IM UNIT 2 Equations and Inequalities Teacher Guide

Support for Students with Disabilities

Representation: Develop Language and Symbols. Create a display of the four inequality
symbols , , , and . During the launch take time to review each symbol and
review some common phrases that translate into inequality symbols. Invite students
to suggest language or diagrams to include that will support their understanding of
inequality symbols.
Supports accessibility for: Conceptual processing; Language

Anticipated Misconceptions

Some students may struggle with the second set of questions because they do not recall

how to write an equation for a vertical line or a horizontal line. Suggest that they write the

coordinates of several points on the line and look for a pattern. For example, some points

on the vertical line are , and . Noticing that the -value is always 3,

regardless of the -value, may help remind students that the equation is .

Student Task Statement .
1. Here is a graph that represents solutions to the equation

Sketch 4 quick graphs representing the solutions to each of these inequalities:

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2. For each graph, write an inequality whose solutions are represented by the
shaded part of the graph.

AB

CD

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Student Response
1. Four sketches:

2. Four inequalities:
A:
B:
C: or
D: or

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Are You Ready for More?

1. The points and are both in the solution region of the inequality
. for both of these

a. Compute
points.

b. Which point comes closest to

satisfying the equation

? That is, for which

pair is closest to 3?

2. The points and are also in the solution region. Which of these

points comes closest to satisfying the equation ?

3. Find a point in the solution region that comes even closer to satisfying the

equation . What is the value of ?

4. For the points and , . Find another point in the solution
region for which .

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5. Find for the point . Then find two other points that give the same

answer.

Student Response .. will work.
1. a. 1 and -3 . Any point on the line
b.
2. -1 and 1.
3. Sample response:
4. Sample response:

5. -1. Sample points: (from previous work), . Any point on the line
will work.

Activity Synthesis

Select students to share their sketched graphs for the first set of questions and the
inequalities they wrote for the second set of questions. Use their work and explanations to
help the class synthesize the new ideas in this lesson.

See Lesson Synthesis for discussion questions.

Lesson Synthesis

Refer to the work students have done in the last activity. Discuss with students how they
made decisions about the solution region and boundary line for the given inequalities, and
about the inequality symbol for the given graphs. Ask questions such as:

• "Once you knew where the boundary line is, how did you decide which side of the

line represents the solution region?"

• "How did you decide whether the boundary line should be solid or dashed?"

• "When you have the graph showing the solution region, how did you determine the

inequality symbol to use?"

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Some students might incorrectly conclude that an inequality with a < symbol will be
shaded below the boundary line and that an inequality with a > symbol will be shaded
above it. Any inequality in the first question can be used to show that this is not the case.

Take for example. We're looking for coordinate pairs that has a value of less than

5 when is subtracted from . Let's see if meets this condition: gives

, which is a true statement. This means that , which is above the graph of

, is in the solution region. If we test a point below the line, say, , we

would see that is greater than 5, not less than 5. This means that the region below

the line is for non-solutions.

Emphasize that we cannot assume that the < or symbol means shading below a line. It is
important to test points on either side of the line to see if the pair of values make the
inequality ture, or to reason carefully about the inequality statement and think about pairs
of values that would satisfy the inequality.

21.4 Pick a Graph

Cool Down: 5 minutes
Addressing

• HSA-REI.D.12

Student Task Statement . Which graph represents ?

1. The line in each graph represents B

A

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

CD

2. Explain your reasons for choosing that graph.

Student Response
1. Graph C

2. Sample response: I substituted the coordinates of a few points above the line into the

inequality and found that they are all not solutions. The point , which is on the

line, is also not a solution. I concluded that the points on and above the line are not

solutions, and the region below the line represents the solutions.

Student Lesson Summary

The equation is an equation in two variables. Its solution is any pair of

and whose sum is 7. The pairs and are two examples.

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We can represent all the solutions to by

graphing the equation on a coordinate plane.

The graph is a line. All the points on the line are

solutions to .

The inequality is an inequality in two variables. Its solution is any pair of

and whose sum is 7 or less than 7.

