Big Ideas Math
Integrated Mathematics
Table of Contents
BIG IDEAS
MATH
®
Integrated Mathematics I
Chapter 1
Solving Linear Equations
Chapter 7
Data Analysis and Displays
Chapter 2
Solving Linear Inequalities
Chapter 8
Basics of Geometry
Chapter 3
Graphing Linear Functions
Chapter 9
Reasoning and Proofs
Chapter 4
Writing Linear Functions
Chapter 10
Parallel and Perpendicular Lines
Chapter 5
Solving Systems of Linear Equations
Chapter 11
Transformations
Chapter 6
Exponential Functions and Sequences
Chapter 12
Congruent Triangles
Integrated Mathematics II
Chapter 1
Functions and Exponents
Chapter 7
Quadrilaterals and Other Polygons
Chapter 2
Polynomial Equations and Factoring
Chapter 8
Similarity
Chapter 3
Graphing Quadratic Functions
Chapter 9
Right Triangles and Trigonometry
Chapter 4
Solving Quadratic Equations
Chapter 10
Circles
Chapter 5
Probability
Chapter 11
Circumference, Area, and Volume
Chapter 6
Relationships Within Triangles
Integrated Mathematics III
Chapter 1
Geometric Modeling
Chapter 6
Rational Functions
Chapter 2
Linear and Quadratic Functions
Chapter 7
Sequences and Series
Chapter 3
Polynomial Functions
Chapter 8
Chapter 4
Rational Exponents and Radical
Functions
Trigonometric Ratios and
Functions
Chapter 9
Trigonometric Identities
Chapter 10
Data Analysis and Statistics
Chapter 5
Exponential and Logarithmic
Functions
Ron Larson
Laurie Boswell
An A
American Company
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INTRODUCING
®
INTEGRATED MATHEMATICS
by Ron Larson and Laurie Boswell
Big Ideas Learning is pleased to introduce a new high school program, Big Ideas Math Integrated Mathematics
I, II, and III. The program was written by renowned authors Ron Larson and Laurie Boswell and was developed using
the consistent, dependable learning and instructional theory that have become synonymous with Big Ideas Math.
Students will gain a deeper understanding of mathematics by narrowing their focus to fewer topics at each grade level.
They will also master content through inductive reasoning opportunities, engaging explorations, concise stepped-out
examples, and rich thought-provoking exercises.
The Big Ideas Math Integrated Mathematics research-based curriculum features a continual development of
concepts that have been previously taught while integrating algebra, geometry, probability, and statistics topics
throughout each course.
In Integrated Mathematics I, students will study linear and exponential equations and functions. Students will use
linear regression and perform data analysis. They will also learn about geometry topics such as simple proofs,
congruence, and transformations.
Integrated Mathematics II expands into quadratic, absolute value, and other functions. Students will also explore
polynomial equations and factoring, and probability and its applications. Coverage of geometry topics extends to
polygon relationships, proofs, similarity, trigonometry, circles, and three-dimensional figures.
In Integrated Mathematics III, students will expand their understanding of area and volume with geometric modeling,
which students will apply throughout the course as they learn new types of functions. Students will study polynomial,
radical, logarithmic, rational, and trigonometric functions. They will also learn how visual displays and statistics relate
to different types of data and probability distributions.
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About the Authors
Ron Larson, Ph.D., is well known as the lead author of a
comprehensive program for mathematics that spans middle
school, high school, and college courses. He holds the distinction
of Professor Emeritus from Penn State Erie, The Behrend College,
where he taught for nearly 40 years. He received his Ph.D. in
mathematics from the University of Colorado. Dr. Larson’s
numerous professional activities keep him actively involved
in the mathematics education community and allow him to
fully understand the needs of students, teachers, supervisors,
and administrators.
Laurie Boswell, Ed.D., is the Head of School and a
mathematics teacher at the Riverside School in Lyndonville,
Vermont. Dr. Boswell is a recipient of the Presidential Award
for Excellence in Mathematics Teaching and has taught
mathematics to students at all levels, from elementary through
college. Dr. Boswell was a Tandy Technology Scholar and
served on the NCTM Board of Directors from 2002 to 2005.
She currently serves on the board of NCSM and is a popular
national speaker.
Dr. Ron Larson and Dr. Laurie Boswell began writing together in 1992. Since that time,
they have authored over two dozen textbooks. In their collaboration, Ron is primarily
responsible for the student edition while Laurie is primarily responsible for the
teaching edition.
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Program Resources
Print
Technology
Student Edition
Student Edition
Designed to the UDL Guidelines
Teaching Edition
Laurie's Notes
Student Journal
Available in English and Spanish
With complete English and Spanish audio
Dynamic eBook App
Dynamic Solutions Tool
Dynamic Investigations
Lesson Tutorial Videos
Dynamic Classroom
Vocabulary Flash Cards
Resources by Chapter
Start Thinking
Warm Up
Cumulative Review Warm Up
Practice A and B
Enrichment and Extension
Puzzle Time
Family Communication Letters
Available in English and Spanish
Assessment Book
Performance Tasks
Prerequisite Skills Test with Item Analysis
Quarterly Standards Based Tests
Quizzes
Chapter Tests
Alternative Assessments with
Scoring Rubrics
Pre-Course Test with Item Analysis
Post Course Test with Item Analysis
Worked-Out Solutions
Extra Examples
Warm Up and Closure Activities
Dynamic Teaching Tools
Interactive Whiteboard Lesson Library
• Compatible with SMART®, Promethean®, and
Mimio® technology
Real-Life STEM Videos
Editable Online Resources
•
•
•
•
Lesson Plans
Assessment Book
Resources by Chapter
Differentiating the Lesson
Answer Presentation Tool
Dynamic Assessment and
Progress Monitoring Tool
Assessment Creation and Delivery
Progress Monitoring
Multilingual Glossary
Key mathematical vocabulary terms
in 14 languages
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Program Overview
Program Philosophy: Rigor and Balance
with Real-Life Applications
Essential Question
How can you use a formula for one
measurement to write a formula for a different measurement?
The Big Ideas Math® program balances
conceptual understanding with procedural
fluency. Real-life applications help turn
mathematical learning into an engaging
and meaningful way to see and explore
the real world.
Using an Area Formula
Work with a partner.
a. Write a formula for the area A of
a parallelogram.
h = 5 in.
b. Substitute the given values into the
formula. Then solve the equation
for b. Justify each step.
Explorations and guiding
Essential Questions
encourage conceptual
understanding.
1.5 Lesson
A = 30 in.2
b
c. Solve the formula in part (a) for b without first substituting values into the formula.
Justify each step.
d. Compare how you solved the equations in parts (b) and (c). How are the processes
similar? How are they different?
What You Will Learn
Rewrite literal equations.
Core Vocabul
Vocabulary
larry
literal equation, p. 36
formula, p. 37
Previous
surface area
Rewrite and use formulas for area.
Rewrite and use other common formulas.
Rewriting Literal Equations
An equation that has two or more variables is called a literal equation. To rewrite a
literal equation, solve for one variable in terms of the other variable(s).
Rewriting a Literal Equation
Solve the literal equation 3y + 4x = 9 for y.
SOLUTION
3y + 4x = 9
Write the equation.
3y + 4x − 4x = 9 − 4x
Subtract 4x from each side.
3y = 9 − 4x
Simplify.
3y 9 − 4x
—=—
3
3
Divide each side by 3.
4
y = 3 − —x
3
Real-life applications
provide students with
opportunities to connect
classroom lessons to
realistic scenarios.
Simplify.
4
The rewritten literal equation is y = 3 − — x.
3
Direct instruction lessons
allow for procedural
fluency and provide the
opportunity to use clear,
precise mathematical
language.
Using the Formula for Simple Interest
You deposit $5000 in an account that earns simple interest. After 6 months, the
account earns $162.50 in interest. What is the annual interest rate?
COMMON ERROR
The unit of t is years. Be
sure to convert months
to years.
1 yr
12 mo
—
⋅ 6 mo = 0.5 yr
SOLUTION
To find the annual interest rate, solve the simple interest formula for r.
I = Prt
I
—=r
Pt
162.50
(5000)(0.5)
—=r
0.065 = r
Write the simple interest formula.
Divide each side by Pt to solve for r.
Substitute 162.50 for I, 5000 for P, and 0.5 for t.
Simplify.
The annual interest rate is 0.065, or 6.5%.
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Dynamic Technology Package
The Big Ideas Math program includes a comprehensive technology package that enhances the
curriculum and allows students to engage with the underlying mathematics in the text.
Student Edition
The Dynamic Student Edition gives students access to the complete textbook and robust embedded
digital resources. Interactive investigations, direct links to remediation, and additional resources are
linked at point-of-use. Students can customize their Dynamic Student Editions through note taking and
bookmarking, and it can also be accessed offline after it has been downloaded to a device. Audio
support and Lesson Tutorial Videos are also included in English and Spanish.
Investigations
Dynamic Investigations are powered by Desmos® and GeoGebra®. These interactivities expand on the
Explorations in the program and allow students to learn mathematics through a hands-on approach.
Teachers and students can integrate these investigations into their discovery learning.
Real-Life STEM Videos
Every chapter in the program contains a Real-Life STEM Video that ties directly to a Performance Task.
Students can explore topics like the speed of light, natural disasters, and wind power while applying
their knowledge to a comprehensive project or task.
Classroom
The Dynamic Classroom is an online interactive version of the Student Edition that can be used as a
lesson presentation tool. Teachers can present their lessons and have point-of-use access to all of the
online resources available that supplement every section of the program.
Teaching Tools
These tools include an Interactive Whiteboard
Lesson Library that includes customizable lessons
and templates for every section in the program.
Lessons are compatible with SMART®,
Promethean®, and Mimio® whiteboards. The
Answer Presentation Tool can be used to display
worked-out solutions to homework and test
problems from Big Ideas Math content.
Assessment and
Progress Monitoring Tool
This online tool allows teachers to create tests by
standard or by Big Ideas Math content. Teachers
can assign any exercise from the student textbook,
problems from the program’s ancillary pieces, or
additional items from the item bank, and students
can complete their assignments within the tool’s interface.
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Big Ideas Math Integrated
Mathematics Table of Contents
Integrated Mathematics I
Chapter 1
Solving Linear Equations
Chapter 7
Data Analysis and Displays
Chapter 2
Solving Linear Inequalities
Chapter 8
Basics of Geometry
Chapter 3
Graphing Linear Functions
Chapter 9
Reasoning and Proofs
Chapter 4
Writing Linear Functions
Chapter 10
Parallel and Perpendicular Lines
Chapter 5
Solving Systems of Linear Equations
Chapter 11
Transformations
Chapter 6
Exponential Functions and Sequences
Chapter 12
Congruent Triangles
Integrated Mathematics II
Chapter 1
Functions and Exponents
Chapter 7
Quadrilaterals and Other Polygons
Chapter 2
Polynomial Equations and Factoring
Chapter 8
Similarity
Chapter 3
Graphing Quadratic Functions
Chapter 9
Right Triangles and Trigonometry
Chapter 4
Solving Quadratic Equations
Chapter 10
Circles
Chapter 5
Probability
Chapter 11
Circumference, Area, and Volume
Chapter 6
Relationships Within Triangles
Integrated Mathematics III
Chapter 1
Geometric Modeling
Chapter 6
Rational Functions
Chapter 2
Linear and Quadratic Functions
Chapter 7
Sequences and Series
Chapter 3
Polynomial Functions
Chapter 8
Chapter 4
Rational Exponents and Radical
Functions
Trigonometric Ratios and
Functions
Chapter 9
Trigonometric Identities
Chapter 10
Data Analysis and Statistics
Chapter 5
Exponential and Logarithmic
Functions
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Integrated Mathematics I
Student Edition
Chapter 1
Solving Linear Equations
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1
1.1
1.2
1.3
1.4
1.5
Solving Linear Equations
Solving Simple Equations
Solving Multi-Step Equations
Solving Equations with Variables on Both Sides
Solving Absolute Value Equations
Rewriting Equations and Formulas
Density of Pyrite (p. 41)
SEE the Big Idea
Cheerleading Competition (p. 29)
Boat (p. 22)
Biking (p. 14)
Average Speed (p. 6)
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Maintaining Mathematical Proficiency
Adding and Subtracting Integers
Example 1
(Grade 7)
Evaluate 4 + (−12).
∣ −12 ∣ > ∣ 4 ∣. So, subtract ∣ 4 ∣ from ∣ −12 ∣.
4 + (−12) = −8
Use the sign of −12.
Example 2 Evaluate −7 − (−16).
−7 − (−16) = −7 + 16
Add the opposite of −16.
=9
Add.
Add or subtract.
1. −5 + (−2)
2. 0 + (−13)
3. −6 + 14
4. 19 − (−13)
5. −1 − 6
6. −5 − (−7)
7. 17 + 5
8. 8 + (−3)
9. 11 − 15
Multiplying and Dividing Integers
Example 3
(Grade 7)
⋅
Evaluate −3 (−5).
The integers have the same sign.
⋅
−3 (−5) = 15
The product is positive.
Example 4
Evaluate 15 ÷ (−3).
The integers have different signs.
15 ÷ (−3) = −5
The quotient is negative.
Multiply or divide.
⋅
⋅
10. −3 (8)
11. −7 (−9)
13. −24 ÷ (−6)
14. —
15. 12 ÷ (−3)
17. 36 ÷ 6
18. −3(−4)
⋅
16. 6 8
−16
2
12. 4 (−7)
19. ABSTRACT REASONING Summarize the rules for (a) adding integers, (b) subtracting integers,
(c) multiplying integers, and (d) dividing integers. Give an example of each.
Dynamic Solutions available at BigIdeasMath.com
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Mathematical
Practices
Mathematically proficient students carefully specify units of measure.
Specifying Units of Measure
Core Concept
Operations and Unit Analysis
Addition and Subtraction
When you add or subtract quantities, they must have the same units of measure.
The sum or difference will have the same unit of measure.
Example
Perimeter of rectangle
= (3 ft) + (5 ft) + (3 ft) + (5 ft)
3 ft
= 16 feet
5 ft
When you add feet,
you get feet.
Multiplication and Division
When you multiply or divide quantities, the product or quotient will have a
different unit of measure.
Example
Area of rectangle = (3 ft) × (5 ft)
= 15 square feet
When you multiply feet, you get
feet squared, or square feet.
Specifying Units of Measure
You work 8 hours and earn $72. What is your hourly wage?
SOLUTION
dollars per hour
dollars per hour
Hourly wage
= $72 ÷ 8 h
($ per h)
= $9 per hour
The units on each side of the
equation balance. Both are
specified in dollars per hour.
Your hourly wage is $9 per hour.
Monitoring Progress
Solve the problem and specify the units of measure.
1. The population of the United States was about 280 million in 2000 and about
310 million in 2010. What was the annual rate of change in population from
2000 to 2010?
2. You drive 240 miles and use 8 gallons of gasoline. What was your car’s gas mileage
(in miles per gallon)?
3. A bathtub is in the shape of a rectangular prism. Its dimensions are 5 feet by 3 feet by
18 inches. The bathtub is three-fourths full of water and drains at a rate of 1 cubic foot
per minute. About how long does it take for all the water to drain?
2
Chapter 1
Solving Linear Equations
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1.1
Solving Simple Equations
Essential Question
How can you use simple equations to solve
real-life problems?
Measuring Angles
Work with a partner. Use a protractor to measure the angles of each quadrilateral.
Copy and complete the table to organize your results. (The notation m∠ A denotes the
measure of angle A.) How precise are your measurements?
a.
A
b.
B
B
c.
A
A
B
C
D
UNDERSTANDING
MATHEMATICAL
TERMS
Quadrilateral
A conjecture is an
a.
unproven statement
about a general
b.
mathematical concept.
c.
After the statement is
proven, it is called a
rule or a theorem. You will
learn more about these
terms in Chapter 9.
