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ACTIVITY 2 Continued
Lesson 2-2
Graphing Systems of Inequalities
ACTIVITY 2
continued
PLAN
My Notes
Learning Targets:
• Represent constraints by equations or inequalities.
• Use a graph to determine solutions of a system of inequalities.
Pacing: 1 class period
Chunking the Lesson
#1
#2
#3
#4–5
#6
Check Your Understanding
#11
#12–13 #14–15
Check Your Understanding
Lesson Practice
SUGGESTED LEARNING STRATEGIES: Think-Pair-Share, Interactive
Word Wall, Create Representations, Work Backward, Discussion Groups,
Close Reading, Debriefing, Activating Prior Knowledge
Work with your group on Items 1 through 5. As needed, refer to the Glossary
to review translations of key terms. Incorporate your understanding into group
discussions to confirm your knowledge and use of key mathematical language.
1. Roy’s spending money depends on both the number of tickets t and the
number of meals m. Determine whether each option is feasible for Roy
and provide a rationale in the table below.
Tickets
(t)
Meals
(m)
Total
Cost
Is it
feasible?
Rationale
6
16
1240
Yes
1240 ≤ 1360
8
14
1360
Yes
1360 = 1360
10
12
1480
No
1480 > 1360
No
You cannot buy half a
ticket.
4.5
11
890
2. Construct viable arguments. For all the ordered pairs (t, m) that
are feasible options, explain why each statement below must be true.
a. All coordinates in the ordered pairs are integer values.
© 2015 College Board. All rights reserved.
© 2015 College Board. All rights reserved.
Sample explanation: Both meals and tickets are integral values, so
only coordinate pairs that are integers will be solutions.
b. If graphed in the coordinate plane, all ordered pairs would fall either
in the first quadrant or on the positive m-axis.
Sample explanation: Tickets must be greater than or equal to zero,
and meals must be greater than zero.
3. Write a linear inequality that represents all ordered pairs (t, m) that are
feasible options for Roy.
100t + 40m ≤ 1360 for t ≥ 0 and m > 0
4. If Roy buys exactly two meals each day, determine the total number of
tickets that he could purchase in five days. Show your work.
9 tickets. Sample answer:
100t + 40(10) ≤ 1360
100t ≤ 960
t ≤ 9.6
Because you cannot purchase part of a ticket, at most 9 tickets can
be purchased.
Lesson 2-2
#7
TEACH
ACADEMIC VOCABULARY
The term feasible means that
something is possible in a given
situation.
DISCUSSION GROUP TIPS
As you share your ideas, be sure to
use mathematical terms and
academic vocabulary precisely.
Make notes as you listen to group
members to help you remember
the meaning of new words and
how they are used to describe
mathematical concepts. Ask and
answer questions clearly to aid
comprehension and to ensure
understanding of all group
members’ ideas.
MATH TERMS
A linear inequality is an inequality
that can be written in one of these
forms, where A and B are not both
equal to 0:
Ax + By < C, Ax + By > C,
Ax + By ≤ C, or Ax + By ≥ C.
Activity 2 • Graphing to Find Solutions
21
Bell-Ringer Activity
Write the statement “You must be at
least 13 years old and at least 54 inches
tall to ride this ride.” on the board. Write
the general ordered pair (age, height) on
the board. Ask students which of the
following people could ride the ride:
Anna(11, 48); Ben(13, 58);
Candice(14, 41); Danielle(10, 55);
Ed(15, 60). Have students discuss which
constraint prevents students from riding
the ride.
1 Interactive Word Wall Introduce
students to the word feasible as it applies
to the solution sets of equations and
inequalities. When using algebraic
expressions to find solution sets within
an applied setting, students must always
interpret the solution set for
reasonableness within the context of the
applied setting. To further illustrate the
word feasible, pose the following to
students: You and a friend are playing a
game of Tic-Tac-Toe. What are the
feasible results of the game? [you win, you
lose, you tie]
2 Think-Pair-Share This item returns
to the domain constraints of the
problem. Discuss how those constraints
affect the feasible options for the
conditions given in Item 1. The statement
in Item 2b assumes that Roy will buy
some number of meals greater than zero.
3 Create Representations,
Debriefing For this item, assume that
Roy has no additional expenses.