This means it includes all the pairs that are solutions to the equation , but
also many other pairs of and that add up to a value less than 7. The
pairs and are two examples.

On a coordinate plane, the solution to

includes the line that represents . If we plot a

few other pairs that make the inequality true,

such as and , we see that these points fall

on one side of the line. (In contrast, pairs that

make the inequality false fall on the other side of the

line.)

We can shade that region on one side of the line to
indicate that all points in it are solutions.

What about the inequality ?

The solution is any pair of and whose sum is less than 7. This means pairs like
and are not solutions.

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On a coordinate plane, the solution does not include

points on the line that represent (because

those points are and pairs whose sum is 7).

To exclude points on that boundary line, we can use a
dashed line.

All points below that line are pairs that make

true. The region on that side of the line can

be shaded to show that it contains the solutions.

Lesson 21 Practice Problems

1. Problem 1

Statement .

Here is a graph of the equation

a. Are the points and solutions to

the equation? Explain or show how you know.

b. Check if each of these points is a solution to

the inequality :

c. Shade the region that represents the solution

set to the inequality .

d. Are the points on the line included in the
solution set? Explain how you know.

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Solution

a. Yes, both points are solutions to the equation.

Sample reasoning: is , which

equals 1, and is , which equals 1.

b. and are solutions. and

are not solutions.

c. See graph.

d. No. Sample reasoning: The point , which is

on the line, is not a solution to the inequality. We

can check with another point on the line,

say, . That point is also not a soution.

2. Problem 2 .

Statement

Select all coordinate pairs that are solutions to the inequality
A.
B.
C.
D.
E.
F.
G.

Solution

["A", "B", "G"]

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3. Problem 3

Statement .

Consider the linear equation

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

is a solution to the equation.

b. Are and solutions to the inequality ? Explain how

you know.

c. Explain how to use the answers to the previous questions to graph the

solution set to the inequality .

Solution

a. Sample responses:

b. is a solution, but is not. Sample reasoning: is -10, which
is less than 5, but is 5, which is not less than 5.

c. Sample response: Use the points from the first question to graph the equation.

Then, shade the side of the line that contains because that point is a

solution to the inequality. The point is on the line and is not a solution, so

the line should be dashed.

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4. Problem 4 . Write an

Statement

The boundary line on the graph represents the equation
inequality that is represented by the graph.

Solution or

Sample responses:

5. Problem 5

Statement

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Choose the inequality whose solution set is represented by this graph.

A.
B.
C.
D.

Solution

A

6. Problem 6
Statement

Solve each system of equations without graphing.

a.

b.

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Solution

a.
b.

(From Unit 2, Lesson 14.)

7. Problem 7

Statement

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Mai and Tyler are selling items to earn money for their elementary school. The
school earns dollars for every wreath sold and dollars for every potted
plant sold. Mai sells 14 wreaths and 3 potted plants and the school earns
$70.50. Tyler sells 10 wreaths and 7 potted plants and the school earns $62.50.

This situation is represented by this system of

equations:

Explain why it makes sense in this situation that the solution of this system is

also a solution to .

Solution

Sample responses:

◦ The equation represents the difference between Mai's sales

and Tyler's sales, or the result of subtracting Tyler's sales from Mai's. The left

side shows the difference in terms of the items sold. The right side shows the

difference in dollars. The cost per wreath, , and the cost per potted plant,

, haven't changed.

◦ The equation is the result of subtracting the second equation

in the system from the first equation. Because is equal to 62.50,

subtracting from the left side of and subtracting

62.50 from its right side keep the two sides equal. The pair that is a

solution to the original system still works in the third equation.

(From Unit 2, Lesson 15.)

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8. Problem 8
Statement

Elena is planning to go camping for the weekend and has already spent $40 on
supplies. She goes to the store and buys more supplies.
Which inequality represents , the total amount in dollars that Elena spends
on supplies?

A.
B.
C.
D.

Solution

A

(From Unit 2, Lesson 18.)

9. Problem 9
Statement

Solve this inequality:

Solution

(From Unit 2, Lesson 19.)