D
m∠A
(degrees)
m∠B
(degrees)
D
C
m∠C
(degrees)
C
m∠A + m∠B
+ m∠C + m∠D
m∠D
(degrees)
Making a Conjecture
Work with a partner. Use the completed table in Exploration 1 to write a conjecture
about the sum of the angle measures of a quadrilateral. Draw three quadrilaterals that
are different from those in Exploration 1 and use them to justify your conjecture.
Applying Your Conjecture
CONNECTIONS TO
GEOMETRY
You will learn more
about angle measures
of quadrilaterals in a
future course.
Work with a partner. Use the conjecture you wrote in Exploration 2 to write an
equation for each quadrilateral. Then solve the equation to find the value of x. Use
a protractor to check the reasonableness of your answer.
a.
b.
85°
c.
78°
30°
100°
x°
90°
80°
x°
72°
x°
60°
90°
Communicate Your Answer
4. How can you use simple equations to solve real-life problems?
5. Draw your own quadrilateral and cut it out. Tear off the four corners of
the quadrilateral and rearrange them to affirm the conjecture you wrote in
Exploration 2. Explain how this affirms the conjecture.
Section 1.1
Solving Simple Equations
3
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1.1
Lesson
What You Will Learn
Solve linear equations using addition and subtraction.
Solve linear equations using multiplication and division.
Core Vocabul
Vocabulary
larry
Use linear equations to solve real-life problems.
conjecture, p. 3
rule, p. 3
theorem, p. 3
equation, p. 4
linear equation
in one variable, p. 4
solution, p. 4
inverse operations, p. 4
equivalent equations, p. 4
Previous
expression
Solving Linear Equations by Adding or Subtracting
An equation is a statement that two expressions are equal. A linear equation in one
variable is an equation that can be written in the form ax + b = 0, where a and b are
constants and a ≠ 0. When you solve an equation, you use properties of real numbers
to find a solution, which is a value that makes the equation true.
Inverse operations are two operations that undo each other, such as addition
and subtraction. When you perform the same inverse operation on each side of an
equation, you produce an equivalent equation. Equivalent equations are equations
that have the same solution(s).
Core Concept
Addition, Subtraction, and Substitution Properties of Equality
CONNECTIONS TO
GEOMETRY
Let a, b, and c be real numbers.
Segment lengths and
angle measures are real
numbers. You will use these
properties later in the book
to write logical arguments
about geometric figures.
Addition Property of Equality
If a = b, then a + c = b + c.
Subtraction Property of Equality
If a = b, then a − c = b − c.
Substitution Property of Equality
If a = b, then a can be substituted for b
(or b for a) in any equation or expression.
Solving Equations by Addition or Subtraction
Solve each equation. Justify each step. Check your answer.
a. x − 3 = −5
b. 0.9 = y + 2.8
SOLUTION
a. x − 3 = −5
+3
Addition Property of Equality
Write the equation.
+3
Check
x − 3 = −5
?
− 2 − 3 = −5
Add 3 to each side.
x = −2
Simplify.
−5 = −5
The solution is x = −2.
b.
Subtraction Property of Equality
0.9 = y + 2.8
− 2.8
− 2.8
−1.9 = y
Write the equation.
Check
Subtract 2.8 from each side.
0.9 = y + 2.8
?
0.9 = −1.9 + 2.8
Simplify.
The solution is y = −1.9.
Monitoring Progress
✓
0.9 = 0.9
✓
Help in English and Spanish at BigIdeasMath.com
Solve the equation. Justify each step. Check your solution.
1. n + 3 = −7
4
Chapter 1
1
2
2. g − —3 = −—3
3. −6.5 = p + 3.9
Solving Linear Equations
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Solving Linear Equations by Multiplying or Dividing
Just as addition and subtraction are inverse operations, multiplication and division are
also inverse operations.
Core Concept
Multiplication and Division Properties of Equality
Let a, b, and c be real numbers.
Multiplication Property of Equality
Division Property of Equality
⋅
⋅
If a = b, then a c = b c, c ≠ 0.
a b
If a = b, then — = — , c ≠ 0.
c c
Solving Equations by Multiplication or Division
Solve each equation. Justify each step. Check your answer.
n
a. −— = −3
5
b. π x = −2π
c. 1.3z = 5.2
SOLUTION
n
−— = −3
5
a.
Multiplication Property of Equality
−5
⋅( )
Write the equation.
⋅
n
− — = −5 (− 3)
5
Check
n
−— = −3
5
15 ?
−— = −3
5
−3 = −3
Multiply each side by − 5.
n = 15
Simplify.
The solution is n = 15.
b. π x = −2π
Division Property of Equality
πx
π
Write the equation.
−2π
π
—=—
Divide each side by π.
x = −2
✓
Check
π x = −2π
?
π (−2) = −2π
− 2π = −2π
Simplify.
The solution is x = −2.
c. 1.3z = 5.2
Division Property of Equality
1.3z 5.2
—=—
1.3
1.3
z=4
Write the equation.
Check
1.3z = 5.2
?
1.3(4) = 5.2
Divide each side by 1.3.
Simplify.
5.2 = 5.2
The solution is z = 4.
Monitoring Progress
✓
✓
Help in English and Spanish at BigIdeasMath.com
Solve the equation. Justify each step. Check your solution.
y
3
4. — = −6
5. 9π = π x
Section 1.1
6. 0.05w = 1.4
Solving Simple Equations
5
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Solving Real-Life Problems
MODELING WITH
MATHEMATICS
Mathematically proficient
students routinely check
that their solutions make
sense in the context of a
real-life problem.
Core Concept
Four-Step Approach to Problem Solving
1.
Understand the Problem What is the unknown? What information is being
given? What is being asked?
2.
Make a Plan This plan might involve one or more of the problem-solving
strategies shown on the next page.
3.
Solve the Problem Carry out your plan. Check that each step is correct.
4.
Look Back Examine your solution. Check that your solution makes sense in
the original statement of the problem.
Modeling with Mathematics
In the 2012 Olympics, Usain Bolt won the
200-meter dash with a time of 19.32 seconds. Write
and solve an equation to find his average speed to
the nearest hundredth of a meter per second.
REMEMBER
SOLUTION
The formula that relates
distance d, rate or speed r,
and time t is
d = rt.
1. Understand the Problem You know the
winning time and the distance of the race.
You are asked to find the average speed to
the nearest hundredth of a meter per second.
2. Make a Plan Use the Distance Formula to write
an equation that represents the problem. Then
solve the equation.
3. Solve the Problem
⋅
200 = r ⋅ 19.32
d=r t
REMEMBER
The symbol ≈ means
“approximately equal to.”
200
19.32
19.32r
19.32
Write the Distance Formula.
Substitute 200 for d and 19.32 for t.
—=—
Divide each side by 19.32.
10.35 ≈ r
Simplify.
Bolt’s average speed was about 10.35 meters per second.
4. Look Back Round Bolt’s average speed to 10 meters per second. At this speed,
it would take
200 m
10 m/sec
— = 20 seconds
to run 200 meters. Because 20 is close to 19.32, your solution is reasonable.
Monitoring Progress
Help in English and Spanish at BigIdeasMath.com
7. Suppose Usain Bolt ran 400 meters at the same average speed that he ran the
200 meters. How long would it take him to run 400 meters? Round your answer
to the nearest hundredth of a second.
6
Chapter 1
Solving Linear Equations
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Core Concept
Common Problem-Solving Strategies
Use a verbal model.
Guess, check, and revise.
Draw a diagram.
Sketch a graph or number line.
Write an equation.
Make a table.
Look for a pattern.
Make a list.
Work backward.
Break the problem into parts.
Modeling with Mathematics
On January 22, 1943, the temperature in Spearfish, South Dakota, fell from 54°F
at 9:00 a.m. to − 4°F at 9:27 a.m. How many degrees did the temperature fall?
SOLUTION
1. Understand the Problem You know the temperature before and after the
temperature fell. You are asked to find how many degrees the temperature fell.
2. Make a Plan Use a verbal model to write an equation that represents the problem.
Then solve the equation.
3. Solve the Problem
Words
Number of degrees
Temperature
Temperature
=
−
the temperature fell
at 9:27 a.m.
at 9:00 a.m.
Variable
Let T be the number of degrees the temperature fell.
Equation
−4
=
−
54
−4 = 54 − T
T
Write the equation.
−4 − 54 = 54 − 54 − T
−58 = − T
Subtract 54 from each side.
Simplify.
58 = T
Divide each side by − 1.
The temperature fell 58°F.
REMEMBER
The distance between two
points on a number line is
always positive.
4. Look Back The temperature fell from 54 degrees above 0 to 4 degrees below 0.
You can use a number line to check that your solution is reasonable.
58
−8 −4
0
4
8 12 16 20 24 28 32 36 40 44 48 52 56 60
Monitoring Progress
Help in English and Spanish at BigIdeasMath.com
8. You thought the balance in your checking account was $68. When your bank
statement arrives, you realize that you forgot to record a check. The bank
statement lists your balance as $26. Write and solve an equation to find the
amount of the check that you forgot to record.
Section 1.1
Solving Simple Equations
7
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1.1
Exercises
Dynamic Solutions available at BigIdeasMath.com
Vocabulary and Core Concept Check
1. VOCABULARY Which of the operations +, −, ×, and ÷ are inverses of each other?
2. VOCABULARY Are the equations − 2x = 10 and −5x = 25 equivalent? Explain.
3. WRITING Which property of equality allows you to check a solution of an equation? Explain.
4. WHICH ONE DOESN’T BELONG? Which expression does not belong with the other three? Explain
your reasoning.
x
8=—
2
3=x÷4
x
3
—=9
x−6=5
Monitoring Progress and Modeling with Mathematics
In Exercises 5–14, solve the equation. Justify each step.
Check your solution. (See Example 1.)
USING TOOLS The sum of the angle measures of a
5. x + 5 = 8
6. m + 9 = 2
quadrilateral is 360°. In Exercises 17–20, write and
solve an equation to find the value of x. Use a protractor
to check the reasonableness of your answer.
7. y − 4 = 3
8. s − 2 = 1
17.
9. w + 3 = −4
10. n − 6 = −7
11. −14 = p − 11
12. 0 = 4 + q
13. r + (−8) = 10
x°
x°
150°
100°
77°
48°
120°
100°
14. t − (−5) = 9
15. MODELING WITH MATHEMATICS A discounted
18.
19. 76°
amusement park ticket costs $12.95 less than the
original price p. Write and solve an equation to find
the original price.
x°
20.
115°
122°
85°
x°
92°
60°
In Exercises 21–30, solve the equation. Justify each step.
Check your solution. (See Example 2.)
16. MODELING WITH MATHEMATICS You and a friend
are playing a board game. Your final score x is
12 points less than your friend’s final score. Write
and solve an equation to find your final score.
ROUND
9
ROUND
10
FINAL
SCORE
Your Friend
You
8
Chapter 1
21. 5g = 20
22. 4q = 52
23. p ÷ 5 = 3
24. y ÷ 7 = 1
25. −8r = 64
26. x ÷ (−2) = 8
x
6
w
−3
27. — = 8
28. — = 6
29. −54 = 9s
30. −7 = —
t
7
Solving Linear Equations
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In Exercises 31– 38, solve the equation. Check your
solution.
31.
3
—2
+t=
1
—2
32. b −
3
—
16
=
45. REASONING Identify the property of equality that
makes Equation 1 and Equation 2 equivalent.
5
—
16
33. —37 m = 6
34. −—5 y = 4
Equation 1
1 x
x−—=—+3
2 4
35. 5.2 = a − 0.4
36. f + 3π = 7π
Equation 2
4x − 2 = x + 12
37. − 108π = 6π j
38. x ÷ (−2) = 1.4
2
46. PROBLEM SOLVING Tatami mats are used as a floor
ERROR ANALYSIS In Exercises 39 and 40, describe and
correct the error in solving the equation.
39.
✗
covering in Japan. One possible layout uses four
identical rectangular mats and one square mat, as
shown. The area of the square mat is half the area of
one of the rectangular mats.
−0.8 + r = 12.6
r = 12.6 + (−0.8)
r = 11.8
40.
✗
3
Total area = 81 ft2
m
−— = −4
3
m
−— = 3 (−4)
3
m = −12
⋅( )
⋅
41. ANALYZING RELATIONSHIPS A baker orders 162 eggs.
Each carton contains 18 eggs. Which equation can
you use to find the number x of cartons? Explain your
reasoning and solve the equation.
A 162x = 18
○
x
B — = 162
○
18
C 18x = 162
○
D x + 18 = 162
○
MODELING WITH MATHEMATICS In Exercises 42– 44,
write and solve an equation to answer the question.
(See Examples 3 and 4.)
42. The temperature at 5 p.m. is 20°F. The temperature
at 10 p.m. is −5°F. How many degrees did the
temperature fall?
a. Write and solve an equation to find the area of
one rectangular mat.
b. The length of a rectangular mat is twice the
width. Use Guess, Check, and Revise to find
the dimensions of one rectangular mat.
47. PROBLEM SOLVING You spend $30.40 on 4 CDs.
Each CD costs the same amount and is on sale for
80% of the original price.
a. Write and solve an
equation to find how
much you spend on
each CD.
b. The next day, the CDs
are no longer on sale.
You have $25. Will you
be able to buy 3 more CDs?
Explain your reasoning.
48. ANALYZING RELATIONSHIPS As c increases, does
the value of x increase, decrease, or stay the same
for each equation? Assume c is positive.
43. The length of an
American flag is
1.9 times its width.
What is the width of
the flag?
Equation
9.5 ft
Value of x
x−c=0
44. The balance of an investment account is $308 more
cx = 1
than the balance 4 years ago. The current balance
of the account is $4708. What was the balance
4 years ago?
cx = c
x
c
—=1
Section 1.1
Solving Simple Equations
9
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49. USING STRUCTURE Use the values −2, 5, 9, and 10
MATHEMATICAL CONNECTIONS In Exercises 53–56, find
the height h or the area of the base B of the solid.
to complete each statement about the equation
ax = b − 5.
53.
54.
a. When a = ___ and b = ___, x is a positive integer.
h
b. When a = ___ and b = ___, x is a negative integer.
7 in.
B = 147 cm2
B
50. HOW DO YOU SEE IT? The circle graph shows the
Volume = 84π in.3
percents of different animals sold at a local pet store
in 1 year.
55.
Hamster: 5%
Volume = 1323 cm3
56.
5m
h
Rabbit:
9%
Bird:
7%
B = 30 ft 2
B
Dog:
48%
Volume = 15π m3
Cat:
x%
Volume = 35 ft3
57. MAKING AN ARGUMENT In baseball, a player’s
batting average is calculated by dividing the number
of hits by the number of at-bats. The table shows
Player A’s batting average and number of at-bats for
three regular seasons.
a. What percent is represented by the entire circle?
b. How does the equation 7 + 9 + 5 + 48 + x = 100
relate to the circle graph? How can you use this
equation to find the percent of cats sold?
Season
Batting average
At-bats
2010
.312
596
2011
.296
446
2012
.295
599
51. REASONING One-sixth of the girls and two-sevenths
of the boys in a school marching band are in the
percussion section. The percussion section has 6 girls
and 10 boys. How many students are in the marching
band? Explain.
a. How many hits did Player A have in the 2011
regular season? Round your answer to the nearest
whole number.
b. Player B had 33 fewer hits in the 2011 season than
Player A but had a greater batting average. Your
friend concludes that Player B had more at-bats in
the 2011 season than Player A. Is your friend
correct? Explain.
52. THOUGHT PROVOKING Write a real-life problem
that can be modeled by an equation equivalent to the
equation 5x = 30. Then solve the equation and write
the answer in the context of your real-life problem.
Maintaining Mathematical Proficiency
Reviewing what you learned in previous grades and lessons
Use the Distributive Property to simplify the expression.
(Skills Review Handbook)
(
1
59. —56 x + —2 + 4
58. 8(y + 3)
)
60. 5(m + 3 + n)
61. 4(2p + 4q + 6)
Copy and complete the statement. Round to the nearest hundredth, if necessary.