Students may incorrectly assume that
everyone eats three meals per day.
Therefore, they may be confused by m
as a variable for the number of meals
that Roy may eat. From their
perspective, Roy will eat three meals per
day, or 15 meals, during his entire stay
in New York City.
4–5 Think-Pair-Share, Work
Backward, Discussion Groups In
Item 4, some students may give t ≤ 9.6
as the answer. Although this is a correct
solution to the inequality, it does not
answer the question asked, which
requires an integer answer.
Activity 2 • Graphing to Find Solutions
21
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ACTIVITY 2 Continued
Lesson 2-2
Graphing Systems of Inequalities
ACTIVITY 2
continued
My Notes
21 meals. Sample answer:
100(5) + 40m ≤ 1360
40m ≤ 860
m ≤ 21.5
Because you cannot purchase part of a meal, at most 21 meals can be
purchased.
Recall how to graph linear
inequalities. First, graph the
corresponding linear equation.
Then choose a test point not on
the line to determine which
half-plane contains the set of
solutions to the inequality. Finally,
shade the half-plane that contains
the solution set.
6. To see what the feasible options are, you can use a visual display of the
values on a graph.
a. Attend to precision. Graph your inequality from Item 3 on the
grid below.
m
Number of Meals
6 Create Representations,
Activating Prior Knowledge,
Quickwrite Review how to graph a
linear inequality. Later, in Item 8,
students will graph all the constraint
inequalities on one grid. For now, the
focus is on one inequality. This makes it
easier to determine whether or not
students have any misunderstandings
about the procedure for graphing linear
inequalities. If students need additional
help, assign Mini-Lesson: Graphing
Linear Inequalities.
MATH TIP
36
34
32
30
28
26
24
22
20
18
16
14
12
10
8
6
4
2
2 4 6 8 10 12 14 16
t
Number of Tickets
b. What is the boundary line of the graph?
100t + 40m = 1360
c. Which half-plane is shaded? How did you decide?
The lower half-plane is shaded.
Sample explanation: The test point (0, 0) satisfies the inequality.
22 SpringBoard® Mathematics Algebra 2, Unit 1 • Equations, Inequalities, Functions
MINI-LESSON: Graphing Linear Inequalities
If students need additional help with graphing linear inequalities,
a mini-lesson is available to provide practice.
See the Teacher Resources at SpringBoard Digital for a student
page for this mini-lesson.
22
SpringBoard® Mathematics Algebra 2, Unit 1 • Equations, Inequalities, Functions
© 2015 College Board. All rights reserved.
ELL Support
Support students whose first language
is not English by pairing them with
more-fluent speakers for Item 6. Pairs
can practice their listening and
speaking skills as they take turns
describing the process of graphing a
linear inequality. Encourage students
to use precise mathematical language
in their discussions.
5. If Roy buys exactly one ticket each day, find the maximum number of
meals that he could eat in the five days. Show your work.
© 2015 College Board. All rights reserved.
4–5 (continued) Likewise, in Item 5,
some students may give m ≤ 21.5 as the
answer. Although this is a solution to
the inequality, the question asks for a
specific number of meals, which also
must be an integer.
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ACTIVITY 2 Continued
Lesson 2-2
Graphing Systems of Inequalities
ACTIVITY 2
continued
My Notes
d. Write your response for each item as points in the form (t, m).
Item 4
Item 5
(9, 10)
(5, 21)
e. Are both those points in the shaded region of your graph? Explain.
Yes. Sample explanation: Both are near but below the boundary line.
7. Use appropriate tools strategically. Now follow these steps to
graph the inequality on a graphing calculator.
a. Replace t with x, and replace m with y. Then solve the inequality
for y. Enter this inequality into your graphing calculator.
100t + 40m ≤ 1360 →
100x + 40y ≤ 1360
40y ≤ 1360 − 100x
y ≤ 34 − 2.5x
TECHNOLOGY TIP
To enter an equation in a graphing
calculator, start with Y= .
b. Use the left arrow key to move the cursor to the far left of the equation
you entered. Press ENTER until the symbol to the left of Y1 changes
to . What does this symbol indicate about the graph?
Sample answer: This symbol indicates that the half-plane below the
boundary line will be shaded.
© 2015 College Board. All rights reserved.