10. Problem 10
Statement

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Which graph represents the solution to ?

A.
B.
C.
D.

Solution

A

(From Unit 2, Lesson 19.)

11. Problem 11

Statement

Solve . Explain how to find the solution set.

Solution

. Sample explanation: The only way for the opposite of a number to be less
than -3 is for the number to be more than 3. To get a number whose opposite is
“more negative” than -3, you need a number that is greater than 3.

(From Unit 2, Lesson 20.)

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Lesson 22: Graphing Linear Inequalities in Two
Variables (Part 2)

Goals
• Find the solution to a two-variable inequality by graphing a related two-variable

equation and determining the correct region for the solution.

• Interpret, in context, points on the graphs of equations and in the solution region of

inequalities in two variables.

• Write inequalities in two variables to represent the constraints in a situation and

identify possible solutions by reasoning.

Learning Targets
• Given a two-variable inequality that represents a situation, I can interpret points in

the coordinate plane and decide if they are solutions to the inequality.

• I can find the solutions to a two-variable inequality by using the graph of a related

two-variable equation.

• I can write inequalities to describe the constraints in a situation.

Lesson Narrative

In a previous lesson, students learned to graphically represent the set of solutions to a
linear inequality in two variables. They made a connection between the solutions to a
linear inequality and the solutions to a related linear equation.

In this lesson, students deepen their understanding of the solutions to linear inequalities
by studying them in context. They write inequalities in two variables to represent
constraints, and interpret the points on a boundary line and on either side of it in terms of
the situation.

The work here illustrates that the solution region represents the set of values that satisfy
the constraint in a situation (MP2). Interpreting the solutions contextually also engages
students in an aspect of mathematical modeling (MP4). It enables students to see that,
while some values might make an inequality true, they might not be feasible or
appropriate in the situation. The activity Rethinking Landscaping is an opportunity to make a
generalization based on repeated reasoning (MP8), since students first find numbers that

Algebra1 Unit 2 Lesson 22 CC BY 2019 by Illustrative Mathematics 1

meet a constraint, and then use variables in place of those numbers to write an equation
and an inequality.

Because reasoning about the solution region of an inequality is important here, graphing
technology should not be used. Students will have opportunities to use graphing
technology to solve inequalities in two variables in upcoming lessons.

Alignments
Addressing

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

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

• HSA-REI.D.10: Understand that the graph of an equation in two variables is the set of

all its solutions plotted in the coordinate plane, often forming a curve (which could be
a line).

• 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

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

Student Learning Goals

• Let’s write inequalities in two variables and make sense of the solutions by

reasoning and by graphing.

22.1 Landscaping Options

Warm Up: 5 minutes
In this lesson, students will be writing linear inequalities that represent constraints in
situations and graphing the solution regions. To prepare for that work, students
review writing and graphing an equation that represents a situation.

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Addressing

• HSA-CED.A.3
• HSA-REI.D.10

Launch
If needed, explain or show additional images of grass sod and flower beds to students who
might be unfamiliar with these landscaping terms.

Student Task Statement
A homeowner is making plans to
landscape her yard. She plans to hire
professionals to install grass sod in some
parts of the yard and flower beds in other
parts.

Grass sod installation costs $2 per square
foot and flower bed installation costs $12
per square foot. Her budget for the
project is $3,000.

1. Write an equation that represents the square feet of grass sod, , and the
square feet of flower beds, , that she could afford if she used her entire
budget.

2. On the coordinate plane, sketch a graph that represents your equation. Be
prepared to explain your reasoning.

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Student Response
1. (or equivalent)

2. Sample response:

Activity Synthesis

Invite students to share their equation and graph. Discuss with students:

• "In this situation, what does a point on the line mean?" (A combination of square feet

of grass sod and square feet of flower beds that the homeowner could have if she
spent her entire budget.)

Algebra1 Unit 2 Lesson 22 CC BY 2019 by Illustrative Mathematics 4

• "What does the vertical intercept of the graph mean?" (The square feet of flower beds

she could have if she installs no grass sod.)