(Skills Review Handbook)
5L
min
L
h
63. — ≈ —
7 gal
min
qt
sec
65. — ≈ —
62. — = —
64. — ≈ —
10
Chapter 1
68 mi
h
mi
sec
8 km
min
h
mi
Solving Linear Equations
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1.2
Solving Multi-Step Equations
Essential Question
How can you use multi-step equations to solve
real-life problems?
Solving for the Angle Measures of a Polygon
Work with a partner. The sum S of the angle measures of a polygon with n sides can
be found using the formula S = 180(n − 2). Write and solve an equation to find each
value of x. Justify the steps in your solution. Then find the angle measures of each
polygon. How can you check the reasonableness of your answers?
a.
b.
c.
(30 + x)°
50°
(2x + 30)°
(x + 10)°
9x°
JUSTIFYING
CONCLUSIONS
To be proficient in math,
you need to be sure your
answers make sense in
the context of the
problem. For instance,
if you find the angle
measures of a triangle,
and they have a sum that
is not equal to 180°, then
you should check your
work for mistakes.
30°
(2x + 20)°
(x + 20)°
50°
x°
d.
(x − 17)°
e.
(x + 35)°
(5x + 2)°
(3x + 5)°
(2x + 8)°
f.
(3x + 16)°
(8x + 8)°
(5x + 10)°
(x + 42)°
x°
(4x − 18)°
(3x − 7)°
(4x + 15)°
(2x + 25)°
Writing a Multi-Step Equation
Work with a partner.
CONNECTIONS TO
GEOMETRY
You will learn more about
angle measures of polygons
in a future course.
a. Draw an irregular polygon.
b. Measure the angles of the polygon. Record the measurements on
a separate sheet of paper.
c. Choose a value for x. Then, using this value, work backward to assign a
variable expression to each angle measure, as in Exploration 1.
d. Trade polygons with your partner.
e. Solve an equation to find the angle measures of the polygon your partner
drew. Do your answers seem reasonable? Explain.
Communicate Your Answer
3. How can you use multi-step equations to solve real-life problems?
4. In Exploration 1, you were given the formula for the sum S of the angle measures
of a polygon with n sides. Explain why this formula works.
5. The sum of the angle measures of a polygon is 1080º. How many sides does the
polygon have? Explain how you found your answer.
Section 1.2
Solving Multi-Step Equations
11
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1.2 Lesson
What You Will Learn
Solve multi-step linear equations using inverse operations.
Use multi-step linear equations to solve real-life problems.
Core Vocabul
Vocabulary
larry
Use unit analysis to model real-life problems.
Previous
inverse operations
mean
Solving Multi-Step Linear Equations
Core Concept
Solving Multi-Step Equations
To solve a multi-step equation, simplify each side of the equation, if necessary.
Then use inverse operations to isolate the variable.
Solving a Two-Step Equation
Solve 2.5x − 13 = 2. Check your solution.
SOLUTION
2.5x − 13 =
+ 13
Undo the subtraction.
Write the equation.
2
+ 13
2.5x =
Add 13 to each side.
2.5x
15
—= —
2.5
2.5
Undo the multiplication.
Check
Simplify.
15
2.5x − 13 = 2
?
2.5(6) − 13 = 2
Divide each side by 2.5.
x=6
2=2
Simplify.
✓
The solution is x = 6.
Combining Like Terms to Solve an Equation
Solve −12 = 9x − 6x + 15. Check your solution.
SOLUTION
Undo the addition.
Undo the multiplication.
−12 = 9x − 6x + 15
Write the equation.
−12 = 3x + 15
Combine like terms.
− 15
Subtract 15 from each side.
− 15
−27 = 3x
Simplify.
−27 3x
—=—
3
3
Divide each side by 3.
−9 = x
Check
Simplify.
− 12 = − 12
The solution is x = −9.
Monitoring Progress
− 12 = 9x − 6x + 15
?
− 12 = 9(− 9) − 6(− 9) + 15
✓
Help in English and Spanish at BigIdeasMath.com
Solve the equation. Check your solution.
1. −2n + 3 = 9
12
Chapter 1
2. −21 = —12 c − 11
3. −2x − 10x + 12 = 18
Solving Linear Equations
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Using Structure to Solve a Multi-Step Equation
Solve 2(1 − x) + 3 = − 8. Check your solution.
SOLUTION
Method 1 One way to solve the equation is by using the Distributive Property.
2(1 − x) + 3 = −8
2(1) − 2(x) + 3 = −8
REMEMBER
2 − 2x + 3 = −8
The Distributive Property
states the following for
real numbers a, b, and c.
−2x + 5 = −8
−5
Sum
a(b + c) = ab + ac
−5
Distributive Property
Multiply.
Combine like terms.
Subtract 5 from each side.
−2x = −13
Simplify.
−2x
−2
Divide each side by −2.
−13
−2
—=—
Difference
a(b − c) = ab − ac
Write the equation.
x = 6.5
The solution is x = 6.5.
Simplify.
Check
2(1 − x) + 3 = − 8
?
2(1− 6.5) + 3 = − 8
−8 = −8
✓
Method 2 Another way to solve the equation is by interpreting the expression
1 − x as a single quantity.
2(1 − x) + 3 = −8
LOOKING FOR
STRUCTURE
−3
First solve for the
expression 1 − x, and
then solve for x.
Write the equation.
−3
Subtract 3 from each side.
2(1 − x) = −11
Simplify.
2(1 − x)
2
Divide each side by 2.
−11
2
—=—
1 − x = −5.5
−1
−1
Simplify.
Subtract 1 from each side.
−x = −6.5
Simplify.
−x
−1
Divide each side by − 1.
−6.5
−1
—=—
x = 6.5
Simplify.
The solution is x = 6.5, which is the same solution obtained in Method 1.
Monitoring Progress
Help in English and Spanish at BigIdeasMath.com
Solve the equation. Check your solution.
4. 3(x + 1) + 6 = −9
5. 15 = 5 + 4(2d − 3)
6. 13 = −2(y − 4) + 3y
7. 2x(5 − 3) − 3x = 5
8. −4(2m + 5) − 3m = 35
9. 5(3 − x) + 2(3 − x) = 14
Section 1.2
Solving Multi-Step Equations
13
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Solving Real-Life Problems
Modeling with Mathematics
U the table to find the number of miles x
Use
yyou need to bike on Friday so that the mean
nnumber of miles biked per day is 5.
SOLUTION
S
Day
Miles
Monday
3.5
Tuesday
5.5
Wednesday
0
Thursday
5
Friday
x
11. Understand the Problem You know how
many miles you biked Monday through
Thursday. You are asked to find the number
of miles you need to bike on Friday so that
the mean number of miles biked per day is 5.
2.
2 Make a Plan Use the definition of mean to write an equation that represents the
problem. Then solve the equation.
3.
3 Solve the Problem The mean of a data set is the sum of the data divided by the
number of data values.
3.5 + 5.5 + 0 + 5 + x
5
Write the equation.
14 + x
5
Combine like terms.
—— = 5
—=5
14 + x
5 —=5 5
5
⋅
14 + x =
− 14
⋅
Multiply each side by 5.
25
Simplify.
− 14
Subtract 14 from each side.
x = 11
Simplify.
You need to bike 11 miles on Friday.
4. Look Back Notice that on the days that you did bike, the values are close to
the mean. Because you did not bike on Wednesday, you need to bike about
twice the mean on Friday. Eleven miles is about twice the mean. So, your
solution is reasonable.
Monitoring Progress
Help in English and Spanish at BigIdeasMath.com
1
10. The formula d = —2 n + 26 relates the nozzle pressure n (in pounds per square
inch) of a fire hose and the maximum horizontal distance the water reaches d
(in feet). How much pressure is needed to reach a fire 50 feet away?
d
14
Chapter 1
Solving Linear Equations
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REMEMBER
When you add miles to
miles, you get miles.
But, when you divide
miles by days, you
get miles per day.
Using Unit Analysis to Model Real-Life Problems
When you write an equation to model a real-life problem, you should check that the
units on each side of the equation balance. For instance, in Example 4, notice how
the units balance.
miles
miles per day
3.5 + 5.5 + 0 + 5 + x
5
mi
day
—— = 5
per
mi
day
—=—
✓
days
Solving a Real-Life Problem
Your school’s drama club charges $4 per person for admission to a play. The club
borrowed $400 to pay for costumes and props. After paying back the loan, the club
has a profit of $100. How many people attended the play?
SOLUTION
1. Understand the Problem You know how much the club charges for admission.
You also know how much the club borrowed and its profit. You are asked to find
how many people attended the play.
2. Make a Plan Use a verbal model to write an equation that represents the problem.
Then solve the equation.
3. Solve the Problem
REMEMBER
When you multiply dollars
per person by people, you
get dollars.
⋅ who attended
Words
Ticket
price
Variable
Let x be the number of people who attended.
Equation
—
$4
person
Number of people
−
Amount
= Profit
of loan
⋅ x people − $400 = $100
4x − 400 = 100
$=$
✓
Write the equation.
4x − 400 + 400 = 100 + 400
Add 400 to each side.
4x = 500
Simplify.
4x
4
Divide each side by 4.
500
4
—=—
x = 125
Simplify.
So, 125 people attended the play.
4. Look Back To check that your solution is reasonable, multiply $4 per person by
125 people. The result is $500. After paying back the $400 loan, the club has $100,
which is the profit.
Monitoring Progress
Help in English and Spanish at BigIdeasMath.com
11. You have 96 feet of fencing to enclose a rectangular pen for your dog. To provide
sufficient running space for your dog to exercise, the pen should be three times as
long as it is wide. Find the dimensions of the pen.
Section 1.2
Solving Multi-Step Equations
15
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1.2
Exercises
Dynamic Solutions available at BigIdeasMath.com
Vocabulary and Core Concept Check
1. COMPLETE THE SENTENCE To solve the equation 2x + 3x = 20, first combine 2x and 3x because
they are _________.
2. WRITING Describe two ways to solve the equation 2(4x − 11) = 10.
Monitoring Progress and Modeling with Mathematics
In Exercises 3−14, solve the equation. Check your
solution. (See Examples 1 and 2.)
23. −3(3 + x) + 4(x − 6) = − 4
3. 3w + 7 = 19
4. 2g − 13 = 3
24. 5(r + 9) − 2(1 − r) = 1
5. 11 = 12 − q
6. 10 = 7 − m
USING TOOLS In Exercises 25−28, find the value of the
z
−4
a
3
7. 5 = — − 3
8. — + 4 = 6
h+6
5
d−8
−2
9. — = 2
10. — = 12
11. 8y + 3y = 44
variable. Then find the angle measures of the polygon.
Use a protractor to check the reasonableness of
your answer.
25.
26.
a°
2k°
12. 36 = 13n − 4n
45°
2a° 2a°
k°
Sum of angle
measures: 180°
13. 12v + 10v + 14 = 80
a°
Sum of angle
measures: 360°
14. 6c − 8 − 2c = −16
15. MODELING WITH MATHEMATICS The altitude a
(in feet) of a plane t minutes after liftoff is given by
a = 3400t + 600. How many minutes
after liftoff is the plane
at an altitude of
21,000 feet?
16. MODELING WITH MATHEMATICS A repair bill for
your car is $553. The parts cost $265. The labor cost
is $48 per hour. Write and solve an equation to find
the number of hours of labor spent repairing the car.
In Exercises 17−24, solve the equation. Check your
solution. (See Example 3.)
17. 4(z + 5) = 32
18. − 2(4g − 3) = 30
19. 6 + 5(m + 1) = 26
20. 5h + 2(11 − h) = − 5
27.
b°
3
b°
2
(b + 45)°
(2b − 90)° 90°
Sum of angle
measures: 540°
28.
120°
x°
100°
120°
120°
(x + 10)°
Sum of angle
measures: 720°
In Exercises 29−34, write and solve an equation to find
the number.
29. The sum of twice a number and 13 is 75.
30. The difference of three times a number and 4 is −19.
31. Eight plus the quotient of a number and 3 is −2.
32. The sum of twice a number and half the number is 10.
21. 27 = 3c − 3(6 − 2c)
33. Six times the sum of a number and 15 is − 42.
22. −3 = 12y − 5(2y − 7)
34. Four times the difference of a number and 7 is 12.
16
Chapter 1
Solving Linear Equations
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USING EQUATIONS In Exercises 35−37, write and solve
ERROR ANALYSIS In Exercises 40 and 41, describe and
an equation to answer the question. Check that the units
on each side of the equation balance. (See Examples 4
and 5.)
correct the error in solving the equation.
40.
35. During the summer, you work 30 hours per week at
a gas station and earn $8.75 per hour. You also work
as a landscaper for $11 per hour and can work as
many hours as you want. You want to earn a total of
$400 per week. How many hours must you work as
a landscaper?
deep
end
shallow
end
d
9 ft
−2(7 − y) + 4 = −4
−14 − 2y + 4 = −4
−10 − 2y = −4
−2y = 6
y = −3
36. The area of the surface of the swimming pool is
210 square feet. What is the length d of the deep
end (in feet)?
✗
41.
✗
1
4
—(x − 2) + 4 = 12
1
4
—(x − 2) = 8
x−2=2
x=4
10 ft
MATHEMATICAL CONNECTIONS In Exercises 42−44,
37. You order two tacos and a salad. The salad costs
$2.50. You pay 8% sales tax and leave a $3 tip. You
pay a total of $13.80. How much does one taco cost?
write and solve an equation to answer the question.
42. The perimeter of the tennis court is 228 feet. What are
the dimensions of the court?
JUSTIFYING STEPS In Exercises 38 and 39, justify each
step of the solution.
1
2
38. −—(5x − 8) − 1 = 6
w
Write the equation.
1
−—(5x − 8) = 7
2
2w + 6
43. The perimeter of the Norwegian flag is 190 inches.
What are the dimensions of the flag?
5x − 8 = −14
5x = −6
y
6
x = −—
5
39.
2(x + 3) + x = −9
11
y
8
Write the equation.
2(x) + 2(3) + x = −9
44. The perimeter of the school crossing sign is
102 inches. What is the length of each side?
2x + 6 + x = −9
s+6
3x + 6 = −9
s+6
3x = −15
s
x = −5
s
2s
Section 1.2
Solving Multi-Step Equations
17
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45. COMPARING METHODS Solve the equation
2(4 − 8x) + 6 = −1 using (a) Method 1 from
Example 3 and (b) Method 2 from Example 3.
Which method do you prefer? Explain.
46. PROBLEM SOLVING An online ticket agency charges
the amounts shown for basketball tickets. The total
cost for an order is $220.70. How many tickets
are purchased?
49. REASONING An even integer can be represented by
the expression 2n, where n is any integer. Find three
consecutive even integers that have a sum of 54.
Explain your reasoning.
50. HOW DO YOU SEE IT? The scatter plot shows the
attendance for each meeting of a gaming club.
Charge
Amount
Ticket price
$32.50 per ticket
Convenience charge
$3.30 per ticket
Processing charge
$5.90 per order
Students
Gaming Club Attendance
y
25
20
15
10
5
0
18
1
48. THOUGHT PROVOKING You teach a math class and
assign a weight to each component of the class. You
determine final grades by totaling the products of the
weights and the component scores. Choose values for
the remaining weights and find the necessary score on
the final exam for a student to earn an A (90%) in the
class, if possible. Explain your reasoning.
Component
Student’s
Weight
score
Score × Weight
92% × 0.20
= 18.4%
3
4
x
a. The mean attendance for the first four meetings
is 20. Is the number of students who attended
the fourth meeting greater than or less than 20?
Explain.
b. Estimate the number of students who attended
the fourth meeting.
c. Describe a way you can check your estimate in
part (b).
REASONING In Exercises 51−56, the letters a, b, and c
represent nonzero constants. Solve the equation for x.
