ACTIVITY 9 Continued
Lesson 9-2
Graphing Inequalities in Two Variables
ACTIVITY 9
continued
PLAN
My Notes
Pacing: 1 class period
Chunking the Lesson
Example A
Example B
#1–2
#3–4
Check Your Understanding
Lesson Practice
The solutions of a linear inequality in two variables can be represented in the
coordinate plane.
Bell-Ringer Activity
Example A
MATH TERMS
A line that separates the coordinate
plane into two regions is a
boundary line. The two regions
are half-planes.
Graph the linear inequality y ≤ 2x + 3.
Step 1: Graph the corresponding linear
equation y = 2x + 3. The line
you graphed is the boundary
line.
10
y
8
6
4
2
–10 –8 –6 –4 –2
–2
2
4
6
8
10
x
–4
–6
–8
–10
MATH TIP
The origin (0, 0) is usually an easy
point to test if it is not on the
boundary line.
Step 2: Test a point in one of the half-planes to see if it is a solution of the
inequality. Using (0, 0), 0 ≤ 2(0) + 3 is a true statement. So (0, 0)
is a solution.
y
Step 3: If the point you tested is a
10
solution, shade the half-plane
8
in which it lies. If it is not,
6
shade the other half-plane. (0, 0)
4
is a solution, so shade the
2
half-plane containing
x
–10 –8 –6 –4 –2
2 4 6 8 10
the solution point (0, 0).
–2
–4
–6
–8
–10
MATH TERMS
A closed half-plane includes the
points on the boundary line.
In Example A, the solution set includes the points on the boundary line. The
solution set is a closed half-plane. In an inequality containing < or > the
solution set does not include the points on the boundary line, so the boundary
line is dashed, and the solution set is an open half-plane.
An open half-plane does not
include the points on the boundary
line.
154
154
SpringBoard® Integrated Mathematics I, Unit 2 • Linear Functions
SpringBoard® Integrated Mathematics I, Unit 2 • Linear Functions
© 2017 College Board. All rights reserved.
Example A Interactive Word Wall,
Vocabulary Organizer, Activating
Prior Knowledge, Create
Representations, Identify a
Subtask Guide students through this
first example of how to graph a linear
inequality in two variables. Ask students
to discuss how graphing a linear
inequality is similar to and how it is
different from graphing a linear
equation. Allow students to make the
connection between using solid and
open circles when graphing inequalities
on the number line and using solid and
dashed lines in the coordinate plane.
Graph on a coordinate plane the solutions of a linear inequality in two
• variables.
• Interpret the graph of the solutions of a linear inequality in two variables.
SUGGESTED LEARNING STRATEGIES: Vocabulary Organizer,
Interactive Word Wall, Create Representations, Identify a Subtask,
Construct an Argument
TEACH
Ask students to think of all the places
where they know about or have seen
boundary lines. These might include
football fields, baseball diamonds, tennis
courts, property lines and state lines.
Ask students to write a working
definition of a boundary line based on
their examples.
Learning Targets:
© 2017 College Board. All rights reserved.
Lesson 9-2
Lesson 9-2
Graphing Inequalities in Two Variables
ACTIVITY 9
continued
My Notes
Example B
Graph x + 2y > 8.
Step 1: Solve the inequality for y.
x − x + 2 y > 8− x
Subtract x from each side.
2 y >−x + 8
2 y −x + 8
>
Divide each side by 2.
2
2
1
y >− x + 4
Simplify.
2
Step 2: Graph the boundary line,
y = − 1 x + 4. Since the
2
inequality uses the >
symbol, the solution set
is an open half-plane.
Draw a dashed line.
ACTIVITY 9 Continued
Example B Activating Prior
Knowledge, Create Representations,
Identify a Subtask Example B
requires the additional step of solving
the inequality for y. Ask students if and
why this step is or is not necessary.
Try These A–B
Answers
a.
y
10
8
6
4
2
y 10
8
6
–10
4
–6
2
–10
–8
–6
–4
2
–2
4
6
8
10
x
–2
–2
–2
–4
–6
–8
–10
2
6
10
2
6
10
2
6
10
2
6
10
x
–4
b.
–6
y
10
8
6
4
2
–8
–10
© 2017 College Board. All rights reserved.
© 2017 College Board. All rights reserved.
Step 3: Check the test point (0, 0):
0 > − 1 (0) + 4 is false.
2
Shade the half-plane that
does not contain the
point (0, 0).
y 10
8
–10
6
–6
4
2
–10
–8
–6
–4
2
–2
4
6
8
10
x
–2
–4
c.
–6
–8
Try These A–B
MATH TIP
b. y < 3x − 2
d. y ≤ −4x + 5
When multiplying or dividing each
side of an inequality by a negative
number, you must reverse the
inequality symbol.
1. When graphing a linear inequality, is it possible for a test point located
on the boundary line to determine which half-plane should be shaded?
Explain. No; this will determine only if the line itself is part of the
–10
–6
–2
–2
–4
–6
–8
–10
d.
solution set, which it is for ≤ and ≥ inequalities. To determine which
half-plane to shade, the test point must be above or below the
boundary line.