• "What does the horizontal intercept of the graph tell us?" (The square feet of grass

sod she could install if she installs no flower beds.)

22.2 Rethinking Landscaping

20 minutes
This is the first of a series of activities in which students write an inequality to represent a
constraint in a situation. Students graph a related equation, interpret the coordinate pairs
of points on the graph and on either side of the graph, and test the pairs of values to see if
they make the inequality true. They then use these observations to determine the solution
region to the inequality.

Students also consider whether all the points in the solution region are necessarily
meaningful or feasible in the situation. In doing so, they reason quantitatively and
abstractly (MP2) and practice evaluating the reasonableness of their solutions in context
(MP4).

Graphing technology should not be used in this activity and the other activities in the
lesson.

Addressing

• HSA-CED.A.3
• HSA-REI.D.12

Instructional Routines

• Aspects of Mathematical Modeling
• MLR6: Three Reads

Launch
Briefly explain to students who are unfamiliar with landscape materials what artificial turf
and gravel are (or show additional pictures).

Consider arranging students in groups of 2 and giving them quiet time to work on the first
set of questions, followed by time to share their thinking before moving on to the rest of
the activity.

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Support for English Language Learners

Reading: 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 (a homeowner is
considering alternatives to maintaining her grass lawn and flower beds). Use the
second read to identify quantities and relationships. Ask students what can be
counted or measured without focusing on the values (cost per square foot of artificial
turf and gravel, total budget). After the third read, ask students to brainstorm possible
strategies to answer the question “What combinations of turf and gravel meet the
homeowner’s constraints?” This helps students connect the language in the word
problem and the reasoning needed to solve the problem while maintaining the
cognitive demand of the task.
Design Principle: Support sense-making

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 to their strategies for exploring solution sets of inequalities.
Supports accessibility for: Organization; Visual-spatial processing

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Student Task Statement .
The homeowner is worried about the mean?
work needed to maintain a grass lawn
and flower beds, so she is now looking at
some low-maintenance materials.

She is considering artificial turf, which
costs $15 per square foot to install, and
gravel, which costs $3 per square foot.
She may use a combination of the two
materials in different parts of the yard.
Her budget is still $3,000.

Here is a graph representing some
constraints in this situation.

1. The graph shows a line going through

a. In this situation, what does the point

b. Write an equation that the line represents.

c. What do the solutions to the equation mean?

2. The point is located to the right and above the line.

a. Does that combination of turf and gravel meet the homeowner’s
constraints? Explain or show your reasoning.

b. Choose another point in the same region (to the right and above the
line). Check if the combination meet the homeowner’s constraints.

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3. The point is located to the left and below the line.

a. Does that combination of turf and gravel meet the homeowner’s
constraints? Explain or show your reasoning.

b. Choose another point in the same region (to the left and below the line).
Check if the combination meets the homeowner’s constraints.

4. Write an inequality that represents the constraints in this situation. Explain
what the solutions mean and show the solution region on the graph.

Student Response

1. Sample response:
a. The cost of installing 500 square feet of gravel and 100 square feet of artificial
turf is $3,000.

b.

c. The solutions are all combinations of square feet of gravel and artificial turf that
would cost $3,000 to install.

2. a. No. Sample reasoning: If she installs 600 square feet of gravel and 200 square

feet of turf, the cost is , which is or $4,800.

b. Sample response: . This combination would cost more than $3,000 to
install.

3. a. Yes. Sample reasoning: . This amount is less than
$3,000.

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b. Sample response: . This combination would cost under $3,000 to
install.

4. . Sample explanation: The solutions represent all possible
combinations of square feet of gravel and turf that would cost no more than $3,000
to install.

Activity Synthesis

Select students to share their interpretations of the two points on either side of the line.
Make sure students understand that one point represents a combination of gravel and
artificial turf that meets the budget constraint and that the other point does not. The
region in which each point belongs can be interpreted in the same way.

Also make sure students understand why points on the boundary line are included in the

solution set of .