Class
Participation
92%
Homework
95%
52. x + a = —4
Midterm
Exam
88%
53. ax − b = 12.5
0.20
2
17
Meeting
47. MAKING AN ARGUMENT You have quarters and
dimes that total $2.80. Your friend says it is possible
that the number of quarters is 8 more than the number
of dimes. Is your friend correct? Explain.
21
51. bx = −7
3
Final Exam
54. ax + b = c
Total
1
55. 2bx − bx = −8
56. cx − 4b = 5b
Maintaining Mathematical Proficiency
Simplify the expression.
57. 4m + 5 − 3m
Reviewing what you learned in previous grades and lessons
(Skills Review Handbook)
58. 9 − 8b + 6b
59. 6t + 3(1 − 2t) − 5
Determine whether (a) x = −1 or (b) x = 2 is a solution of the equation.
18
(Skills Review Handbook)
60. x − 8 = − 9
61. x + 1.5 = 3.5
62. 2x − 1 = 3
63. 3x + 4 = 1
64. x + 4 = 3x
65. − 2(x − 1) = 1 − 3x
Chapter 1
Solving Linear Equations
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1.3
Solving Equations with
Variables on Both Sides
Essential Question
How can you solve an equation that has
variables on both sides?
Perimeter
Work with a partner. The two polygons have the same perimeter. Use this
information to write and solve an equation involving x. Explain the process you
used to find the solution. Then find the perimeter of each polygon.
3
x
2
x
5
5
3
5
2
2
x
CONNECTIONS TO
GEOMETRY
4
Perimeter and Area
Work with a partner.
You will learn about finding
perimeters and areas of
polygons in the coordinate
plane in Chapter 8.
•
Each figure has the unusual property that the value of its perimeter (in feet) is equal
to the value of its area (in square feet). Use this information to write an equation for
each figure.
•
•
Solve each equation for x. Explain the process you used to find the solution.
a.
Find the perimeter and area of each figure.
5
3
b.
5
4
LOOKING FOR
STRUCTURE
To be proficient in math,
you need to visualize
complex things, such as
composite figures, as
being made up of simpler,
more manageable parts.
c.
1
2
6
2
x
x
x
Communicate Your Answer
3. How can you solve an equation that has variables on both sides?
4. Write three equations that have the variable x on both sides. The equations should
be different from those you wrote in Explorations 1 and 2. Have your partner
solve the equations.
Section 1.3
Solving Equations with Variables on Both Sides
19
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1.3 Lesson
What You Will Learn
Solve linear equations that have variables on both sides.
Core Vocabul
Vocabulary
larry
Identify special solutions of linear equations.
Use linear equations to solve real-life problems.
identity, p. 21
Previous
inverse operations
Solving Equations with Variables on Both Sides
Core Concept
Solving Equations with Variables on Both Sides
To solve an equation with variables on both sides, simplify one or both sides of the
equation, if necessary. Then use inverse operations to collect the variable terms on
one side, collect the constant terms on the other side, and isolate the variable.
Solving an Equation with Variables on Both Sides
Solve 10 − 4x = −9x. Check your solution.
SOLUTION
10 − 4x = −9x
+ 4x
Write the equation.
+ 4x
Add 4x to each side.
10 = − 5x
Simplify.
10
−5x
—=—
−5
−5
Divide each side by −5.
−2 = x
Simplify.
Check
10 − 4x = −9x
?
10 − 4(−2) = −9(−2)
18 = 18
✓
The solution is x = −2.
Solving an Equation with Grouping Symbols
1
Solve 3(3x − 4) = —(32x + 56).
4
SOLUTION
3(3x − 4) =
9x − 12 =
— (32x + 56)
1
4
Write the equation.
8x + 14
Distributive Property
+ 12
Add 12 to each side.
+ 12
9x =
− 8x
8x + 26
Simplify.
− 8x
Subtract 8x from each side.
x = 26
Simplify.
The solution is x = 26.
Monitoring Progress
Help in English and Spanish at BigIdeasMath.com
Solve the equation. Check your solution.
1. −2x = 3x + 10
20
Chapter 1
2. —12 (6h − 4) = −5h + 1
3
3. −—4 (8n + 12) = 3(n − 3)
Solving Linear Equations
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Identifying Special Solutions of Linear Equations
Core Concept
Special Solutions of Linear Equations
Equations do not always have one solution. An equation that is true for all values
of the variable is an identity and has infinitely many solutions. An equation that
is not true for any value of the variable has no solution.
Identifying the Number of Solutions
REASONING
The equation 15x + 6 = 15x
is not true because the
number 15x cannot be
equal to 6 more than itself.
Solve each equation.
a. 3(5x + 2) = 15x
b. −2(4y + 1) = −8y − 2
SOLUTION
a. 3(5x + 2) =
15x
Write the equation.
15x + 6 =
15x
Distributive Property
− 15x
− 15x
6=0
Subtract 15x from each side.
✗
Simplify.
The statement 6 = 0 is never true. So, the equation has no solution.
b. −2(4y + 1) = −8y − 2
READING
Write the equation.
−8y − 2 = −8y − 2
Distributive Property
+ 8y
Add 8y to each side.
All real numbers are
solutions of an identity.
+ 8y
−2 = −2
Simplify.
The statement −2 = −2 is always true. So, the equation is an identity and has
infinitely many solutions.
Monitoring Progress
Help in English and Spanish at BigIdeasMath.com
Solve the equation.
5
4. 4(1 − p) = −4p + 4
5. 6m − m = —6 (6m − 10)
6. 10k + 7 = −3 − 10k
7. 3(2a − 2) = 2(3a − 3)
Concept Summary
Steps for Solving Linear Equations
Here are several steps you can use to solve a linear equation. Depending on the
equation, you may not need to use some steps.
Step 1 Use the Distributive Property to remove any grouping symbols.
STUDY TIP
To check an identity, you
can choose several different
values of the variable.
Step 2 Simplify the expression on each side of the equation.
Step 3 Collect the variable terms on one side of the equation and the constant
terms on the other side.
Step 4 Isolate the variable.
Step 5 Check your solution.
Section 1.3
Solving Equations with Variables on Both Sides
21
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Solving Real-Life Problems
Modeling with Mathematics
A boat leaves New Orleans and travels upstream on the Mississippi River for 4 hours.
The return trip takes only 2.8 hours because the boat travels 3 miles per hour faster
T
ddownstream due to the current. How far does the boat travel upstream?
SOLUTION
1. Understand the Problem You are given the amounts of time the boat travels and
the difference in speeds for each direction. You are asked to find the distance the
boat travels upstream.
22. Make a Plan Use the Distance Formula to write expressions that represent the
problem. Because the distance the boat travels in both directions is the same, you
can use the expressions to write an equation.
33. Solve the Problem Use the formula (distance) = (rate)(time).
Words
Distance upstream = Distance downstream
Variable
Let x be the speed (in miles per hour) of the boat traveling upstream.
Equation
x mi
1h
—
(x + 3) mi
⋅4 h = —
⋅ 2.8 h
1h
(mi = mi)
✓
4x = 2.8(x + 3)
Write the equation.
4x = 2.8x + 8.4
Distributive Property
− 2.8x − 2.8x
Subtract 2.8x from each side.
1.2x = 8.4
Simplify.
1.2x
1.2
Divide each side by 1.2.
8.4
1.2
—=—
x=7
Simplify.
So, the boat travels 7 miles per hour upstream. To determine how far the boat
travels upstream, multiply 7 miles per hour by 4 hours to obtain 28 miles.
4. Look Back To check that your solution is reasonable, use the formula for
distance. Because the speed upstream is 7 miles per hour, the speed downstream
is 7 + 3 = 10 miles per hour. When you substitute each speed into the Distance
Formula, you get the same distance for upstream and downstream.
Upstream
⋅
7 mi
Distance = — 4 h = 28 mi
1h
Downstream
⋅
10 mi
Distance = — 2.8 h = 28 mi
1h
Monitoring Progress
Help in English and Spanish at BigIdeasMath.com
8. A boat travels upstream on the Mississippi River for 3.5 hours. The return trip
only takes 2.5 hours because the boat travels 2 miles per hour faster downstream
due to the current. How far does the boat travel upstream?
22
Chapter 1
Solving Linear Equations
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1.3
Exercises
Dynamic Solutions available at BigIdeasMath.com
Vocabulary and Core Concept Check
1. VOCABULARY Is the equation − 2(4 − x) = 2x + 8 an identity? Explain your reasoning.
2. WRITING Describe the steps in solving the linear equation 3(3x − 8) = 4x + 6.
Monitoring Progress and Modeling with Mathematics
In Exercises 3–16, solve the equation. Check your
solution. (See Examples 1 and 2.)
3. 15 − 2x = 3x
4. 26 − 4s = 9s
5. 5p − 9 = 2p + 12
6. 8g + 10 = 35 + 3g
7. 5t + 16 = 6 − 5t
In Exercises 19–24, solve the equation. Determine
whether the equation has one solution, no solution, or
infinitely many solutions. (See Example 3.)
19. 3t + 4 = 12 + 3t
20. 6d + 8 = 14 + 3d
21. 2(h + 1) = 5h − 7
22. 12y + 6 = 6(2y + 1)
8. −3r + 10 = 15r − 8
23. 3(4g + 6) = 2(6g + 9)
9. 7 + 3x − 12x = 3x + 1
1
24. 5(1 + 2m) = —2 (8 + 20m)
10. w − 2 + 2w = 6 + 5w
11. 10(g + 5) = 2(g + 9)
ERROR ANALYSIS In Exercises 25 and 26, describe and
correct the error in solving the equation.
12. −9(t − 2) = 4(t − 15)
25.
13. —23 (3x + 9) = −2(2x + 6)
✗
3
14. 2(2t + 4) = —4 (24 − 8t)
2c − 6 = 4
2c = 10
c=5
15. 10(2y + 2) − y = 2(8y − 8)
16. 2(4x + 2) = 4x − 12(x − 1)
26.
17. MODELING WITH MATHEMATICS You and your
friend drive toward each other. The equation
50h = 190 − 45h represents the number h of hours
until you and your friend meet. When will you meet?
18. MODELING WITH MATHEMATICS The equation
1.5r + 15 = 2.25r represents the number r of movies
you must rent to spend the same amount at each
movie store. How many movies must you rent to
spend the same amount at each movie store?
VIDEO
CITY
MEM
M
MBE
ERSH
HIP
Membership Fee: $15
5c − 6 = 4 − 3c
✗
6(2y + 6) = 4(9 + 3y)
12y + 36 = 36 + 12y
12y = 12y
0=0
The equation has no solution.
27. MODELING WITH MATHEMATICS Write and solve an
equation to find the month when you would pay the
same total amount for each Internet service.
Installation fee
Price per month
Company A
$60.00
$42.95
Company B
$25.00
$49.95
Membership Fee: Free
Section 1.3
Solving Equations with Variables on Both Sides
23
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28. PROBLEM SOLVING One serving of granola provides
37. REASONING Two times the greater of two
4% of the protein you need daily. You must get the
remaining 48 grams of protein from other sources.
How many grams of protein do you need daily?
consecutive integers is 9 less than three times the
lesser integer. What are the integers?
38. HOW DO YOU SEE IT? The table and the graph show
information about students enrolled in Spanish and
French classes at a high school.
USING STRUCTURE In Exercises 29 and 30, find the value
of r.
29. 8(x + 6) − 10 + r = 3(x + 12) + 5x
Students enrolled
this year
30. 4(x − 3) − r + 2x = 5(3x − 7) − 9x
MATHEMATICAL CONNECTIONS In Exercises 31 and 32,
the value of the surface area of the cylinder is equal to
the value of the volume of the cylinder. Find the value of
x. Then find the surface area and volume of the cylinder.
32.
2.5 cm
355
French
229
1
5
7 ft
x ft
x cm
y
400
350
300
250
200
150
0
33. MODELING WITH MATHEMATICS A cheetah that
speed for only about 20 seconds. If an antelope is
too far away for a cheetah to catch it in 20 seconds,
the antelope is probably safe. Your friend claims the
antelope in Exercise 33 will not be safe if the cheetah
starts running 650 feet behind it. Is your friend
correct? Explain.
1 2 3 4 5 6 7 8 9 10 x
39. WRITING EQUATIONS Give an example of a linear
equation that has (a) no solution and (b) infinitely
many solutions. Justify your answers.
REASONING In Exercises 35 and 36, for what value of a
is the equation an identity? Explain your reasoning.
40. THOUGHT PROVOKING Draw
35. a(2x + 3) = 9x + 15 + x
a different figure that has
the same perimeter as the
triangle shown. Explain
why your figure has the
same perimeter.
36. 8x − 8 + 3ax = 5ax − 2a
Maintaining Mathematical Proficiency
x+3
2x + 1
3x
Reviewing what you learned in previous grades and lessons
(Skills Review Handbook)
41. 9, ∣ −4 ∣, −4, 5, ∣ 2 ∣
42.
43. −18, ∣ −24 ∣, −19, ∣ −18 ∣, ∣ 22 ∣
44. −∣ −3 ∣, ∣ 0 ∣, −1, ∣ 2 ∣, −2
Chapter 1
Spanish
a. Use the graph to determine after how many years
there will be equal enrollment in Spanish and
French classes.
b. How does the equation 355 − 9x = 229 + 12x
relate to the table and the graph? How can you
use this equation to determine whether your
answer in part (a) is reasonable?
34. MAKING AN ARGUMENT A cheetah can run at top
24
French
Years from now
is running 90 feet per second is 120 feet behind an
antelope that is running 60 feet per second. How
long will it take the cheetah to catch up to the
antelope? (See Example 4.)
Order the values from least to greatest.
9 fewer students
each year
12 more students
each year
Predicted Language
Class Enrollment
Students enrolled
31.
Spanish
Average rate
of change
∣ −32 ∣, 22, −16, −∣ 21 ∣, ∣ −10 ∣
Solving Linear Equations
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1.1–1.3
What Did You Learn?
Core Vocabulary
conjecture, p. 3
rule, p. 3
theorem, p. 3
equation, p. 4
linear equation in one variable, p. 4
solution, p. 4
inverse operations, p. 4
equivalent equations, p. 4
identity, p. 21
Core Concepts
Section 1.1
Addition Property of Equality, p. 4
Subtraction Property of Equality, p. 4
Substitution Property of Equality, p. 4
Multiplication Property of Equality, p. 5
Division Property of Equality, p. 5
Four-Step Approach to Problem Solving, p. 6
Common Problem-Solving Strategies, p. 7
Section 1.2
Solving Multi-Step Equations, p. 12
Unit Analysis, p. 15
Section 1.3
Solving Equations with Variables on Both Sides, p. 20
Special Solutions of Linear Equations, p. 21
Mathematical Practices
1.
How did you make sense of the relationships between the quantities in Exercise 46 on page 9?
2.
What is the limitation of the tool you used in Exercises 25–28 on page 16?
3.
What definition did you use in your reasoning in Exercises 35 and 36 on page 24?
Using the Features of Yourr Textbook
d Tests
ests
to Prepare for Quizzes and
• Read and understand the core vocabulary and
the contents of the Core Concept boxes.
• Review the Examples and the Monitoring
Progress questions. Use the tutorials at
BigIdeasMath.com for additional help.
• Review previously completed homework
assignments.
25
5
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1.1–1.3
Quiz
Solve the equation. Justify each step. Check your solution. (Section 1.1)
1. x + 9 = 7
2. 8.6 = z − 3.8
3. 60 = −12r
4. —34 p = 18
Solve the equation. Check your solution. (Section 1.2)
5. 2m − 3 = 13
6. 5 = 10 − v
7. 5 = 7w + 8w + 2
8. −21a + 28a − 6 = −10.2
1
9. 2k − 3(2k − 3) = 45
10. 68 = —5 (20x + 50) + 2
Solve the equation. (Section 1.3)
11. 3c + 1 = c + 1
12. −8 − 5n = 64 + 3n
13. 2(8q − 5) = 4q
14. 9(y − 4) − 7y = 5(3y − 2)
15. 4(g + 8) = 7 + 4g
16. −4(−5h − 4) = 2(10h + 8)
17. To estimate how many miles you are from a thunderstorm, count the seconds between
when you see lightning and when you hear thunder. Then divide by 5. Write and solve an
equation to determine how many seconds you would count for a thunderstorm that is
2 miles away. (Section 1.1)
18. You want to hang three equally-sized travel posters on a wall so that the posters on the ends
are each 3 feet from the end of the wall. You want the spacing between posters to be equal.
