a blank number line for all to see. Ask students to share some values on the
number line that are and are not solutions. Use different colors or different symbols to
mark on the number line the solutions and non-solutions.

Emphasize that if we solve an inequality by using a related equation, it is important to
make sure that the solution to the equation is correct because that solution gives us a
boundary from which we could check the solutions to the inequality. If the boundary value
is off, we may not be able to correctly find the solution set to the inequality.

Algebra1 Unit 2 Lesson 19 CC BY 2019 by Illustrative Mathematics 18

19.5 More or Less?

Optional: 15 minutes (there is a digital version of this activity)
This optional activity allows students to visualize an inequality in one variable in another
way—by graphing the expressions on each side and comparing the values of the
expressions at different values of the variable. Doing so allows them to see the values at
which the inequality is satisfied.

This activity works best when each student has access to devices that can run the Desmos
applet because students will benefit from seeing the relationship in a dynamic way. If
students don't have individual access, projecting the applet would be helpful during the
synthesis. (Students can still graph the equations in the activity using the graphing
technology available in the classroom.)

Addressing

• HSA-REI.B.3

Instructional Routines

• Graph It

Launch

Consider projecting for all to see the applet in the digital version of this activity. Enter an

equation, say, . Ask students:

• “From the graph, can you tell what -value gives a -value of 4? In other words, what

-value makes the value of exactly 4?” (1)

• “Which -values give a -value that is greater than 4?” ( -values less than 1)

• “Which -values give a -value that is less than 10?” ( -values greater than -1)

• “Which -values give a -value that is greater than -5?” ( -values less than 4)

If students have individual access to Desmos or another tool with a slider function,
consider demonstrating how moving the slider for in the applet could help them see the
answers to these questions more clearly. Otherwise, consider showing the slider during
discussion (after students have analyzed the graphs and estimated the values visually).

Anticipated Misconceptions

Some students may need help parsing the phrase “for what values of is the -value...” Ask

them: “When is 0, what is the -value in ? What about in ? Which

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