2-1
Graphing and Writing
Inequalities
Going Deeper
Essential question: How can you represent relationships using inequalities?
INTRODUCE
Standards for
Mathematical Content
Students routinely have problems remembering
which inequality symbol to use to write an
inequality. They may know what the symbols mean,
but may have trouble selecting which symbol to
use when translating from a verbal sentence. Ask
students what it means if they have a cell phone
plan under which they can send no more than 100
text messages per month. Ask them to write this as
an inequality. Use this example as a springboard to
list other verbal phrases associated with the various
inequality symbols. For each phrase, ask students to
give the related inequality symbol.
N-Q.1.2 Define appropriate quantities for the
purpose of descriptive modeling.*
A-CED.1.1 Create … inequalities in one variable
and use them to solve problems.*
A-CED.1.3 Represent constraints by …inequalities
… and interpret solutions as viable or nonviable
options in a modeling context.*
Vocabulary
inequality
solution of an inequality
Prerequisites
TEACH
Writing Expressions
Writing Equations
Solving Equations
1
example
Questioning Strategies
• Which inequality symbol can be used to
represent “at least”? ≥
Math Background
Students have previously evaluated, simplified, and
written algebraic expressions and solved equations.
In this lesson, students write inequalities.
Inequalities compare two expressions that are not
strictly equal by using inequality symbols.
• Can partial dollar amounts be part of the solution
set? Explain. Yes; any amount from $0 to $43 is in
• Can any amounts greater than $43 be part of the
solution set? Explain. Yes; since 1.15(43) = 49.45,
One of the basic building blocks of mathematics is
the Law of Trichotomy. This axiom states that given
any two real numbers, a and b, exactly one of the
following is true:
any amount that has a result from 49.45 to 50 is
part of the complete solution set. $43.48 is the
greatest amount since 1.15(43.48) = 50.00.
a = b, a < b, or a > b
extra example
Sarah can afford to spend at most $45, including
8% sales tax, on a present for her mother. Write
an equation to represent the situation. What is the
greatest whole dollar amount that she can spend
before the tax is added, and what is the cost including
tax? p + 0.08p ≤ 45; 1.08p ≤ 45; $41; $44.28
In the first case, a and b are related by an equation.
In the second and third cases, a and b are related by
an inequality.
A solution of an inequality is the value or values
that make the statement true. Thus, the solution set
of x > 5 contains every value on a number line to
the right of 5 and is an infinite set of numbers.
continued
Chapter 2 73
Lesson 1
© Houghton Mifflin Harcourt Publishing Company
the solution set.
Name
Class
Date
Notes
2-1
Graphing and Writing Inequalities
Going Deeper
Essential question: How can you represent relationships using inequalities?
An inequality is a statement that compares two expressions that are
not strictly equal by using one of the following inequality signs.
Symbol
Meaning
<
is less than
≤
is less than or equal to
>
is greater than
≥
is greater than or equal to
≠
is not equal to
A solution of an inequality is any value of the variable that makes the inequality true.
You can find solutions by making a table.
A-CED.1.1
1
EXAMPLE
Writing and Solving Inequalities
© Houghton Mifflin Harcourt Publishing Company
Kristin can afford to spend at most $50 for a birthday dinner at a restaurant, including a
15% tip. Describe some costs that are within her budget.
A
Which inequality symbol can be used to represent “at most”?
B
Complete the verbal model for the situation.
Cost before tip
(dollars)
+
15%
Cost before
tip (dollars)
Chapter 2
Budget limit
(dollars)
Lesson 1
73
Complete the table to find some costs that are within Kristin’s budget.
Cost
Substitute
Compare
Solution?
$47
1.15(47) ≤ 50
54.05 ≤ 50 ✗
No
$45
1.15(45) ≤ 50
51.75 ≤ 50 ✗
No
$43
1.15(43) ≤ 50
49.45 ≤ 50 ✓
Yes
$41
1.15(41) ≤ 50
47.15 ≤ 50 ✓
Yes
Kristin can spend $43, $41, or any lesser amount and stay within budget.
REFLECT
1a. Can Kristin spend $40 on the meal before the tip? Explain.
Yes; substituting 40 into the inequality you get 1.15(40) ≤ 50, or 46 ≤ 50,
which is a true statement. So, Kristin can spend $40 on the meal.
1b. What whole dollar amount is the most Kristin can spend before the tip? Explain.
$43; If Kristin spends $44, the cost with tip will be 1.15(44) = 50.6, which
is over her budget of $50.
1c. The solution set of an equation or inequality consists of all values that make the
statement true. Describe the whole dollar amounts that are in the solution set for
this situation.
Whole dollar amounts from $0 to $43 are in the solution set.
1d. Suppose Kristin also has to pay a 6% meal tax. Write an inequality to represent the
new situation. Then identify two solutions.
© Houghton Mifflin Harcourt Publishing Company
© Houghton Mifflin Harcourt Publishing Company
≤
Write and simplify an inequality for the model. c + 0.15c ≤ 50; 1.15c ≤ 50
C
D
≤
1.21c ≤ 50; $30, $41
PRACTICE
Tell whether each value of the variable is a solution of the inequality 4p < 64.
Show your reasoning.
1. p = 40
2. p = 45
no; 4(40) = 160; 160 > 64
3. p = 5
no; 4(45) = 180; 180 > 64
4. p = 22
yes; 4(5) = 20; 20 < 64
Chapter 2
Chapter 2
no; 4(22) = 88; 88 > 64
74
Lesson 1
74
Lesson 1
1 EXAMPLE
CLOS E
continued
Essential Question
How can you represent relationships using
inequalities?