To encourage students to evaluate the reasonableness of their solutions in terms of the
situation being modeled, discuss questions such as:

• "Both and are in the solution region. Both points mean a total of

150 square feet of landscape materials. Does it make a difference which option is
chosen?" (It does not make a difference in terms of the total area, but it does in terms
of cost. It might also make a difference to the plants and to the overall appearance of
the yard.)

• “All the points in the shaded region represent amounts of gravel and turf that are

within the homeowner's budget. Are all these options equally good and desirable?"

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(Most likely not. For example, if the homeowner needs both materials, pairs such as

or are probably not desirable because they mean buying only one

material but not the other. The point is also in the solution region, but buying

no materials is probably also not an option because it would not help the

homeowner with her landscaping goals.)

22.3 The Saturday Market

10 minutes
In this activity, students continue to work with inequalities in two variables in context. They
write an inequality that represents the constraints in a situation, graph its solutions, and
interpret points in the solution region.

Earlier, students saw that some points in the solution region might satisfy an inequality
(mathematically) but might not work in the given situation because other considerations
were at play. Here, they see another reason that some values that satisfy an inequality
might be unfeasible in the situation—namely, that the pair of values must be whole
numbers. When students think about the plausibility of these values in context, they
engage in an aspect of mathematical modeling (MP4).

As students work, look for those who plot discrete points to represent the solution region
and those who shade a part of the plane.

Addressing

• HSA-CED.A.3
• HSA-REI.D.12

Instructional Routines

• Aspects of Mathematical Modeling
• Think Pair Share

Launch
Arrange (or keep) students in groups of 2. Give students a few minutes of quiet time to
work on the first three questions, and then time to discuss their responses before they
continue with the last two questions.

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Support for Students with Disabilities

Representation: Internalize Comprehension. To support working memory, provide
students with sticky notes or mini whiteboards. Some students may need extra time
to explore possible combinations for multiples of 9 and 5. Encourage them to model
these combinations by drawing pictures or listing multiples of each number using the
whiteboard or sticky notes.
Supports accessibility for: Memory; Organization

Anticipated Misconceptions
Some students may have trouble graphing the line that delineates the solution region
from non-solution region because they are used to solving for the variable , but here the
variables are and . Ask these students to decide which variable to solve for based on
the graph that has been set up. Ask them to notice which quantity is represented by the
vertical axis.

Student Task Statement
A vendor at the Saturday Market makes $9 profit on each necklace she sells and $5
profit on each bracelet.

1. Find a combination of necklaces and bracelets that she could sell and make:

a. exactly $100 profit

b. more than $100 profit

2. Write an equation whose solution is the combination of necklaces and
bracelets she could sell and make exactly $100 profit.

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3. Write an inequality whose solutions are the combinations of necklaces and
bracelets she could sell and make more than $100 profit.

4. Graph the solutions to your inequality.

5. Is a solution to the inequality? Explain your reasoning.

Student Response
1. Possible responses:
a. 10 necklaces and 2 bracelets
b. 0 necklaces and 20 bracelets
2. , where represents number of necklaces sold and represents
number of bracelets.
3.

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4.

5. Sample responses: is in the shaded region.

◦ Yes, it is. The point

◦ No, it is not. The vendor can’t sell 18.6 bracelets.

Are You Ready for More?

1. Write an inequality using two variables and where the solution would be
represented by shading the entire coordinate plane.

2. Write an inequality using two variables and where the solution would be
represented by not shading any of the coordinate plane.

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Student Response
Sample responses:

1.

2.

Students may first come up with good reasons why this can’t be done (with linear

inequalities). They may also respond with or something similar. Both of these

tacks deserve praise, but tell students that this problem really is possible using real

numbers. Students may ultimately need the hint that they will need to leave the realm of

linear inequalities to answer these questions.

Activity Synthesis

If some students plotted discrete points and some shaded the region, choose one of each
and display these for all to see. Ask students why each one might be an appropriate
representation of the solutions to the inequality.

Highlight that the discrete points represent the situation more accurately, because it is
impossible to sell a fraction of a bracelet or 2.75 necklaces. It is, however, tedious to plot a
bunch of points to show the solution region. It is much easier to shade the entire region,
but with the understanding that, in this situation, only whole-number values make sense
as solutions.