Activity 9 • Writing and Graphing Inequalities
x
y
10
8
6
4
2
–10
Graph each inequality.
a. 3x − 4y ≥ −12
c. −2y > 6 + x
–2
–2
–4
–6
–8
–10
x
y
10
8
6
4
2
155
–10
–6
–2
–2
–4
–6
–8
–10
Activity 9 • Writing and Graphing Inequalities
x
155
ACTIVITY 9 Continued
Lesson 9-2
Graphing Inequalities in Two Variables
ACTIVITY 9
continued
My Notes
2. Critique the reasoning of others. Axl tried to graph the inequality
2x + 4y < 8. He first graphed the linear equation 2x + 4y = 8. He then
chose the test point (2, 1) and used his result to shade above the line.
Explain Axl’s mistake and how he should correct it.
Axl’s test point (2, 1) lies on the boundary line: 2(2) + 4(1) = 8. Axl
must choose a test point that is not on the boundary line to
determine the half-plane to shade. Axl could use (0, 0); then the
inequality would give 2(0) + 4(0) < 8, which is true. Since (0, 0) lies
below the graph of 2x + 4y = 8, Axl should shade below the line.
3. Axl and Aneeza try a new plan in which they can upload no more than
350 terabytes per month. On the grids below, the x-axis represents the
number of terabytes that Axl can upload and the y-axis represents the
number of terabytes that Aneeza can upload.
y
a. Suppose Aneeza does
not upload anything
360
for a month. Write
330
an inequality that
300
represents the amount 270
of data that Axl can
240
upload during that
210
month. Graph the
inequality on the grid. 180
3–4 Discussion Groups, Identify a
Subtask, Create Representations,
Debriefing These items bring students
back to the original problem context and
mirror earlier questions, providing
reinforcement. Students may determine
equations and inequalities using different
methods. Some may use the slope and
y-intercept as determined from the
graph, while some may choose points on
the line to determine the slope. Monitor
group discussions carefully to be sure
students understand the reasons why
the graphs are restricted to the first
quadrant. To make connections to
future learning, a discussion of how the
context limits the graph of a linear
inequality may be revisited.
x ≤ 350
150
120
90
60
30
x
y
© 2017 College Board. All rights reserved.
y ≤ 350
0
36
0
33
0
30
0
27
0
24
0
21
0
18
0
15
0
12
90
60
30
b. Suppose Axl does not
upload anything for a
month. Write an
inequality that
represents the amount
of data that Aneeza
can upload during
that month. Graph the
inequality on the grid.
360
330
300
270
240
210
180
150
120
90
60
30
x
0
36
0
33
0
30
0
27
0
24
0
21
0
18
0
15
0
12
90
60
30
156
SpringBoard® Integrated Mathematics I, Unit 2 • Linear Functions
MINI-LESSON: Equations and Inequalities in One Variable
Follow these steps to explore equations and inequalities in one variable.
1. On grid paper, graph y = 1, which means for any value of x, y = 1.
2. Graph x = 3, which means for any value of y, x = 3.
3. When the x-coordinate of each point on a line is the same, the line is vertical.
4. When the y-coordinate of each point on a line is the same, the line is horizontal.
5. Which parts of the coordinate plane can be described with inequalities
in one variable? Explain your answer. (When the one variable is y,
a region above or below a horizontal line is shaded. When the one
variable is x, a region to the left or right of a vertical line is shaded.)
156
SpringBoard® Integrated Mathematics I, Unit 2 • Linear Functions
© 2017 College Board. All rights reserved.
1–2 Activating Prior Knowledge,
Create Representations, Think-PairShare, Construct an Argument These
items lead students to understand that a
boundary line and the two half-planes it
divides are separate entities. The union
of the two half-planes and the boundary
line makes up the entire coordinate
plane. It is important for students to
understand that while the points on the
boundary line are sometimes solutions
to the inequality, these points cannot be
used to determine in which half-plane
the other solutions lie.
ACTIVITY 9 Continued
Lesson 9-2
Graphing Inequalities in Two Variables
ACTIVITY 9
3–4 (continued) As this portion of the
activity is debriefed, discuss how the
equation of the boundary line is related
to the inequality. Encourage students to
describe the meaning of the x-and
y-intercepts of the graph within the
provided context.
continued
My Notes
4. The graph below represents another plan that Axl and Aneeza
considered. The x-axis represents the number of terabytes of data from
photos that can be uploaded each month. The y-axis represents the
number of terabytes of data from text files.
y
420
Differentiating Instruction
360
Support students who forget to
change the direction of the inequality
symbol when multiplying or dividing
both sides by a negative number. For
the following inequalities, have
students predict the direction of the
inequality symbol when solved for y
with y appearing on the left side of
the inequality. Then have them
confirm their predictions by solving
the inequalities for y.