Avoid Common Errors
Some students may use the greater than or equal to
symbol to represent at most, because they see the
word “most” and correlate it to “greater than”. Help
students understand that “at most” means the value
given is the greatest the result can be. All other values
must be less than the given value. Therefore, at most
is represented by the less than or equal to symbol,
and at least is represented by the greater than or
equal to symbol.
Relationships between expressions that involve
“at least”, “at most”, “less than”, or “greater than”
can be expressed as inequalities. Inequalities
representing “at least” or “at most” use the
≥ and ≤ symbols respectively. Inequalities
representing “less than” or “greater than” use the
< and > symbols respectively.
Summarize
Have students describe in their journal the
differences and similarities between expressions,
equations, and inequalities.
Technology
Students can use a graphing calculator to create a
table to find solutions to an inequality. Enter the
expression x + 0.15x into the Y = function of the
calculator. Press 2nd then TBLSET and enter 40 after
TblStart and 1 after Tbl. Then press 2nd TABLE.
The column labeled X will show the cost without
the 15% tip, and the column labeled Y1 will show
the cost including the 15% tip. Using a calculator
allows students to scroll through a large number of
values to find that the inequality in 1 EXAMPLE
is true for integer values less than or equal to 43.
The Tbl value can be changed to 0.01 to find the
maximum value to the nearest cent.
PR ACTICE
Where skills are
taught
1 EXAMPLE
Where skills are
practiced
EXS. 1–16
Highlighting
the Standards
Chapter 2
© Houghton Mifflin Harcourt Publishing Company
1 EXAMPLE , its Reflect questions, and
Questioning Strategies offer an opportunity
to address Mathematical Practice Standard
2 (Reason abstractly and quantitatively).
Students take a situation and represent it
symbolically by writing an inequality. Then
students analyze the solution set in the
parameters of whole dollars. They are then
asked if it is possible to expand the solution
set to include partial dollar amounts and to
explain why this is possible. By analyzing what
values are in the solution set if it is restricted
to whole dollars, and then expanding the
solution set to include partial dollar amounts,
students should understand that the solution
set changes depending on the units that can
be used (dollars or cents).
75
Lesson 1
Notes
Tell whether each value of the variable is a solution of the inequality 7p ≥ 105.
Show your reasoning.
5. p = 6
6. p = 21
no; 7(6) = 42; 42 < 105
yes; 7(21) = 147; 147 > 105
7. p = 4
8. p = 15
no; 7(4) = 28; 28 < 105
yes; 7(15) = 105; 105 = 105
Tell whether the value is a solution of the inequality. Explain.
9. x = 36; 3x < 100
10. m = 12; 5m + 4 > 50
no; 3(36) = 108
yes; 5(12) + 4 = 64
11. b = 5; 60 - 10b ≤ 20
12. y = -4; 7y + 6 < -20
yes; 60 - 10(5) = 10
yes; 7(-4) + 6 = -22
13. n = -4; 18 - 2n ≥ 26
14. d = -6; 27 + 8d > -14
no; 27 + 8(-6) = -21
yes; 18 - 2(-4) = 26
15. Brent is ordering books for a reading group. Each book costs $11.95. If he orders at
least $200 worth of books, he will get free shipping.
a. Complete the verbal model for the situation.
© Houghton Mifflin Harcourt Publishing Company
Price per
book
Number of
books
≥
Amount for
free shipping
b. Choose a variable for the unknown quantity. Include units.
b
Let
c.
represent the
number of books
.
Write an inequality from the verbal model.
11.95b ≥ 200
d. Complete the table to find some numbers of books Brent can order and receive
free shipping.
Books
Substitute
Compare
15
11.95(15) ≥ 200
179.25 ≥ 200 Solution?
No
16
11.95(16) ≥ 200
191.20 ≥ 200 No
17
11.95(17) ≥ 200
203.15 ≥ 200 Yes
18
11.95(18) ≥ 200
215.10 ≥ 200 Yes
Brent must order 17 books or more to get free shipping.
Chapter 2
75
Lesson 1
16. Farzana has a prepaid cell phone that costs $1 per day plus $.10 per minute she
uses. She has a daily budget of $5 for phone costs.
a. Write an inequality to represent the situation.
b. What is the maximum number of minutes Farzana can use and still stay within
her daily budget? Show your reasoning.
40 min; 1 + 0.10(35) = 4.5, 1 + 0.10(40) = 5, 1 + 0.10(45) = 5.5
c.
Describe the solution set of the inequality.
Any amount of time between 0 and 40 min is in the solution set.
© Houghton Mifflin Harcourt Publishing Company
© Houghton Mifflin Harcourt Publishing Company
Let m be the daily minutes used; 1 + 0.10m ≤ 5
Chapter 2
Chapter 2
76
Lesson 1
76
Lesson 1
ADD I T I O NA L P R AC TI C E
AND PRO BL E M S O LV I N G
Assign these pages to help your students practice
and apply important lesson concepts. For
additional exercises, see the Student Edition.
Answers
Additional Practice
1. yes
2. no
3. no
4. yes
5. no
6. no
7. yes
8. yes
9. s = hours of sleep; s > 7
10. w = wage; w ≥ 0.75
11. g = cost of calculator; g ≤ 63
© Houghton Mifflin Harcourt Publishing Company
12. s = number of cartons; s > 6
Problem Solving
1. a = age of person; a ≥ 35
2. w = weight; w ≤ 2500
3. f = percent forested; f ≤ 30 where f is
nonnegative.
4. w = weight; w ≥ 125
5. C
6. G
7. A
8. G
Chapter 2
77
Lesson 1
Name
Class
Notes
2-1
Date
Additional Practice
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Chapter 2
Chapter 2
78
Lesson 1
78
Lesson 1