22.4 Charity Concerts

Optional: 15 minutes
This optional activity offers an additional opportunity for writing a linear inequality in two
variables to represent constraints and for graphing and interpreting the solutions.

In previous activities, the solution region of an inequality lies below a

boundary line and the solution region of an inequality lies above a boundary

line. In this activity, students encounter an example in which the symbol does not

correspond to a solution region above the boundary line. (In the given situation, pairs of

values that generate more revenue for the concert are points below the graph of

.)

The work here reinforces the importance of reasoning about points on either side of a
boundary line, rather than simply assuming that < or means shading below the line and
> or means shading above the line. It allows students to practice reasoning
quantitatively and abstractly (MP2).

Algebra1 Unit 2 Lesson 22 CC BY 2019 by Illustrative Mathematics 14

Addressing

• HSA-CED.A.3
• HSA-REI.D.12

Launch

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. Use this opportunity to show students an
example of reading for understanding. Demonstrate pausing after each sentence, and
rereading to stress important information.
Supports accessibility for: Language; Conceptual processing

Anticipated Misconceptions

Some students may find it challenging to graph the boundary line ( )

by identifying the intercepts. The horizontal intercept is fairly easy to find, but the graph

intersects the vertical axis at a negative value (and a negative number of concerts does not

make sense in this situation). Ask students to find at least one other point (besides the

vertical intercept) that is a solution to the equation.

Students who rewrite the equation in slope-intercept form and find the slope to be 0.02
may also find it difficult to interpret. Ask them to try writing the slope as a fraction ( ).

Student Task Statement
A popular band is trying to raise at least $20,000 for charity by holding multiple
concerts at a park. It plans to sell tickets at $25 each. For each 2-hour concert, the
band would need to pay the park $1,250 in fees for security, cleaning, and traffic
services.

The band needs to find the combinations of number of tickets sold, , and number
of concerts held, , that would allow it to reach its fundraising goal.

1. Write an inequality to represent the constraints in this situation.

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

3. Name two possible combinations of number of tickets sold and number of
concerts held that would allow the band to meet its goal.

4. Which combination of tickets and concerts would mean more money for
charity:
a. 1,300 tickets and 10 concerts, or 1,300 tickets and 5 concerts?
b. 1,600 tickets and 16 concerts, or 1,200 tickets and 9 concerts?
c. 2,000 tickets and 4 concerts, or 2,500 tickets and 10 concerts?

Algebra1 Unit 2 Lesson 22 CC BY 2019 by Illustrative Mathematics 16

Student Response
1.

2.

3. Sample response: and

4. a. 1,300 tickets and 5 concerts

b. The two combinations would raise exactly the same amount of money
($20,000).

c. 2,000 tickets and 4 concerts

Activity Synthesis
Invite students to share their inequality and the graph of the solution region.

Focus the discussion on the last two questions—on how students knew which
combinations of tickets and concerts would enable the band to meet its goal and would
raise more money. Highlight responses that involve testing pairs of values to see if they
satisfy the inequality or to make comparisons.

If not mentioned in students' explanations, point out that even though the inequality has a
symbol, the solution region is below the boundary line, not above it. Stress the

importance of not blindly shading a region based on the symbol in the inequality.

Lesson Synthesis

Display the inequalities and graphs from both activities in the lesson.

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Ask students to compare the solution regions and think about what they tell us about the
constraints of the situation they represent. Discuss questions such as:

• "How are the solution regions of the two inequalities alike?" (Each region represents

all pairs of values that meet a money-related constraint in a situation. They cover a

part of the plane. They stop at a line.)

• "How do we know where the boundary line would be for each graph?" (It is the graph

of a related equation.)

• "For each inequality, how did we find out which side of the line contains the

solutions?" (We tested one or more pairs of values on each side and see if—when

substituted for the variables—they make the inequality true.)