© 2015 College Board. All rights reserved.
c. Now press GRAPH . Depending on your window settings, you may or
may not be able to see the boundary line. Press WINDOW and adjust the
viewing window so that it matches the graph from Item 6. Then press
GRAPH again.
Check students’ graphs.
d. Describe the graph.
Sample answer: The graph shows a line with negative slope, with
shading below and to the left of the line.
TECHNOLOGY TIP
To graph an inequality that
includes ≥ or ≤, you would use
the symbol or . You need to
indicate whether the half-plane
above or below the boundary line
will be shaded.
Activity 2 • Graphing to Find Solutions
Technology Tip
If students are using TI-Nspire
technology, provide the following
directions for how to graph the
inequality given in Item 7:
Step 1: Choose Graphs&Geometry
from the home screen.
Step 2: Change the equal sign to a less
than sign by using the CLEAR key
followed by the < key located in the
leftmost column of white keys.
Step 3: Enter the function f1(x) as
34 − 2.5 X (use the green letter key
for x).
Step 4: Adjust the viewing window as
needed to view the graph.
For additional technology resources,
visit SpringBoard Digital.
7 Activating Prior Knowledge,
Interactive Word Wall,
Debriefing Part a provides an
opportunity to review the concept of
independent and dependent variables.
Discuss that t is replaced with x because
t (the number of tickets) is the
independent variable and m (the
number of meals) depends on the
number of tickets. Students should
discuss why replacing t with y and m
with x does not work. In part c, students
can adjust the viewing window by
pressing WINDOW and entering an Xmin
of 0, an Xmax of 16, an Xscl of 1, a Ymin
of 0, a Ymax of 36, and a Yscl of 1. Note
that due to the height of the graph on
the student page, the graph displayed on
the calculator will not completely match,
but students should be able to relate one
to the other.
23
Activity 2 • Graphing to Find Solutions
23
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ACTIVITY 2 Continued
My Notes
Check Your Understanding
8. Compare and contrast the two graphs of the linear inequality: the one
you made using paper and pencil and the one on your graphing
calculator. Describe an advantage of each graph compared to the other.
Answers
9. a. Solutions on the boundary line
represent solutions for which Roy
would have no money left over.
For these solutions, the total cost
of the tickets plus the total cost of
the meals is equal to $1360, which
is the amount of money Roy has.
b. Solutions below the boundary
line represent solutions for which
Roy would have money left over.
For these solutions, the total cost
of the tickets plus the total cost
of the meals is less than $1360,
which is the amount of money
Roy has.
MATH TIP
Use a solid boundary line for
inequalities that include ≥ or ≤.
Use a dashed boundary line for
inequalities that include > or <.
MATH TERMS
Constraints are the conditions or
inequalities that limit a situation.
10. Explain how you would graph the inequality 2x + 3y < 12, either by
using paper and pencil or by using a graphing calculator.
11. Roy realized that some other conditions or constraints apply. Write an
inequality for each constraint described below.
a. Roy eats lunch and dinner the first day. On the remaining four days,
Roy eats at least one meal each day, but he never eats more than three
meals each day.
6 ≤ m ≤ 14
b. There are only 10 performances playing that Roy actually wants to
see while he is in New York City, but he may not be able to attend all
of them.
0 ≤ t ≤ 10
c. Roy wants the number of meals that he eats to be no more than twice
the number of performances that he attends.
m ≤ 2t
10. Sample answer: Write the related
linear equation. Then choose
several values of x and substitute
them into the equation to find the
corresponding values of y. Graph
the ordered pairs and draw a dashed
line through them. Then choose a
test point not on the boundary line
and use it to determine which
half-plane to shade.
11 Activating Prior Knowledge,
Create Representations Students
must work with the language of
inequalities in such phrases as at least,
at most, not more than, less than, more
than, and so on. Ask students to
distinguish between not more than three
and less than three, and to distinguish
between at least one and more than one.
For each pair of phrases, ask students to
use a number line to identify numbers
that satisfy both phrases and then
numbers that satisfy just one phrase but
not the other. Have students represent
each phrase by writing an inequality.
9. a. What part of your graphs represents solutions for which Roy would
have no money left over? Explain.
b. What part of your graphs represents solutions for which Roy would
have money left over? Explain.