Write and solve an equation to determine how much space you should leave between the
posters. (Section 1.2)
3 ft
2 ft
2 ft
2 ft
3 ft
15 ft
19. You want to paint a piece of pottery at an art studio. The total cost is the cost of the piece
plus an hourly studio fee. There are two studios to choose from. (Section 1.3)
a. After how many hours of painting are the total costs the same at both studios? Justify
your answer.
b. Studio B increases the hourly studio fee by $2. How does this affect your answer in part (a)? Explain.
26
Chapter 1
Solving Linear Equations
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1.4
Solving Absolute Value Equations
Essential Question
How can you solve an absolute value equation?
Solving an Absolute Value
Equation Algebraically
Work with a partner. Consider the absolute value equation
∣ x + 2 ∣ = 3.
MAKING SENSE
OF PROBLEMS
To be proficient in math,
you need to explain to
yourself the meaning of
a problem and look for
entry points to its solution.
a. Describe the values of x + 2 that make the equation true. Use your description
to write two linear equations that represent the solutions of the absolute value
equation.
b. Use the linear equations you wrote in part (a) to find the solutions of the absolute
value equation.
c. How can you use linear equations to solve an absolute value equation?
Solving an Absolute Value
Equation Graphically
Work with a partner. Consider the absolute value equation
∣ x + 2 ∣ = 3.
a. On a real number line, locate the point for which x + 2 = 0.
−10 −9 −8 −7 −6 −5 −4 −3 −2 −1
0
1
2
3
4
5
6
7
8
9 10
b. Locate the points that are 3 units from the point you found in part (a). What do you
notice about these points?
c. How can you use a number line to solve an absolute value equation?
Solving an Absolute Value
Equation Numerically
Work with a partner. Consider the absolute value equation
∣ x + 2 ∣ = 3.
a. Use a spreadsheet, as shown,
to solve the absolute value equation.
b. Compare the solutions you found using
the spreadsheet with those you found
in Explorations 1 and 2. What do
you notice?
c. How can you use a spreadsheet to
solve an absolute value equation?
Communicate Your Answer
1
2
3
4
5
6
7
8
9
10
11
A
x
-6
-5
-4
-3
-2
-1
0
1
2
B
|x + 2|
4
abs(A2 + 2)
4. How can you solve an absolute value equation?
5. What do you like or dislike about the algebraic, graphical, and numerical methods
for solving an absolute value equation? Give reasons for your answers.
Section 1.4
Solving Absolute Value Equations
27
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1.4 Lesson
What You Will Learn
Solve absolute value equations.
Solve equations involving two absolute values.
Core Vocabul
Vocabulary
larry
Identify special solutions of absolute value equations.
absolute value equation, p. 28
extraneous solution, p. 31
Previous
absolute value
opposite
Solving Absolute Value Equations
An absolute value equation is an equation that contains an absolute value expression.
You can solve these types of equations by solving two related linear equations.
Core Concept
Properties of Absolute Value
Let a and b be real numbers. Then the following properties are true.
1. ∣ a ∣ ≥ 0
2. ∣ −a ∣ = ∣ a ∣
3.
∣ ∣
∣a∣
a
4. — = —, b ≠ 0
b
∣b∣
∣ ab ∣ = ∣ a ∣ ∣ b ∣
Solving Absolute Value Equations
To solve ∣ ax + b ∣ = c when c ≥ 0, solve the related linear equations
ax + b = c
or
ax + b = − c.
When c < 0, the absolute value equation ∣ ax + b ∣ = c has no solution because
absolute value always indicates a number that is not negative.
Solving Absolute Value Equations
Solve each equation. Graph the solutions, if possible.
a. ∣ x − 4 ∣ = 6
b. ∣ 3x + 1 ∣ = −5
SOLUTION
a. Write the two related linear equations for ∣ x − 4 ∣ = 6. Then solve.
x−4=6
or
x − 4 = −6
x = 10
Write related linear equations.
x = −2
Add 4 to each side.
The solutions are x = 10 and x = −2.
−4
−2
0
2
4
6
6
8
10
12
6
Each solution is 6 units from 4.
Property of Absolute Value
b. The absolute value of an expression must be greater than or equal to 0. The
expression ∣ 3x + 1 ∣ cannot equal −5.
So, the equation has no solution.
Monitoring Progress
Help in English and Spanish at BigIdeasMath.com
Solve the equation. Graph the solutions, if possible.
1. ∣ x ∣ = 10
28
Chapter 1
2. ∣ x − 1 ∣ = 4
3. ∣ 3 + x ∣ = −3
Solving Linear Equations
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Solving an Absolute Value Equation
Solve ∣ 3x + 9 ∣ − 10 = −4.
SOLUTION
First isolate the absolute value expression on one side of the equation.
ANOTHER WAY
∣ 3x + 9 ∣ − 10 = −4
Using the product
property of absolute
value, |ab| = |a| |b|,
you could rewrite the
equation as
∣ 3x + 9 ∣ = 6
Write the equation.
Add 10 to each side.
Now write two related linear equations for ∣ 3x + 9 ∣ = 6. Then solve.
3x + 9 =
3|x + 3| − 10 = −4
and then solve.
6
3x + 9 = −6
or
3x = −3
3x = −15
x = −1
x = −5
Write related linear equations.
Subtract 9 from each side.
Divide each side by 3.
The solutions are x = −1 and x = −5.
Writing an Absolute Value Equation
REASONING
Mathematically proficient
students have the ability
to decontextualize
problem situations.
In a cheerleading competition, the minimum length of a routine is 4 minutes. The
maximum length of a routine is 5 minutes. Write an absolute value equation that
represents the minimum and maximum lengths.
SOLUTION
1. Understand the Problem You know the minimum and maximum lengths. You are
asked to write an absolute value equation that represents these lengths.
2. Make a Plan Consider the minimum and maximum lengths as solutions to an
absolute value equation. Use a number line to find the halfway point between the
solutions. Then use the halfway point and the distance to each solution to write an
absolute value equation.
33. Solve the Problem
4.0
4.1
4.2
4.3
4.4
4.5
4.6
0.5
4.7
4.8
4.9
5.0
0.5
halfway point
distance from halfway point
∣ x − 4.5 ∣ = 0.5
The equation is ∣ x − 4.5 ∣ = 0.5.
4.
4 Look Back To check that your equation is reasonable, substitute the minimum and
maximum lengths into the equation and simplify.
Minimum
∣ 4 − 4.5 ∣ = 0.5
Maximum
✓
∣ 5 − 4.5 ∣ = 0.5
Monitoring Progress
M
✓
Help in English and Spanish at BigIdeasMath.com
Solve the equation. Check your solutions.
S
4. ∣ x − 2 ∣ + 5 = 9
5. 4∣ 2x + 7 ∣ = 16
6. −2∣ 5x − 1 ∣ − 3 = −11
7. For a poetry contest, the minimum length of a poem is 16 lines. The maximum
length is 32 lines. Write an absolute value equation that represents the minimum
and maximum lengths.
Section 1.4
Solving Absolute Value Equations
29
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Solving Equations with Two Absolute Values
If the absolute values of two algebraic expressions are equal, then they must either be
equal to each other or be opposites of each other.
Core Concept
Solving Equations with Two Absolute Values
To solve ∣ ax + b ∣ = ∣ cx + d ∣, solve the related linear equations
ax + b = cx + d
or
ax + b = −(cx + d).
Solving Equations with Two Absolute Values
Solve (a) ∣ 3x − 4 ∣ = ∣ x ∣ and (b) ∣ 4x − 10 ∣ = 2∣ 3x + 1 ∣.
SOLUTION
a. Write the two related linear equations for ∣ 3x − 4 ∣ = ∣ x ∣. Then solve.
Check
3x − 4 =
∣ 3x − 4 ∣ = ∣ x ∣
−x
?
∣ 3(2) − 4 ∣ =
∣2∣
✓
+4
∣ 3x − 4 ∣ = ∣ x ∣
?
∣ 3(1) − 4 ∣ =
∣1∣
?
∣ −1 ∣ =
∣1∣
1=1
✓
3x − 4 = −x
or
−x
2x − 4 =
?
∣2∣ =
∣2∣
2=2
x
+x
+x
4x − 4 =
0
+4
+4
0
+4
2x = 4
4x = 4
—=—
2x
2
4
2
—=—
x=2
x=1
4x
4
4
4
The solutions are x = 2 and x = 1.
b. Write the two related linear equations for ∣ 4x − 10 ∣ = 2∣ 3x + 1 ∣. Then solve.
4x − 10 =
2(3x + 1)
4x − 10 =
6x + 2
− 6x
4x − 10 = 2[−(3x + 1)]
or
4x − 10 = 2(−3x − 1)
− 6x
− 2x − 10 =
+ 10
2
+ 6x
+ 10
−2x =
−2x
−2
4x − 10 = −6x − 2
+ 6x
10x − 10 = −2
+ 10
12
12
−2
—=—
+ 10
10x = 8
10x
10
8
10
—=—
x = −6
x = 0.8
The solutions are x = −6 and x = 0.8.
Monitoring Progress
Help in English and Spanish at BigIdeasMath.com
Solve the equation. Check your solutions.
8. ∣ x + 8 ∣ = ∣ 2x + 1 ∣
30
Chapter 1
9. 3∣ x − 4 ∣ = ∣ 2x + 5 ∣
Solving Linear Equations
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Identifying Special Solutions
When you solve an absolute value equation, it is possible for a solution to be
extraneous. An extraneous solution is an apparent solution that must be rejected
because it does not satisfy the original equation.
Identifying Extraneous Solutions
Solve ∣ 2x + 12 ∣ = 4x. Check your solutions.
SOLUTION
Write the two related linear equations for ∣ 2x + 12 ∣ = 4x. Then solve.
Check
∣ 2x + 12 ∣ = 4x
2x + 12 = 4x
?
∣ 2(6) + 12 ∣ =
4(6)
?
∣ 24 ∣ =
24
24 = 24
12 = 2x
∣ 2x + 12 ∣ = 4x
?
∣ 2(−2) + 12 ∣ =
4(−2)
8 ≠ −8
✗
2x + 12 = −4x
12 = −6x
6=x
✓
?
∣8∣ =
−8
or
−2 = x
Write related linear equations.
Subtract 2x from each side.
Solve for x.
Check the apparent solutions to see if either is extraneous.
The solution is x = 6. Reject x = −2 because it is extraneous.
When solving equations of the form ∣ ax + b ∣ = ∣ cx + d ∣, it is possible that one of the
related linear equations will not have a solution.
Solving an Equation with Two Absolute Values
Solve ∣ x + 5 ∣ = ∣ x + 11 ∣.
SOLUTION
By equating the expression x + 5 and the opposite of x + 11, you obtain
x + 5 = −(x + 11)
Write related linear equation.
x + 5 = −x − 11
Distributive Property
2x + 5 = −11
Add x to each side.
2x = −16
Subtract 5 from each side.
x = −8.
Divide each side by 2.
However, by equating the expressions x + 5 and x + 11, you obtain
x + 5 = x + 11
REMEMBER
Always check your
solutions in the original
equation to make sure
they are not extraneous.
x=x+6
0=6
Write related linear equation.
Subtract 5 from each side.
Subtract x from each side.
which is a false statement. So, the original equation has only one solution.
The solution is x = −8.
Monitoring Progress
Help in English and Spanish at BigIdeasMath.com
Solve the equation. Check your solutions.
10. ∣ x + 6 ∣ = 2x
11.
∣ 3x − 2 ∣ = x
12. ∣ 2 + x ∣ = ∣ x − 8 ∣
13.
∣ 5x − 2 ∣ = ∣ 5x + 4 ∣
Section 1.4
Solving Absolute Value Equations
31
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1.4
Exercises
Dynamic Solutions available at BigIdeasMath.com
Vocabulary and Core Concept Check
1. VOCABULARY What is an extraneous solution?
2. WRITING Without calculating, how do you know that the equation ∣ 4x − 7 ∣ = −1 has no solution?
Monitoring Progress and Modeling with Mathematics
In Exercises 3−10, simplify the expression.
3.
∣ −9 ∣
4. −∣ 15 ∣
5.
∣ 14 ∣ − ∣ −14 ∣
6.
∣ −3 ∣ + ∣ 3 ∣
⋅
8.
∣ −0.8 ⋅ 10 ∣
7. −∣ −5 (−7) ∣
9.
∣ −327 ∣
—
10.
26. WRITING EQUATIONS The shoulder heights of the
shortest and tallest miniature poodles are shown.
∣ −−124 ∣
—
15 in.
10 in.
In Exercises 11−24, solve the equation. Graph the
solution(s), if possible. (See Examples 1 and 2.)
11.
∣w∣ = 6
12.
∣ r ∣ = −2
13.
∣ y ∣ = −18
14.
∣ x ∣ = 13
15.
∣m + 3∣ = 7
16.
∣ q − 8 ∣ = 14
17.
∣ −3d ∣ = 15
18.
∣ ∣
19.
∣ 4b − 5 ∣ = 19
20.
t
— =6
2
∣x − 1∣ + 5 = 2
a. Represent these two heights on a number line.
b. Write an absolute value equation that represents
these heights.
USING STRUCTURE In Exercises 27−30, match the
absolute value equation with its graph without solving
the equation.
27.
∣x + 2∣ = 4
28.
∣x − 4∣ = 2
29.
∣x − 2∣ = 4
30.
∣x + 4∣ = 2
21. −4∣ 8 − 5n ∣ = 13
∣
2
3
∣
A.
22. −3 1 − — v = −9
∣ 14
−10
∣
−4
−8
−6
−4
−2
4
25. WRITING EQUATIONS The minimum distance from
0
2
0
2
4
6
8
8
10
4
C.
−4
Earth to the Sun is 91.4 million miles. The maximum
distance is 94.5 million miles. (See Example 3.)
a. Represent these two distances on a number line.
b. Write an absolute value equation that represents
the minimum and maximum distances.
−2
2
B.
24. 9∣ 4p + 2 ∣ + 8 = 35
Chapter 1
−6
2
23. 3 = −2 — s − 5 + 3
32
−8
−2
0
2
4
4
4
D.
−2
0
2
4
2
6
2
Solving Linear Equations
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In Exercises 31−34, write an absolute value equation
that has the given solutions.
31. x = 8 and x = 18
32. x = −6 and x = 10
33. x = 2 and x = 9
34. x = −10 and x = −5
48. MODELING WITH MATHEMATICS The recommended
weight of a soccer ball is 430 grams. The actual
weight is allowed to vary by up to 20 grams.
a. Write and solve an absolute
value equation to find the
minimum and maximum
acceptable soccer ball weights.
b. A soccer ball weighs 423 grams.
Due to wear and tear, the weight of
the ball decreases by 16 grams. Is the
weight acceptable? Explain.
In Exercises 35−44, solve the equation. Check your
solutions. (See Examples 4, 5, and 6.)
35.
∣ 4n − 15 ∣ = ∣ n ∣
36.
∣ 2c + 8 ∣ = ∣ 10c ∣
37.
∣ 2b − 9 ∣ = ∣ b − 6 ∣
38.
∣ 3k − 2 ∣ = 2∣ k + 2 ∣
39. 4∣ p − 3 ∣ = ∣ 2p + 8 ∣
40. 2∣ 4w − 1 ∣ = 3∣ 4w + 2 ∣
41.