1. 3x − 2y ≤ 4 [ y ≥ 3 x − 2]
2
2. −9x −3y < 8 [ y > −3x − 8 ]
3
3. x − 4y ≥ −12 [ y ≤ 1 x + 3]
4
4. 5x < 2 − y [ y < −5x + 2]
Text Files (TB)
300
240
180
120
60
x
60
120
180 240 300
Photos (TB)
360
420
a. Identify the x-intercept and y-intercept. What do they represent in
this context?
MATH TIP
© 2017 College Board. All rights reserved.
© 2017 College Board. All rights reserved.
The x-intercept is (180, 0); this represents the maximum number of
terabytes of photos that can be uploaded during a month in which
no text files are uploaded. The y-intercept is (0, 360); this represents
the maximum number of terabytes of text files that can be
uploaded during a month in which no photos are uploaded.
The x-intercept of a graph is the
point where the graph crosses the
x-axis. The y-intercept is the point
where the graph crosses the y-axis.
Extend students’ thinking by asking
them to evaluate the following
statement as always, sometimes or
never true and give an explanation
for their response.
b. Determine the equation for the boundary line of the graph. Justify
your response.
The solution of a linear inequality
includes ordered pairs in each of the
four quadrants of a coordinate plane.
y = −2x + 360; The slope of the line is − 360 = −2, and the
180
y-intercept is (0, 360).
Check Your Understanding
Answers
c. Model with mathematics. Write a linear inequality to represent this
situation. Then describe the plan in your own words.
5.
y
2x + y ≤ 360; The plan allows up to 360 terabytes of uploaded
information, but no more than 180 terabytes can be from photos.
10
8
6
4
2
–10
Activity 9 • Writing and Graphing Inequalities
y
7.
157
–6
–2
–2
–4
–6
–8
–10
–2
–2
–4
–6
–8
–10
2
6
10
2
6
10
x
2
6
10
x
y
6.
10
8
6
4
2
10
8
6
4
2
–10
–6
–10
–6
–2
–2
–4
–6
–8
–10
x
8. x < 0
Activity 9 • Writing and Graphing Inequalities
157
ACTIVITY 9 Continued
Lesson 9-2
Graphing Inequalities in Two Variables
ACTIVITY 9
Check Your Understanding
Debrief students’ answers by asking them
how they know which half-plane to shade
when graphing an inequality. Have
students identify the boundary lines for
Items 5–7. Have them share their
boundary line graphs with others before
shading. Pay close attention to the
shading of the half-planes.
continued
My Notes
Check Your Understanding
5. Graph the linear inequality x < −3 on the coordinate plane.
6. Graph the linear inequality y > 2 on the coordinate plane.
7. Graph the linear inequality x ≥ 0 on the coordinate plane.
Answers
8. Write an inequality whose solutions are all points in the second and
third quadrants.
See previous page.
ASSESS
Students’ answers to the Lesson Practice
items will provide a formative assessment
of their understanding of graphing and
interpreting a linear inequality in two
variables, and of students’ ability to apply
their learning.
LESSON 9-2 PRACTICE
Graph each inequality on the coordinate plane.
9. x − y ≤ 4
10. 2x − y > 1
11. y ≥ 3x + 7
Short-cycle formative assessment items
for Lesson 9-2 are also available in the
Assessment section on SpringBoard
Digital.
12. −x + 6 > y
13. Make sense of problems. Write the inequality whose solutions are
shown in the graph.
Refer back to the graphic organizer the
class created when unpacking Embedded
Assessment 3. Ask students to use the
graphic organizer to identify the concepts
or skills they learned in this lesson.
y 3
2
1
LESSON 9-2 PRACTICE
y
9.
–3
–2
–2
–4
–6
–8
–10
3
2
4
x
–2
© 2017 College Board. All rights reserved.
–6
1
–1
–1
10
8
6
4
2
–10
–2
–3
2
6
10
x
Check students’ answers to the Lesson
Practice to ensure that students can
correctly graph an inequality in two
variables on the coordinate plane. If
students are struggling to shade the correct
half-plane, ask them to plot their test point
before they test it. If the test yields a true
statement, they should shade toward the
point they have graphed. If the test results
yield a false statement, they should shade
on the side of the boundary line that does
not contain the test point. Encourage
students to shade lightly so that corrections
can easily be made if necessary.
See the Activity Practice on page 163 and
the Additional Unit Practice in the
Teacher Resources on SpringBoard Digital
for additional problems for this lesson.
You may wish to use the Teacher
Assessment Builder on SpringBoard
Digital to create custom assessments or
additional practice.
158
158 SpringBoard® Integrated Mathematics I, Unit 2 • Linear Functions
y
y
11.
10.
10
8
6
4
2
–10
–6
–2
–2
–4
–6
–8
–10
2
6
10
x
–10
SpringBoard® Integrated Mathematics I, Unit 2 • Linear Functions
–6
–2
–2
–4
–6
–8
–10
2
6
y
12.
10
8
6
4
2
10
x
© 2017 College Board. All rights reserved.
ADAPT
10
8
6
4
2
–10
–6
–2
–2
–4
–6
–8
–10
13. y ≥ −2x + 2
2
6
10
x