12. Model with mathematics. You can use a graph to organize all the
constraints on Roy’s trip to New York City.
a. List the inequalities you found in Items 3 and 11.
6 ≤ m ≤ 14, 0 ≤ t ≤ 10, m ≤ 2t, 100t + 40m ≤ 1360
b. Graph the inequalities from Items 3 and 11 on a single grid.
m 1 • Equations, Inequalities, Functions
Answers
24 SpringBoard® Mathematics Algebra 2, Unit
12. b. The shaded region is the
intersection of the four
inequalities.
26
24
22
20
18
16
14
12
10
8
6
4
2
2 4 6 8 10 12 14 16
Number of Tickets
24
SpringBoard® Mathematics Algebra 2, Unit 1 • Equations, Inequalities, Functions
t
© 2015 College Board. All rights reserved.
8. Sample answer: Both graphs show
the same inequality, but the paper
graph uses the variables t and m,
and the calculator graph uses the
variables x and y. One advantage of
the paper graph is that you can add
titles to the axes to show what the
graph represents. One advantage of
the calculator graph is that you do
not need to determine the
coordinates of points on the
boundary line in order to graph it.
© 2015 College Board. All rights reserved.
Debrief students’ answers to these items
to ensure that they understand concepts
related to graphing linear inequalities
and interpreting solutions to situations
involving linear inequalities.
Lesson 2-2
Graphing Systems of Inequalities
ACTIVITY 2
continued
Number of Meals
Check Your Understanding
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ACTIVITY 2 Continued
Lesson 2-2
Graphing Systems of Inequalities
ACTIVITY 2
continued
13. By looking at your graph, identify two ordered pairs that are feasible
options to all of the inequalities. Confirm that these ordered pairs
satisfy the inequalities listed in Item 12.
Answers will vary based on the points that students pick.
My Notes
ACADEMIC VOCABULARY
When you confirm a statement,
you show that it is true or correct.
a. First ordered pair (t, m):
b. Second ordered pair (t, m):
14. Label the point (6, 10) on the grid in Item 6.
a. Interpret the meaning of this point.
Roy will go to 6 performances and have 10 meals.
b. Construct viable arguments. Is this ordered pair in the solution
region common to all of the inequalities? Explain.
14–15 Quickwrite, Discussion
Groups, Debriefing Have students list
four other points that are solutions and
four points that are not solutions. In
each case, have them identify why the
point is a solution or why it is not. Tie in
the graphic representation to the
real-world context, and for points that
are not solutions, have students write a
few sentences explaining why the point
is not a solution.
Yes. Sample explanation: The ordered pair is in the shaded region
of the graphs of all four inequalities.
15. If Roy uses his prize money to purchase 6 tickets and eat 10 meals, how
much money will he have left over for other expenses? Show your work.
© 2015 College Board. All rights reserved.
$360
Sample answer: 1360 − (6 × 100 + 10 × 40) = 360
© 2015 College Board. All rights reserved.
12–13 Create Representations As
students graph the various inequalities,
be sure that they use the boundary lines
and shading correctly to interpret the
inequalities. Have several students plot
the four inequalities on the board so
that any differences can be discussed
and reconciled. For Item 13, some
students may choose points that are not
actually solutions of the system of
inequalities. Use this as an opportunity
to discuss why graphically represented
information is only as accurate as the
care taken when graphing, and as the
precision that a graphing tool allows.
Reinforce what it means to be a member
of the feasible region and also the
importance of verification by
substituting into all four inequalities.
Check Your Understanding
Check Your Understanding
Debrief students’ answers to these items
to ensure that they understand concepts
related to graphing linear inequalities
and interpreting solutions to situations
involving linear inequalities.
16. Given the set of constraints described earlier, how many tickets could
Roy purchase if he buys 12 meals? Explain.
17. a. If you were Roy, how many meals and how many tickets would you
buy during the 5-day trip?
b. Explain why you made the choices you did, and tell how you know
that this combination of meals and tickets is feasible.
Answers
16. 6, 7, or 8 tickets; Sample
explanation: Substituting 12 for m
and solving m ≤ 2t for t shows that
t ≥ 6, so the number of tickets is at
least 6. Substituting 12 for m and
solving 100t + 40m ≤ 1360 for
t shows that t ≤ 8.8, so the number
of tickets is no more than 8.