∣ 3h + 1 ∣ = 7h
42.
∣ 6a − 5 ∣ = 4a
43.
∣f − 6∣ = ∣f + 8∣
44.
∣ 3x − 4 ∣ = ∣ 3x − 5 ∣
ERROR ANALYSIS In Exercises 49 and 50, describe and
correct the error in solving the equation.
✗
49.
45. MODELING WITH MATHEMATICS Starting from
2x − 1 = −(−9)
2x = 10
x=5
The solutions are x = −4 and x = 5.
✗
∣ 5x + 8 ∣ = x
5x + 8 = x
or
5x + 8 = −x
4x + 8 = 0
47. MODELING WITH MATHEMATICS You randomly
6x + 8 = 0
4x = −8
survey students about year-round school. The results
are shown in the graph.
6x = −8
4
x = −—
3
4
The solutions are x = −2 and x = −—.
3
x = −2
Year-Round School
68%
Oppose
51. ANALYZING EQUATIONS Without solving completely,
32%
Error: ±5%
Favor
0%
or
x = −4
50.
absolute value equation ∣ 3x + 8 ∣ − 9 = −5 has no
solution because the constant on the right side of the
equation is negative. Is your friend correct? Explain.
2x − 1 = −9
2x = −8
300 feet away, a car drives toward you. It then passes
by you at a speed of 48 feet per second. The distance
d (in feet) of the car from you after t seconds is given
by the equation d = ∣ 300 − 48t ∣. At what times is the
car 60 feet from you?
46. MAKING AN ARGUMENT Your friend says that the
∣ 2x − 1 ∣ = −9
20%
40%
60%
place each equation into one of the three categories.
80%
The error given in the graph means that the actual
percent could be 5% more or 5% less than the percent
reported by the survey.
a. Write and solve an absolute value equation to find
the least and greatest percents of students who
could be in favor of year-round school.
b. A classmate claims that —13 of the student body is
actually in favor of year-round school. Does this
conflict with the survey data? Explain.
Section 1.4
No
solution
One
solution
Two
solutions
∣x − 2∣ + 6 = 0
∣x + 3∣ − 1 = 0
∣x + 8∣ + 2 = 7
∣x − 1∣ + 4 = 4
∣ x − 6 ∣ − 5 = −9
∣ x + 5 ∣ − 8 = −8
Solving Absolute Value Equations
33
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52. USING STRUCTURE Fill in the equation
∣x −
60. HOW DO YOU SEE IT? The circle graph shows the
∣=
results of a survey of registered voters the day of
an election.
with a, b, c, or d so that the
equation is graphed correctly.
a
b
d
Which Party’s Candidate
Will Get Your Vote?
c
d
Other: 4%
Green: 2%
Libertarian:
5%
ABSTRACT REASONING In Exercises 53−56, complete
the statement with always, sometimes, or never. Explain
your reasoning.
53. If x 2 = a 2, then ∣ x ∣ is ________ equal to ∣ a ∣.
Democratic:
47%
Republican:
42%
54. If a and b are real numbers, then ∣ a − b ∣ is
_________ equal to ∣ b − a ∣.
Error: ±2%
55. For any real number p, the equation ∣ x − 4 ∣ = p will
The error given in the graph means that the actual
percent could be 2% more or 2% less than the
percent reported by the survey.
________ have two solutions.
56. For any real number p, the equation ∣ x − p ∣ = 4 will
________ have two solutions.
a. What are the minimum and maximum percents
of voters who could vote Republican? Green?
b. How can you use absolute value equations to
represent your answers in part (a)?
c. One candidate receives 44% of the vote. Which
party does the candidate belong to? Explain.
57. WRITING Explain why absolute value equations can
have no solution, one solution, or two solutions. Give
an example of each case.
58. THOUGHT PROVOKING Describe a real-life situation
that can be modeled by an absolute value equation
with the solutions x = 62 and x = 72.
61. ABSTRACT REASONING How many solutions does
the equation a∣ x + b ∣ + c = d have when a > 0
and c = d? when a < 0 and c > d ? Explain
your reasoning.
59. CRITICAL THINKING Solve the equation shown.
Explain how you found your solution(s).
8∣ x + 2 ∣ − 6 = 5∣ x + 2 ∣ + 3
Maintaining Mathematical Proficiency
Reviewing what you learned in previous grades and lessons
Identify the property of equality that makes Equation 1 and Equation 2 equivalent.
62.
Equation 1
3x + 8 = x − 1
Equation 2
3x + 9 = x
Use a geometric formula to solve the problem.
63.
Equation 1
4y = 28
Equation 2
y=7
(Section 1.1)
(Skills Review Handbook)
64. A square has an area of 81 square meters. Find the side length.
65. A circle has an area of 36π square inches. Find the radius.
66. A triangle has a height of 8 feet and an area of 48 square feet. Find the base.
67. A rectangle has a width of 4 centimeters and a perimeter of 26 centimeters. Find the length.
34
Chapter 1
Solving Linear Equations
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1.5
Rewriting Equations and Formulas
Essential Question
How can you use a formula for one
measurement to write a formula for a different measurement?
Using an Area Formula
Work with a partner.
a. Write a formula for the area A of
a parallelogram.
REASONING
QUANTITATIVELY
To be proficient in math,
you need to consider the
given units. For instance,
in Exploration 1, the
area A is given in square
inches and the height h
is given in inches. A unit
analysis shows that the
units for the base b are
also inches, which
makes sense.
A = 30 in.2
h = 5 in.
b. Substitute the given values into the
formula. Then solve the equation
for b. Justify each step.
b
c. Solve the formula in part (a) for b without first substituting values into the formula.
Justify each step.
d. Compare how you solved the equations in parts (b) and (c). How are the processes
similar? How are they different?
Using Area, Circumference, and
Volume Formulas
Work with a partner. Write the indicated formula for each figure. Then write a new
formula by solving for the variable whose value is not given. Use the new formula to
find the value of the variable.
a. Area A of a trapezoid
b. Circumference C of a circle
b1 = 8 cm
h
C = 24π ft
r
A = 63 cm2
b2 = 10 cm
c. Volume V of a rectangular prism
d. Volume V of a cone
V = 75 yd3
V = 24π m3
h
B = 12π m2
h
B = 15 yd2
Communicate Your Answer
3. How can you use a formula for one measurement to write a formula for a
different measurement? Give an example that is different from those given
in Explorations 1 and 2.
Section 1.5
Rewriting Equations and Formulas
35
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1.5 Lesson
What You Will Learn
Rewrite literal equations.
Core Vocabul
Vocabulary
larry
Rewrite and use formulas for area.
Rewrite and use other common formulas.
literal equation, p. 36
formula, p. 37
Rewriting Literal Equations
Previous
surface area
An equation that has two or more variables is called a literal equation. To rewrite a
literal equation, solve for one variable in terms of the other variable(s).
Rewriting a Literal Equation
Solve the literal equation 3y + 4x = 9 for y.
SOLUTION
3y + 4x = 9
Write the equation.
3y + 4x − 4x = 9 − 4x
Subtract 4x from each side.
3y = 9 − 4x
3y
3
Simplify.
9 − 4x
3
—=—
Divide each side by 3.
4
y = 3 − —x
3
Simplify.
4
The rewritten literal equation is y = 3 − — x.
3
Rewriting a Literal Equation
Solve the literal equation y = 3x + 5xz for x.
SOLUTION
y = 3x + 5xz
Write the equation.
y = x(3 + 5z)
Distributive Property
x(3 + 5z)
y
3 + 5z
3 + 5z
y
—=x
3 + 5z
—=—
REMEMBER
Division by 0 is undefined.
Divide each side by 3 + 5z.
Simplify.
y
The rewritten literal equation is x = —.
3 + 5z
3
In Example 2, you must assume that z ≠ −—5 in order to divide by 3 + 5z. In general, if
you have to divide by a variable or variable expression when solving a literal equation,
you should assume that the variable or variable expression does not equal 0.
Monitoring Progress
Help in English and Spanish at BigIdeasMath.com
Solve the literal equation for y.
1. 3y − x = 9
2. 2x − 2y = 5
3. 20 = 8x + 4y
Solve the literal equation for x.
4. y = 5x − 4x
36
Chapter 1
5. 2x + kx = m
6. 3 + 5x − kx = y
Solving Linear Equations
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Rewriting and Using Formulas for Area
A formula shows how one variable is related to one or more other variables.
A formula is a type of literal equation.
Rewriting a Formula for Surface Area
The formula for the surface area S of a rectangular prism is S = 2ℓw + 2ℓh + 2wh.
Solve the formula for the lengthℓ.
SOLUTION
S = 2ℓw + 2ℓh + 2wh
h
w
Write the equation.
S − 2wh = 2ℓw + 2ℓh + 2wh − 2wh
Subtract 2wh from each side.
S − 2wh = 2ℓw + 2ℓh
Simplify.
S − 2wh =ℓ(2w + 2h)
Distributive Property
ℓ(2w + 2h)
2w + 2h
S − 2wh
2w + 2h
— = ——
Divide each side by 2w + 2h.
S − 2wh
2w + 2h
Simplify.
— =ℓ
S − 2wh
When you solve the formula forℓ, you obtainℓ= —.
2w + 2h
Using a Formula for Area
You own a rectangular lot that is 500 feet deep. It has an area of 100,000 square feet.
To pay for a new water system, you are assessed $5.50 per foot of lot frontage.
a. Find the frontage of your lot.
b. How much are you assessed for the new water system?
SOLUTION
a. In the formula for the area of a rectangle, let the width w represent the lot frontage.
w
frontage
500 ft
A =ℓw
Write the formula for area of a rectangle.
A
ℓ
Divide each side byℓ to solve for w.
—=w
100,000
500
—=w
Substitute 100,000 for A and 500 forℓ.
200 = w
Simplify.
The frontage of your lot is 200 feet.
⋅
$5.50
b. Each foot of frontage costs $5.50, and — 200 ft = $1100.
1 ft
So, your total assessment is $1100.
Monitoring Progress
Help in English and Spanish at BigIdeasMath.com
Solve the formula for the indicated variable.
1
7. Area of a triangle: A = —2 bh; Solve for h.
8. Surface area of a cone: S = πr 2 + π rℓ; Solve for ℓ.
Section 1.5
Rewriting Equations and Formulas
37
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Rewriting and Using Other Common Formulas
Core Concept
Common Formulas
F = degrees Fahrenheit, C = degrees Celsius
Temperature
5
C = — (F − 32)
9
I = interest, P = principal,
r = annual interest rate (decimal form),
t = time (years)
Simple Interest
I = Prt
d = distance traveled, r = rate, t = time
d = rt
Distance
Rewriting the Formula for Temperature
Solve the temperature formula for F.
SOLUTION
5
C = —(F − 32)
9
9
5
— C = F − 32
Write the temperature formula.
9
Multiply each side by —.
5
— C + 32 = F − 32 + 32
9
5
Add 32 to each side.
9
5
Simplify.
— C + 32 = F
The rewritten formula is F = —95 C + 32.
Using the Formula for Temperature
Which has the greater surface temperature: Mercury or Venus?
SOLUTION
Convert the Celsius temperature of Mercury to degrees Fahrenheit.
9
F = — C + 32
5
Mercury
427°C
Venus
864°F
Write the rewritten formula from Example 5.
9
= —(427) + 32
5
Substitute 427 for C.
= 800.6
Simplify.
Because 864°F is greater than 800.6°F, Venus has the greater surface temperature.
Monitoring Progress
Help in English and Spanish at BigIdeasMath.com
9. A fever is generally considered to be a body temperature greater than 100°F. Your
friend has a temperature of 37°C. Does your friend have a fever?
38
Chapter 1
Solving Linear Equations
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Using the Formula for Simple Interest
You deposit $5000 in an account that earns simple interest. After 6 months, the
account earns $162.50 in interest. What is the annual interest rate?
COMMON ERROR
The unit of t is years. Be
sure to convert months
to years.
1 yr
12 mo
—
⋅
SOLUTION
To find the annual interest rate, solve the simple interest formula for r.
I = Prt
I
Pt
—=r
6 mo = 0.5 yr
Write the simple interest formula.
Divide each side by Pt to solve for r.
162.50
(5000)(0.5)
—=r
Substitute 162.50 for I, 5000 for P, and 0.5 for t.
0.065 = r
Simplify.
The annual interest rate is 0.065, or 6.5%.
Solving a Real-Life Problem
A truck driver averages 60 miles per hour while delivering freight to a customer. On
the return trip, the driver averages 50 miles per hour due to construction. The total
driving time is 6.6 hours. How long does each trip take?
SOLUTION
Step 1
Rewrite the Distance Formula to write expressions that represent the two trip
d
d
times. Solving the formula d = rt for t, you obtain t = —. So, — represents
r
60
d
the delivery time, and — represents the return trip time.
50
Step 2
Use these expressions and the total driving time to write and solve an
equation to find the distance one way.
d
60
d
50
— + — = 6.6
The sum of the two trip times is 6.6 hours.
— = 6.6
11d
300
Add the left side using the LCD.
11d = 1980
Multiply each side by 300 and simplify.
d = 180
Divide each side by 11 and simplify.
The distance one way is 180 miles.
Step 3
Use the expressions from Step 1 to find the two trip times.
60 mi
So, the delivery takes 180 mi ÷ — = 3 hours, and the return trip takes
1h
50 mi
180 mi ÷ — = 3.6 hours.
1h
Monitoring Progress
Help in English and Spanish at BigIdeasMath.com
10. How much money must you deposit in a simple interest account to earn $500 in
interest in 5 years at 4% annual interest?
11. A truck driver averages 60 miles per hour while delivering freight and 45 miles
per hour on the return trip. The total driving time is 7 hours. How long does each
trip take?
Section 1.5
Rewriting Equations and Formulas
39
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1.5
Exercises
Dynamic Solutions available at BigIdeasMath.com
Vocabulary and Core Concept Check
π
5
1. VOCABULARY Is 9r + 16 = — a literal equation? Explain.
2. DIFFERENT WORDS, SAME QUESTION Which is different? Find “both” answers.
Solve 3x + 6y = 24 for x.
Solve 24 − 3x = 6y for x.
Solve 6y = 24 − 3x for y in terms of x.
Solve 24 − 6y = 3x for x in terms of y.
Monitoring Progress and Modeling with Mathematics
In Exercises 3–12, solve the literal equation for y.
(See Example 1.)
3. y − 3x = 13
4. 2x + y = 7
5. 2y − 18x = −26
6. 20x + 5y = 15
7. 9x − y = 45
8. 6 − 3y = −6
9. 4x − 5 = 7 + 4y
1
11. 2 + —6 y = 3x + 4
24. MODELING WITH MATHEMATICS The penny
size of a nail indicates the length of the nail.
The penny size d is given by the literal
equation d = 4n − 2, where n is the
length (in inches) of the nail.
n
a. Solve the equation for n.
b. Use the equation from part (a) to find
the lengths of nails with the following
penny sizes: 3, 6, and 10.
10. 16x + 9 = 9y − 2x
1
12. 11 − —2 y = 3 + 6x
ERROR ANALYSIS In Exercises 25 and 26, describe and
In Exercises 13–22, solve the literal equation for x.
(See Example 2.)
13. y = 4x + 8x
14. m = 10x − x
15. a = 2x + 6xz
16. y = 3bx − 7x
17. y = 4x + rx + 6
18. z = 8 + 6x − px
19. sx + tx = r
20. a = bx + cx + d
21. 12 − 5x − 4kx = y
22. x − 9 + 2wx = y
correct the error in solving the equation for x.
25.
✗
−2x = −2(y − x) − 12
x = (y − x) + 6
26.
✗
10 = ax − 3b
10 = x(a − 3b)
10
a − 3b
—=x
23. MODELING WITH MATHEMATICS The total cost
C (in dollars) to participate in a ski club is given by
the literal equation C = 85x + 60, where x is the
number of ski trips you take.