• "In the first situation, some pairs of values that are in the solution region don't make

sense in the situation. Can you explain why a pair such as , which is in the

solution region, might not be a reasonable option for the homeowner?" (It doesn't

quite make sense to cover 1 square foot of the garden with artificial turf and the rest

with gravel. The area of the garden might be a lot greater or a lot less than 801

square feet.)

• "In the second situation, we know that fractional values are not meaningful even

though they are in the shaded region. Can you think of other reasons that

some points in the solution region might not make sense?" (A point like

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

would be in the solution region, but the vendor might not have that many items to
sell.)

Some students may point out that it is possible to reason about the side that contains the
solutions by reasoning about each context. In these particular examples, this can be done
intuitively and correctly. When the boundary line represents a the cost of two quantities
exactly on budget, smaller values of each quantity would lead to costs that are below the
budget. When the boundary line represents a profit of $100 from selling two kinds of
products, a greater number of each product would mean a greater profit.

Emphasize, however, that it is not always the case that the solution region could be
reasoned easily or correctly from the context, so it is always a good idea to verify using
another method. The optional activity (Charity Concert) illustrates this point.

22.5 A Weekend of Games

Cool Down: 5 minutes
Graphing technology should not be used in this cool-down.

Addressing

• HSA-CED.A.3
• HSA-REI.D.12

Student Task Statement
To raise money for after-school programs at an elementary school, a group of
parents is holding a weekend of games in a community center. They charge $8 per
person for entry into the event. The group would like to earn at least $600, after
paying for the cost of renting the space, which is $40 an hour.

1. If represents the number of entry tickets sold and the hours of space
rental, which inequality represents the constraints in the situation?
a.

b.

c.

d.

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

2. The line is the graph of . Select all points whose values

represent the group reaching its fundraising goal. Explain or show your

reasoning.

3. Complete the graph so that it represents solutions to an inequality that
represents this situation. (Be clear about whether you want to use a solid or
dashed line.)

Student Response
1. d

2. Points B and C. Sample explanation: When the coordinates of B and C are substituted
for the and in the equation, they result in a number that is at least 600.

3.

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

Student Lesson Summary

Inequalities in two variables can represent constraints in real-life situations.
Graphing their solutions can enable us to solve problems.

Suppose a café is purchasing coffee and tea from a supplier and can spend up to
$1,000. Coffee beans cost $12 per kilogram and tea leaves costs $8 per kilogram.

Buying pounds of coffee beans and pounds of tea leaves will therefore cost

. To represent the budget constraints, we can write: .

The solution to this inequality is any pair of and that makes the inequality true. In
this situation, it is any combination of the pounds of coffee and tea that the café
can order without going over the $1,000 budget.

We can try different pairs of and to see what combinations satisfy the constraint,

but it would be difficult to capture all the possible combinations this way. Instead,

we can graph a related equation, , and then find out which region

represents all possible solutions.

Here is the graph of that equation.

To determine the solution region, let’s
take one point on the line and one point
on each side of the line, and see if the
pairs of values produce true statements.

A point on the line:

This is true.

Algebra1 Unit 2 Lesson 22 CC BY 2019 by Illustrative Mathematics 21

A point below the line: A point above the line:

This is true. This is false.

The points on the line and in the region
below the line are solutions to the
inequality. Let's shade the solution
region.

It is easy to read solutions from the

graph. For example, without any

computation, we can tell that is a

solution because it falls in the shaded

region. If the café orders 50 kilograms of

coffee and 20 kilograms of tea, the cost

will be less than $1,000.

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

Lesson 22 Practice Problems

1. Problem 1

Statement

To qualify for a loan from a bank, the total in someone’s checking and savings
accounts together must be $500 or more.

a. Which of these inequalities best
represents this situation?

▪
▪
▪
▪

b. Complete the graph so that it
represents solutions to an
inequality representing this
situation.

(Be clear about whether you want
to use a solid or dashed line.)

Algebra1 Unit 2 Lesson 22 CC BY 2019 by Illustrative Mathematics 23

Solution

a.

b.