18. Explain how you would graph this constraint on a coordinate plane:
2 ≤ x ≤ 5.
Activity 2 • Graphing to Find Solutions
25
17. a. Sample answer: 14 meals and
7 tickets
b. Sample answer: If I were Roy,
I would want to eat 3 meals per
day whenever possible, so
I would choose the maximum
number of meals, which is 14.
If Roy eats 14 meals, he can buy
either 7 or 8 tickets and still meet
all of the constraints. If I were
Roy, I would pick 7 tickets so
that I could have some money
left over for souvenirs.
18. Graph the vertical lines x = 2 and
x = 5. Then shade the region
between them.
Activity 2 • Graphing to Find Solutions
25
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ACTIVITY 2 Continued
Lesson 2-2
Graphing Systems of Inequalities
ACTIVITY 2
ASSESS
continued
Students’ answers to Lesson Practice
problems will provide you with a
formative assessment of their
understanding of the lesson concepts
and their ability to apply their learning.
My Notes
19. Graph these inequalities on the same grid, and shade the solution region
that is common to all of the inequalities: y ≥ 2, x ≤ 8, and y ≤ 2 + 1 x .
2
20. Identify two ordered pairs that satisfy the constraints in Item 19 and two
ordered pairs that do not satisfy the constraints.
See the Activity Practice for additional
problems for this lesson. You may assign
the problems here or use them as a
culmination for the activity.
A snack company plans to package a mixture of almonds and peanuts. The
table shows information about these types of nuts. The company wants the
nuts in each package to have at least 60 grams of protein and to cost no more
than $4. Use this information for Items 21–23.
LESSON 2-2 PRACTICE
19.
10
LESSON 2-2 PRACTICE
y
8
Nut
Protein
(g/oz)
Cost
($/oz)
Almonds
6
0.30
Peanuts
8
0.20
6
4
2
–10 –8 –6 –4 –2
–2
2
4
6
8
10
x
–4
MATH TIP
–6
–8
–10
20. Sample answer: ordered pairs that
satisfy the constraints: (4, 3) and
(6, 2); ordered pairs that do not
satisfy the constraints: (5, 1) and
(9, 4)
21. 6x + 8y ≥ 60, 0.30x + 0.20y ≤ 4,
x ≥ 0, y ≥ 0
22. Graph the constraints. Shade the solution region that is common to
all of the inequalities.
23. a. Identify two ordered pairs that satisfy the constraints.
b. Reason quantitatively. Which ordered pair represents the more
expensive mixture? Which ordered pair represents the mixture with
more protein? Explain your answer.
Nut Mixture
y
© 2015 College Board. All rights reserved.
22.
When answering Item 21,
remember that the number of
ounces of each type of nut cannot
be negative.
21. Model with mathematics. Write inequalities that model the
constraints in this situation. Let x represent the number of ounces of
almonds in each package and y represent the number of ounces of
peanuts.
16
12
8
4
4
8
12
16
20
d
© 2015 College Board. All rights reserved.
Ounces of Peanuts
20
Ounces of Almonds
ADAPT
Check students’ answers to the Lesson
Practice to ensure that they understand
how to graph inequalities and identify
the solutions of a system of inequalities.
Watch for students who choose points
that satisfy either inequality as opposed
to all inequalities in a system. Reinforce
students’ understanding of the meaning
of the solution to a system of inequalities.
26a. Answers
SpringBoard
Mathematics
Algebra
Inequalities,
Functions
23.
will vary
but should
be 2, Unit 1 • Equations,
ACTIVITY
PRACTICE
ordered pairs in the solution region.
1. y = − 2 x + 3
3
Sample answer: (10, 2) and (4, 12)
2. y = −x + 2
b. Answers will vary depending on the
answer to part a. Sample answer:
y
10
(10, 2) represents a mixture that
8
would cost $3.40 and have 76 g of
6
protein. (4, 12) represents a mixture
4
that would cost $3.60 and have 120 g
1
2
of protein. So, (4, 12) represents both
–10 –8 –6 –4 –2
2 4 6
the more expensive mixture and the
–2
–4
mixture with more protein.
2
®
–6
–8
–10
26
SpringBoard® Mathematics Algebra 2, Unit 1 • Equations, Inequalities, Functions
8
10
x