12 − 2x = −2(y − x)
In Exercises 27–30, solve the formula for the indicated
variable. (See Examples 3 and 5.)
a. Solve the equation for x.
27. Profit: P = R − C; Solve for C.
b. How many ski trips do
you take if you spend
a total of $315? $485?
28. Surface area of a cylinder: S = 2π r 2 + 2π rh;
Solve for h.
1
29. Area of a trapezoid: A = —2 h(b1 + b2); Solve for b2.
v −v
t
1
0;
30. Average acceleration of an object: a = —
40
Chapter 1
Solving Linear Equations
Solve for v1.
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31. REWRITING A FORMULA A common statistic used in
35. PROBLEM SOLVING You deposit $2000 in an account
professional football is the quarterback rating. This
rating is made up of four major factors. One factor is
the completion rating given by the formula
that earns simple interest at an annual rate of 4%. How
long must you leave the money in the account to earn
$500 in interest? (See Example 7.)
(
C
R = 5 — − 0.3
A
)
36. PROBLEM SOLVING A flight averages 460 miles per
hour. The return flight averages 500 miles per hour
due to a tailwind. The total flying time is 4.8 hours.
How long is each flight? Explain. (See Example 8.)
where C is the number of completed passes and A is
the number of attempted passes. Solve the formula
for C.
32. REWRITING A FORMULA Newton’s law of gravitation
is given by the formula
m1m2
F=G —
d2
( )
37. USING STRUCTURE An athletic facility is building an
where F is the force between two objects of masses
m1 and m2, G is the gravitational constant, and d is
the distance between the two objects. Solve the
formula for m1.
indoor track. The track is composed of a rectangle and
two semicircles, as shown.
x
33. MODELING WITH MATHEMATICS The sale price
S (in dollars) of an item is given by the formula
S = L − rL, where L is the list price (in dollars)
and r is the discount rate (in decimal form).
(See Examples 4 and 6.)
r
r
a. Solve the formula for r.
a. Write a formula for the perimeter of the
indoor track.
b. The list price of the shirt
is $30. What is the
discount rate?
b. Solve the formula for x.
Sale price:
$18
34. MODELING WITH MATHEMATICS The density d of a
c. The perimeter of the track is 660 feet, and r is
50 feet. Find x. Round your answer to the
nearest foot.
38. MODELING WITH MATHEMATICS The distance
m
substance is given by the formula d = —, where m is
V
its mass and V is its volume.
d (in miles) you travel in a car is given by the two
equations shown, where t is the time (in hours) and
g is the number of gallons of gasoline the car uses.
Pyrite
Density: 5.01g/cm3
Volume: 1.2 cm3
d = 55t
d = 20g
a. Write an equation that relates g and t.
b. Solve the equation for g.
c. You travel for 6 hours. How many gallons of
gasoline does the car use? How far do you travel?
Explain.
a. Solve the formula for m.
b. Find the mass of the pyrite sample.
Section 1.5
Rewriting Equations and Formulas
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39. MODELING WITH MATHEMATICS One type of stone
41. MAKING AN ARGUMENT Your friend claims that
formation found in Carlsbad Caverns in New Mexico
is called a column. This cylindrical stone formation
connects to the ceiling and the floor of a cave.
Thermometer A displays a greater temperature than
Thermometer B. Is your friend correct? Explain
yyour reasoning.
g
100
90
80
70
60
50
40
30
20
10
0
−10
column
°F
Thermometer A
Thermometer B
stalagmite
42. THOUGHT PROVOKING Give a possible value for h.
a. Rewrite the formula for the circumference of a
circle, so that you can easily calculate the radius
of a column given its circumference.
Justify your answer. Draw and label the figure using
your chosen value of h.
b. What is the radius (to the nearest tenth of a foot)
of a column that has a circumference of 7 feet?
8 feet? 9 feet?
A = 40 cm2
h
c. Explain how you can find the area of a
cross section of a column when you know its
circumference.
8 cm
40. HOW DO YOU SEE IT? The rectangular prism shown
MATHEMATICAL CONNECTIONS In Exercises 43 and 44,
has bases with equal side lengths.
write a formula for the area of the regular polygon.
Solve the formula for the height h.
43.
44.
center
center
b
h
b
h
b
b
a. Use the figure to write a formula for the surface
area S of the rectangular prism.
REASONING In Exercises 45 and 46, solve the literal
equation for a.
b. Your teacher asks you to rewrite the formula
by solving for one of the side lengths, b orℓ.
Which side length would you choose? Explain
your reasoning.
a+b+c
ab
45. x = —
( a ab− b )
46. y = x —
Maintaining Mathematical Proficiency
Evaluate the expression.
(Skills Review Handbook)
⋅
47. 15 − 5 + 52
48. 18 2 − 42 ÷ 8
⋅
49. 33 + 12 ÷ 3 5
Solve the equation. Graph the solutions, if possible.
51.
42
∣x − 3∣ + 4 = 9
Chapter 1
Reviewing what you learned in previous grades and lessons
52.
∣ 3y − 12 ∣ − 7 = 2
50. 25(5 − 6) + 9 ÷ 3
(Section 1.4)
53. 2∣ 2r + 4 ∣ = −16
54. −4∣ s + 9 ∣ = −24
Solving Linear Equations
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1.4–1.5
What Did You Learn?
Core Vocabulary
absolute value equation, p. 28
extraneous solution, p. 31
literal equation, p. 36
formula, p. 37
Core Concepts
Section 1.4
Properties of Absolute Value, p. 28
Solving Absolute Value Equations, p. 28
Solving Equations with Two Absolute Values, p. 30
Special Solutions of Absolute Value Equations, p. 31
Section 1.5
Rewriting Literal Equations, p. 36
Common Formulas, p. 38
Mathematical Practices
1.
How did you decide whether your friend’s argument in Exercise 46 on page 33 made sense?
2.
How did you use the structure of the equation in Exercise 59 on page 34 to rewrite the equation?
3.
What entry points did you use to answer Exercises 43 and 44 on page 42?
Performance Task:
Dead Reckoning
Have you ever wondered how sailors navigated the oceans
before the Global Positioning System (GPS)? One method
sailors used is called dead reckoning. How does dead
reckoning use mathematics to track locations? Could you
use this method today?
To explore the answers to these questions and
more, check out the Performance Task and
Real-Life STEM video at BigIdeasMath.com.
43
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1
Chapter Review
1.1
Dynamic Solutions available at BigIdeasMath.com
Solving Simple Equations (pp. 3–10)
a. Solve x − 5 = −9. Justify each step.
x − 5 = −9
Addition Property of Equality
+5
Write the equation.
+5
Add 5 to each side.
x = −4
Simplify.
The solution is x = −4.
b. Solve 4x = 12. Justify each step.
4x = 12
Write the equation.
4x
4
Divide each side by 4.
12
4
—=—
Division Property of Equality
x=3
Simplify.
The solution is x = 3.
Solve the equation. Justify each step. Check your solution.
1. z + 3 = − 6
3.2
3
.2
2
1.2
n
5
3. −— = −2
2. 2.6 = −0.2t
Solving Multi-Step Equations (pp. 11–18)
Solve −6x + 23 + 2x = 15.
−6x + 23 + 2x = 15
Write the equation.
−4x + 23 = 15
Combine like terms.
−4x = −8
Subtract 23 from each side.
x=2
Divide each side by −4.
The solution is x = 2.
Solve the equation. Check your solution.
4. 3y + 11 = −16
7. −4(2z + 6) − 12 = 4
5.
8.
6=1−b
3
—2 (x
6. n + 5n + 7 = 43
1
− 2) − 5 = 19
7
9. 6 = —5 w + —5 w − 4
Find the value of x. Then find the angle measures of the polygon.
10.
110°
5x°
2x°
Sum of angle measures: 180°
44
Chapter 1
(x − 30)°
11.
(x − 30)°
x°
x°
(x − 30)°
Sum of angle measures: 540°
Solving Linear Equations
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1.3
Solving Equations with Variables on Both Sides (pp. 19–24)
Solve 2( y − 4) = −4( y + 8).
2( y − 4) = −4( y + 8)
Write the equation.
2y − 8 = −4y − 32
Distributive Property
6y − 8 = −32
Add 4y to each side.
6y = −24
Add 8 to each side.
y = −4
Divide each side by 6.
The solution is y = −4.
Solve the equation.
12. 3n − 3 = 4n + 1
1.4
1
13. 5(1 + x) = 5x + 5
14. 3(n + 4) = —2 (6n + 4)
Solving Absolute Value Equations (pp. 27–34)
a. Solve ∣ x − 5 ∣ = 3.
x−5=
+5
3
x − 5 = −3
or
+5
+5
x=8
+5
x=2
Write related linear equations.
Add 5 to each side.
Simplify.
The solutions are x = 8 and x = 2.
Check
b. Solve ∣ 2x + 6 ∣ = 4x. Check your solutions.
2x + 6 = 4x
−2x
−2x
or
2x + 6 = −4x
−2x
−2x
Write related linear equations.
Subtract 2x from each side.
6 = 2x
6 = −6x
6 2x
—=—
2
2
6
−6x
—=—
−6
−6
Solve for x.
3=x
−1 = x
Simplify.
Simplify.
∣ 2x + 6 ∣ = 4x
?
∣ 2(3) + 6 ∣ =
4(3)
?
∣ 12 ∣ = 12
12 = 12
∣ 2x + 6 ∣ = 4x
?
∣ 2(−1) + 6 ∣ =
4(−1)
?
∣ 4 ∣ = −4
Check the apparent solutions to see if either is extraneous.
The solution is x = 3. Reject x = −1 because it is extraneous.
∕ −4
4=
Solve the equation. Check your solutions.
15.
∣ y + 3 ∣ = 17
16. −2∣ 5w − 7 ∣ + 9 = − 7
17. ∣ x − 2 ∣ = ∣ 4 + x ∣
18. The minimum sustained wind speed of a Category 1 hurricane is 74 miles per hour. The maximum
sustained wind speed is 95 miles per hour. Write an absolute value equation that represents the
minimum and maximum speeds.
Chapter 1
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1.5
Rewriting Equations and Formulas (pp. 35–42)
a. The slope-intercept form of a linear equation is y = mx + b. Solve the equation for m.
y = mx + b
Write the equation.
y − b = mx + b − b
Subtract b from each side.
y − b = mx
Simplify.
y − b mx
x
x
y−b
—=m
x
—=—
Divide each side by x.
Simplify.
y−b
When you solve the equation for m, you obtain m = —.
x
b. The formula for the surface area S of a cylinder is S = 2𝛑 r 2 + 2𝛑 rh. Solve the formula
for the height h.
2π r 2 + 2πrh
S=
− 2πr 2
− 2πr 2
Write the equation.
Subtract 2πr 2 from each side.
S − 2πr 2 = 2πrh
Simplify.
—=—
S − 2πr 2
2πr
Divide each side by 2πr.
S − 2πr 2
2πr
Simplify.
2πrh
2πr
—=h
S − 2π r 2
When you solve the formula for h, you obtain h = —.
2πr
Solve the literal equation for y.
19. 2x − 4y = 20
20. 8x − 3 = 5 + 4y
21. a = 9y + 3yx
1
22. The volume V of a pyramid is given by the formula V = —3 Bh, where B is the area of the
base and h is the height.
a. Solve the formula for h.
b. Find the height h of the pyramid.
V = 216 cm3
B = 36 cm2
9
23. The formula F = —5 (K − 273.15) + 32 converts a temperature from kelvin K to degrees
Fahrenheit F.
a. Solve the formula for K.
b. Convert 180°F to kelvin K. Round your answer to the nearest hundredth.
46
Chapter 1
Solving Linear Equations
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1
Chapter Test
Solve the equation. Justify each step. Check your solution.
2. —23 x + 5 = 3
1. x − 7 = 15
3. 11x + 1 = −1 + x
Solve the equation.
4. 2∣ x − 3 ∣ − 5 = 7
5.
∣ 2x − 19 ∣ = 4x + 1
6. −2 + 5x − 7 = 3x − 9 + 2x
7. 3(x + 4) − 1 = −7
8.
∣ 20 + 2x ∣ = ∣ 4x + 4 ∣
9. —13 (6x + 12) − 2(x − 7) = 19
Describe the values of c for which the equation has no solution. Explain your reasoning.
10. 3x − 5 = 3x − c
11.
∣x − 7∣ = c
12. A safety regulation states that the minimum height of a handrail is 30 inches. The
maximum height is 38 inches. Write an absolute value equation that represents the
minimum and maximum heights.
13. The perimeter P (in yards) of a soccer field is represented by the formula P = 2ℓ+ 2w,
whereℓ is the length (in yards) and w is the width (in yards).
a. Solve the formula for w.
b. Find the width of the field.
c. About what percent of the field
is inside the circle?
P = 330 yd
10 yd
= 100 yd
14. Your car needs new brakes. You call a dealership and a local
mechanic for prices.
Cost of parts
Labor cost per hour
Dealership
$24
$99
Local Mechanic
$45
$89
a. After how many hours are the total costs the same at both places? Justify your answer.
b. When do the repairs cost less at the dealership? at the local mechanic? Explain.
15. Consider the equation ∣ 4x + 20 ∣ = 6x. Without calculating, how do you know that x = −2 is an
extraneous solution?
16. Your friend was solving the equation shown and was confused by the result
“−8 = −8.” Explain what this result means.
4(y − 2) − 2y = 6y − 8 − 4y
4y − 8 − 2y = 6y − 8 − 4y
2y − 8 = 2y − 8
−8 = −8
Chapter 1
Chapter Test
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1
Cumulative Assessment
1. A mountain biking park has 48 trails, 37.5% of which are beginner trails. The rest are
divided evenly between intermediate and expert trails. How many of each kind of trail
are there?
A 12 beginner, 18 intermediate, 18 expert
○
B 18 beginner, 15 intermediate, 15 expert
○
C 18 beginner, 12 intermediate, 18 expert
○
D 30 beginner, 9 intermediate, 9 expert
○
2. Which of the equations are equivalent to cx − a = b?
cx − a + b = 2b
0 = cx − a + b
b
2cx − 2a = —
2
b
x−a=—
c
a+b
x=—
c
b + a = cx
3. Let N represent the number of solutions of the equation 3(x − a) = 3x − 6. Complete
each statement with the symbol <, >, or =.
a. When a = 3, N ____ 1.
b. When a = −3, N ____ 1.
c. When a = 2, N ____ 1.
d. When a = −2, N ____ 1.
e. When a = x, N ____ 1.
f. When a = −x, N ____ 1.
4. You are painting your dining room white and your living room blue. You spend
$132 on 5 cans of paint. The white paint costs $24 per can, and the blue paint costs
$28 per can.
a. Use the numbers and symbols to write an equation that represents how many cans
of each color you bought.
x
132
5
24
28
=
(
)
+
−
×
÷
b. How much would you have saved by switching the colors of the dining room and
living room? Explain.
48
Chapter 1
Solving Linear Equations
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5. Which of the equations are equivalent?
6x + 6 = −14
8x + 6 = −2x − 14
5x + 3 = −7
7x + 3 = 2x − 13
6. The perimeter of the triangle is 13 inches. What is the length of the shortest
side?
(x − 5) in.
A 2 in.
○
B 3 in.
○
x
2
in.
C 4 in.
○
6 in.
D 8 in.
○
7. You pay $45 per month for cable TV. Your friend buys a satellite TV receiver for $99 and
pays $36 per month for satellite TV. Your friend claims that the expenses for a year of
satellite TV are less than the expenses for a year of cable.
a. Write and solve an equation to determine when you and your friend will have paid the
same amount for TV services.
b. Is your friend correct? Explain.