Algebra1 Unit 2 Lesson 22 CC BY 2019 by Illustrative Mathematics 24

2. Problem 2

Statement

The soccer team is selling bags of popcorn for $3 each and cups of lemonade
for $2 each. To make a profit, they must collect a total of more than $120.

a. Write an inequality to represent
the number of bags of popcorn
sold, , and the number of cups
of lemonade sold, , in order to
make a profit.

b. Graph the solution set to the
inequality on the coordinate
plane.

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

Solution

a.

b. See graph.

c. Sample response: The solution is not included
because the coordinates of the points on the line
do not satisfy the inequality. For example,
means 32 bags of popcorn and 12 cups of
lemonade. This would mean they collect a total of
$120, which is not enough to make a profit.

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

3. Problem 3

Algebra1 Unit 2 Lesson 22 CC BY 2019 by Illustrative Mathematics 26

Statement

Tickets to the aquarium are $11 for adults and $6 for children. An after-school
program has a budget of $200 for a trip to the aquarium.

If the boundary line in each graph represents the equation ,
which graph represents the cost constraint in this situation?

Algebra1 Unit 2 Lesson 22 CC BY 2019 by Illustrative Mathematics 27

A.

B.

C.

D.

Solution

C

Algebra1 Unit 2 Lesson 22 CC BY 2019 by Illustrative Mathematics 28

4. Problem 4

Statement

Tyler filled a small jar with quarters and dimes and donated it to his school's
charity club. The club member receiving the jar asked, "Do you happen to
know how much is in the jar?" Tyler said, "I know it's at least $8.50, but I don't
know the exact amount."

a. Write an inequality to represent
the relationship between the
number of dimes, , the number
of quarters, , and the dollar
amount of the money in the jar.

b. Graph the solution set to the
inequality and explain what a
solution means in this situation.

c. Suppose Tyler knew there are 25
dimes in the jar. Write an
inequality that represents how
many quarters could be in the jar.

Solution (or equivalent)

a.

b. See graph. A solution represents a number of
dimes and a number of quarters that together are
worth $8.50 or more.

c. Sample response: or or

Algebra1 Unit 2 Lesson 22 CC BY 2019 by Illustrative Mathematics 29

5. Problem 5

Statement . He first solves a related

Andre is solving the inequality
equation.

This seems strange to Andre. He thinks he probably
made a mistake. What was his mistake?

Solution

Sample response: Andre made a mistake after getting . He divided each side

by , but this is not valid since may be 0 and dividing by 0 gives an undefined result.

(From Unit 2, Lesson 20.)

Algebra1 Unit 2 Lesson 22 CC BY 2019 by Illustrative Mathematics 30

6. Problem 6
Statement

Kiran says, “I bought 2.5 pounds of red and yellow lentils. Both were $1.80 per
pound. I spent a total of $4.05.”

a. Write a system of equations to describe the relationships between the
quantities in Kiran's statement. Be sure to specify what each variable
represents.

b. Elena says, “That can't be right.” Explain how Elena can tell that
something is wrong with Kiran's statement.

c. Kiran says, “Oops, I meant to say I bought 2.25 pounds of lentils.” Revise
your system of equations to reflect this correction.

d. Is it possible to tell for sure how many pounds of each kind of lentil Kiran
might have bought? Explain your reasoning.

Algebra1 Unit 2 Lesson 22 CC BY 2019 by Illustrative Mathematics 31

Solution

a.

b. Sample explanations:

▪ When the equations are graphed, the lines are parallel. Because the two

lines never intersect, there are no values of and that could make both

equations true.

▪ If we multiply both sides of by 1.8, we have .

The expression cannot equal both 2.5 and 4.5.

c. Change to .

d. No. Sample explanations:

▪ The graphs of the two equations are the same line, so there are countless

possible combinations of red and yellow lentils that Kiran could have

bought.

▪ and are equivalent, so they share all the

same solutions. There are many pairs of values that meet both the weight

and price constraints.

(From Unit 2, Lesson 17.)

7. Problem 7

Statement

Algebra1 Unit 2 Lesson 22 CC BY 2019 by Illustrative Mathematics 32