IM UNIT 2 Equations and Inequalities Teacher Guide

1. Choose two ingredients that you like to eat. (You can choose from the
ingredients in the previous activity, or you can look up nutrition information
for other ingredients.)

2. Think about the constraints for your trail mix. What do you want to be true
about its calories, protein, sugar, fat, or fiber?

3. Write inequalities to represent your constraints. Then, graph the inequalities.

4. Is it possible to make trail mix that meets all your constraints using your
ingredients? If not, make changes to your constraints or your ingredients and
record them here.

5. Write a possible combination of ingredients for your trail mix.

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Pause here so your teacher can review your work and give further instructions for
displaying your work.

Student Response
Sample response:

1. Sunflower seeds and raisins

2. The trail mix should have less than 300 calories and more than 5 grams of protein.

3. Let represent amount of sunflower seeds in grams and represent amount of
raisins in grams.

4. Yes

5. 30 grams of sunflower seeds and 40 grams of raisins

Activity Synthesis
Select groups to share their visual displays. Encourage students to ask questions about the
mathematical thinking or design approach that went into creating the display. Here are
questions for discussion, if not already mentioned by students:

• What constraints did every group use?
• How do the graphs of the various mixes compare?

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• Did anyone have to revise or change their model in order to come up with a solution

they could use?

• How did you use the graph to choose a recipe for your mix?

Support for English Language Learners

Representing, Conversing: MLR7 Compare and Connect. Use this routine to prepare
students for the whole-class discussion. At the appropriate time, invite groups to
create a visual display of their work. Students should consider what types of details
(annotations, notes, diagrams, arrows, etc.) to include on their displays that will help
communicate their reasoning. Begin the whole-class discussion by selecting and
arranging 2–4 displays for all to see. Give students 2–3 minutes of quiet think time to
interpret the displays before inviting the authors to present their work.
Design Principle(s): Optimize output; Cultivate conversation

Lesson Synthesis

Allow enough time for students to present their trail mix recipes. Consider a gallery walk
as a way for students to share their display and to ask and answer questions.

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Lesson 26 Practice Problems

1. Problem 1

Statement

The organizers of a conference needs to prepare at least 200 notepads for the
event and have a budget of $160 for the notepads. A store sells notepads in
packages of 24 and packages of 6.

This system of inequalities represent these constraints:

a. Explain what the second inequality in the system tells us about the
situation.

b. Here are incomplete graphs of the
inequalities in the system, showing
only the boundary lines of the
solution regions.
Which graph represents the
boundary line of the second
inequality?

c. Complete the graphs to show the solution set to the system of
inequalities.

d. Find a possible combination of large and small packages of notepads the
organizer could order.

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Solution

a. Sample response: The price for each large package
is $18 and the price for a small package is $5.40.
The total cost of buying large packages and
small packages must be at most $160.

b. The graph that intersects the horizontal axis at 10
represents the second inequality.

c. See graph.

d. Sample response: 10 large packages and 0 small
packages

(From Unit 2, Lesson 25.)

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2. Problem 2
Statement

A certain stylist charges $15 for a haircut and $30 for hair coloring. A haircut
takes on average 30 minutes, while coloring takes 2 hours. The stylist works up
to 8 hours in a day, and she needs to make a minimum of $150 a day to pay
for her expenses.

a. Create a system of inequalities that describes the constraints in this
situation. Be sure to specify what each variable represents.

b. Graph the inequalities and show the solution set.

c. Identify a point that represents a combination of haircuts and and
hair-coloring jobs that meets the stylist’s requirements.

d. Identify a point that is a solution to the system of inequalities but is not
possible or not likely in the situation. Explain why this solution is
impossible or unlikely.

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Solution

a. Let represent the number of haircuts and the number of coloring
appointments.

b. Sample graph:

c. Sample response: The point represents 11 haircuts and 1 coloring job,

which will take less than 8 hours and generate more than $150 in income.

d. Sample response: is in the solution region of the system of inequalities,

▪ The point

but it is not possible to have a fractional number of haircut or

hair-coloring appointments.

▪ The point is a solution to the system, but it is impossible to have a

negative number of hair-coloring jobs.

▪ The point is a solution to the system, but it is very unlikely that the

stylist does 16 haircuts for 8 hours without taking a break.

(From Unit 2, Lesson 25.)

3. Problem 3

Statement

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Choose the graph that shows the solution to this system:

A.

B.

C.

D.

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Solution

D

(From Unit 2, Lesson 24.)

4. Problem 4

Statement

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Match each inequality to the graph of its solution.
1.

2.
A.

3.

4.
B.

5.

C.

D.

E.

Solution

5. A: 4

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6. B: 3
7. C: 1
8. D: 5
9. E: 2

(From Unit 2, Lesson 23.)

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