8. Place each equation into one of the four categories.
No solution
One solution
Two solutions
Infinitely many solutions
∣ 8x + 3 ∣ = 0
−6 = 5x − 9
3x − 12 = 3(x − 4) + 1
−2x + 4 = 2x + 4
0 = ∣ x + 13 ∣ + 2
9 = 3∣ 2x − 11 ∣
−4(x + 4) = −4x − 16
12x − 2x = 10x − 8
7 − 2x = 3 − 2(x − 2)
9. A car travels 1000 feet in 12.5 seconds. Which of the expressions do not represent the
average speed of the car?
second
80 —
feet
feet
80 —
second
80 feet
second
—
—
second
80 feet
Chapter 1
Cumulative Assessment
49
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Integrated Mathematics I
Teaching Edition
Chapter 1
Solving Linear Equations
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1
Chapter 1 Pacing Guide
Chapter Opener/
Mathematical Practices
1 Day
Section 1
1 Day
Section 2
2 Days
Section 3
1 Day
Quiz
1 Day
Section 4
2 Days
Section 5
2 Days
Chapter Review/
Chapter Tests
2 Days
Total Chapter 1
12 Days
Year-to-Date
12 Days
1.1
1.2
1.3
1.4
1.5
Solving Linear Equations
Solving Simple Equations
Solving Multi-Step Equations
Solving Equations with Variables on Both Sides
Solving Absolute Value Equations
Rewriting Equations and Formulas
Density of Pyrite (p. 41)
SEE the Big Idea
Cheerleading Competition (p. 29)
Boat (p. 22)
Biking (p. 14)
Average Speed (p. 6)
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Laurie’s Notes
Dynamic Teaching Tools
Dynamic Assessment & Progress Monitoring Tool
Lesson Planning Tool
Chapter Summary
Welcome to a new school year, and for many students, a new school. There is always
great excitement, and students are anxious to start anew. As teachers, we need to
capitalize on the opportunity, establishing norms and routines for student discourse
and classroom climate.
• In this book, students are expected to work together on explorations, to make conjectures,
to construct viable arguments, and to critique the reasoning of others. Take time in this first
chapter to make explicit what classroom productive dialogue sounds like. Listen for students
explaining their thinking, not just their process.
• Chapter 1 presents the foundational skills related to solving linear equations and the
connected skills of solving absolute value equations and rewriting equations and formulas.
• Most students will have prior experience with the Properties of Equality and techniques
presented in the first three sections. It will sound familiar that whatever operation is
performed on one side of the equation, the same operation must be performed on the other
side of the equation to keep equality, or balance.
• The fourth section of the chapter applies the techniques of equation solving to the context of
absolute value equations. Understanding absolute value as a function concept and not simply
two vertical lines can be challenging for students.
• Solving literal equations in the last section requires students to see the structure of equations
and perform operations on variable terms (i.e., 4x) as they would perform operations on
constants (i.e., 4).
• Essential to success in this chapter is accuracy in computation. Feedback to students should
distinguish between an error in computation and a process error.
Interactive Whiteboard Lesson Library
Dynamic Classroom with Dynamic Investigations
Real-Life STEM Videos
Scaffolding in the Classroom
Graphic Organizers: Word Magnet
A Word Magnet can be used to organize
information associated with a vocabulary
word or term. Students write the word or
term inside the magnet. Students write
associated information on the blank
lines that “radiate” from the magnet.
Associated information can include, but is
not limited to: other vocabulary words or
terms, definitions, formulas, procedures,
examples, and visuals. This type of
organizer serves as a good summary tool
because any information related to a topic
can be included.
What Your Students Have Learned
Middle School
• Add, subtract, multiply, and divide rational numbers.
• Find the absolute values of numbers and use absolute value to compare numbers in
real-life situations.
• Use variables to represent quantities in real-life problems.
• Write simple equations to solve real-life problems.
• Solve linear equations using the Distributive Property and combining like terms.
What Your Students Will Learn
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Math I
• Solve multi-step linear equations with variables on one or both sides.
• Solve absolute value equations involving one or two absolute value expressions.
• Identify equations with no solution or infinitely many solutions.
• Identify extraneous solutions.
• Use unit analysis to model and solve real-life problems.
• Rewrite and use literal equations and common formulas.
Chapter 1
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Questioning in the Classroom
But why?
Ask questions that require critical thinking
so that follow-up questions may be asked.
Avoid questions having yes or no answers.
If this cannot be avoided, always follow
with why?
Laurie’s Notes
Maintaining Mathematical Proficiency
Adding and Subtracting Integers
• Remind students how to add integers with the same sign. They should add the absolute
values of the integers and then use the common sign.
• Remind students how to add integers with different signs. They should subtract the lesser
absolute value from the greater absolute value and use the sign of the integer with the
greater absolute value.
COMMON ERROR Students may ignore the signs and just add the integers.
Multiplying and Dividing Integers
• Remind students that the product or quotient of two integers with the same sign is always
positive.
• Remind students that the product or quotient of two integers with different signs is always
negative.
COMMON ERROR When both factors are negative, students may write the product as negative.
Mathematical Practices (continued on page 2)
• The Mathematical Practices page focuses attention on how mathematics is learned—process
versus content. Page 2 demonstrates that focusing on unit analysis leads to labeling answers
with correct units of measures, providing an entry point into solving problems.
• Use the Mathematical Practices page to help students develop mathematical habits of
mind —how mathematics can be explored and how mathematics is thought about.
If students need help...
If students got it...
Student Journal
• Maintaining Mathematical Proficiency
Game Closet at BigIdeasMath.com
Lesson Tutorials
Start the next Section
Int_Math1_PE_01OP.
Skills Review Handbook
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What Your Students
Have Learned
Maintaining Mathematical Proficiency
Adding and Subtracting Integers
Example 1
(Grade 7)
• Add integers with the same sign or with
different signs using absolute value.
Evaluate 4 + (−12).
∣ −12 ∣ > ∣ 4 ∣. So, subtract ∣ 4 ∣ from ∣ −12 ∣.
• Subtract integers by adding the
opposite.
4 + (−12) = −8
Use the sign of −12.
Example 2
• Multiply integers with the same sign or
with different signs.
Evaluate −7 − (−16).
−7 − (−16) = −7 + 16
• Divide integers with the same sign or
with different signs and recognize the
quotient as a rational number.
Add the opposite of −16.
=9
Add.
Add or subtract.
1. −5 + (−2)
2. 0 + (−13)
3. −6 + 14
4. 19 − (−13)
5. −1 − 6
6. −5 − (−7)
ANSWERS
7. 17 + 5
8. 8 + (−3)
9. 11 − 15
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
Multiplying and Dividing Integers
(Grade 7)
⋅
Example 3 Evaluate −3 (−5).
The integers have the same sign.
⋅
−3 (−5) = 15
The product is positive.
Example 4 Evaluate 15 ÷ (−3).
The integers have different signs.
15 ÷ (−3) = −5
The quotient is negative.
Multiply or divide.
⋅
⋅
10. −3 (8)
11. −7 (−9)
12. 4 (−7)
13. −24 ÷ (−6)
−16
14. —
2
15. 12 ÷ (−3)
17. 36 ÷ 6
18. −3(−4)
⋅
16. 6 8
19. ABSTRACT REASONING Summarize the rules for (a) adding integers, (b) subtracting integers,
(c) multiplying integers, and (d) dividing integers. Give an example of each.
Dynamic Solutions available at BigIdeasMath.com
1
−7
−13
8
32
−7
2
22
5
−4
−24
63
−28
4
−8
−4
48
6
12
a. If the signs are the same, add
the absolute values and attach the
sign. If the signs are different,
subtract the absolute values and
attach the sign of the number
with the greatest absolute value;
Sample answer: −6 + 2 = −4
19b–d. See Additional Answers.
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Vocabulary Review
Have students make Information Frames for the following words.
• Integer
• Opposite
• Absolute value
Chapter 1
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Mathematical
Practices
MONITORING PROGRESS
ANSWERS
1. about 3 million per year
2. 30 mi/gal
3. about 17 min
Mathematically proficient students carefully specify units of measure.
Specifying Units of Measure
Core Concept
Operations and Unit Analysis
Addition and Subtraction
When you add or subtract quantities, they must have the same units of measure.
The sum or difference will have the same unit of measure.
Example
Perimeter of rectangle
= (3 ft) + (5 ft) + (3 ft) + (5 ft)
3 ft
= 16 feet
5 ft
When you add feet,
you get feet.
Multiplication and Division
When you multiply or divide quantities, the product or quotient will have a
different unit of measure.
Example
Area of rectangle = (3 ft) × (5 ft)
= 15 square feet
When you multiply feet, you get
feet squared, or square feet.
Specifying Units of Measure
You work 8 hours and earn $72. What is your hourly wage?
SOLUTION
dollars per hour
dollars per hour
The units on each side of the
equation balance. Both are
specified in dollars per hour.
Hourly wage
= $72 ÷ 8 h
($ per h)
= $9 per hour
Your hourly wage is $9 per hour.
Monitoring Progress
Solve the problem and specify the units of measure.
1. The population of the United States was about 280 million in 2000 and about
310 million in 2010. What was the annual rate of change in population from
2000 to 2010?
2. You drive 240 miles and use 8 gallons of gasoline. What was your car’s gas mileage
(in miles per gallon)?
3. A bathtub is in the shape of a rectangular prism. Its dimensions are 5 feet by 3 feet by
18 inches. The bathtub is three-fourths full of water and drains at a rate of 1 cubic foot
per minute. About how long does it take for all the water to drain?
2
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Solving Linear Equations
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Int_Math1_PE_01
(continued from page T-1)
• Allow students time to read through the Core Concept and example.
Ask probing questions to assess students’ understanding of the units of measure being the same
on each side of the equation.
• Students could work with partners or in groups on Monitoring Progress. Allow private think time
before dialogue begins.
2
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1.1–1.3
What Did You Learn?
Dynamic Teaching Tools
Dynamic Assessment & Progress Monitoring Tool
Core Vocabulary
conjecture, p. 3
rule, p. 3
theorem, p. 3
equation, p. 4
linear equation in one variable, p. 4
Interactive Whiteboard Lesson Library
solution, p. 4
inverse operations, p. 4
equivalent equations, p. 4
identity, p. 21
Dynamic Classroom with Dynamic Investigations
ANSWERS
1. Sample answer: Let A represent the
area of a single rectangle. Because
each rectangle has the same area, and
the area of the square is half the area
of a rectangle, use the expression
4A + —12A to represent the total area.
2. Sample answer: A limitation of a
protractor is centering it exactly on
each vertex point.
3. the definition of “identity”
Core Concepts
Section 1.1
Addition Property of Equality, p. 4
Subtraction Property of Equality, p. 4
Substitution Property of Equality, p. 4
Multiplication Property of Equality, p. 5
Division Property of Equality, p. 5
Four-Step Approach to Problem Solving, p. 6
Common Problem-Solving Strategies, p. 7
Section 1.2
Solving Multi-Step Equations, p. 12
Unit Analysis, p. 15
Section 1.3
Solving Equations with Variables on Both Sides, p. 20
Special Solutions of Linear Equations, p. 21
Mathematical Practices
1.
How did you make sense of the relationships between the quantities in Exercise 46 on page 9?
2.
What is the limitation of the tool you used in Exercises 25–28 on page 16?
3.
What definition did you use in your reasoning in Exercises 35 and 36 on page 24?
Using the Features of Yourr Textbook
d Tests
ests
to Prepare for Quizzes and
• Read and understand the core vocabulary and
the contents of the Core Concept boxes.
• Review the Examples and the Monitoring
Progress questions. Use the tutorials at
BigIdeasMath.com for additional help.
• Review previously completed homework
assignments.
25
5
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ANSWERS
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
x = −2; Subtract 9 from each side.
z = 12.4; Add 3.8 to each side.
r = −5; Divide each side by −12.
p = 24; Multiply each side by —43 .
m=8
v=5
w = —51
a = −0.6
k = −9
x = 14
c=0
n = −9
q = —56
y = −2
no solution
infinitely many solutions
s
17. 2 = —; 10 sec
5
18. 12 + 2x = 15; 1—12 ft
19. a. 3 h; The cost at studio A is
10 + 8h and the cost at Studio
B is 16 + 6h. To find when the
costs are the same, set these two
expressions equal and solve for
the time.
b. The costs will never be the same;
Sample answer: The cost for
studio B changes to 16 + 8h, and
the new equation has no solution.
1.1–1.3
Quiz
Solve the equation. Justify each step. Check your solution. (Section 1.1)
1. x + 9 = 7
2. 8.6 = z − 3.8
3. 60 = −12r
4. —34 p = 18
Solve the equation. Check your solution. (Section 1.2)
5. 2m − 3 = 13
6. 5 = 10 − v
7. 5 = 7w + 8w + 2
8. −21a + 28a − 6 = −10.2
1
9. 2k − 3(2k − 3) = 45
10. 68 = —5 (20x + 50) + 2
Solve the equation. (Section 1.3)
11. 3c + 1 = c + 1
12. −8 − 5n = 64 + 3n
13. 2(8q − 5) = 4q
14. 9(y − 4) − 7y = 5(3y − 2)
15. 4(g + 8) = 7 + 4g
16. −4(−5h − 4) = 2(10h + 8)
17. To estimate how many miles you are from a thunderstorm, count the seconds between
when you see lightning and when you hear thunder. Then divide by 5. Write and solve an
equation to determine how many seconds you would count for a thunderstorm that is
2 miles away. (Section 1.1)
18. You want to hang three equally-sized travel posters on a wall so that the posters on the ends
are each 3 feet from the end of the wall. You want the spacing between posters to be equal.
Write and solve an equation to determine how much space you should leave between the
posters. (Section 1.2)
3 ft
2 ft
2 ft
2 ft
3 ft
15 ft
19. You want to paint a piece of pottery at an art studio. The total cost is the cost of the piece
plus an hourly studio fee. There are two studios to choose from. (Section 1.3)
a. After how many hours of painting are the total costs the same at both studios? Justify
your answer.
b. Studio B increases the hourly studio fee by $2. How does this affect your answer in part (a)? Explain.
26
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Laurie’s Notes
Dynamic Teaching Tools
Dynamic Assessment & Progress Monitoring Tool
Lesson Planning Tool
Overview of Section 1.5
Interactive Whiteboard Lesson Library
Dynamic Classroom with Dynamic Investigations
Introduction
• In the study of high school mathematics, students will work with equations involving more
than one variable. In Math I, this occurs when students start to represent linear equations in
two variables, x and y. This chapter presents equation-solving techniques, often referred to as
symbolic manipulation, that apply to equations involving more than one variable.
• This lesson uses formulas that should be familiar to students, in order to practice the skill of
rewriting equations. In addition, students will rewrite an equation in two variables, x and y,
to solve for y. Rewriting an equation in function form is an important skill that you want
students to be secure with now. Any weaknesses in rational number operations will surface,
so it is important to differentiate where students are having a problem in this lesson. Is the
challenge with using inverse operations correctly, or is it problems with rational numbers,
or both?
Common Misconceptions
• When an equation involves more than one variable, students can be uncertain how to
treat the variable(s) that is not being solved for. It is helpful to review vocabulary such as
coefficient, variable term, and constant term before you begin this lesson.
Formative Assessment Tips
• Whiteboarding: Whiteboards can be used to provide individual responses, or used with
small groups to encourage student collaboration and consensus on a problem or solution
method. Whiteboards can be used at the beginning of class for warm-ups or throughout
the lesson to elicit student responses. Unlike writing on scrap paper (individual response) or
chart paper (group response), responses can be erased and modified easily. As understanding
progresses, responses can reflect this growth.
• Use boards for more than quick responses. Sizable boards can be used to communicate
thinking, providing evidence of how a problem was solved. When students display their
whiteboards in the front of the room, classmates can critique their reasoning or method
of solution.
Pacing Suggestion
• The formal lesson is quite long. You might have students complete only Exploration 1 and
then move to Examples 3, 4, 7, and 8 in the formal lesson. That would leave Examples 1 and
2 (literal equations) and Examples 5 and 6 (temperature formulas) for the second day.
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Section 1.5
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What Your Students
Will Learn
• Rewrite literal equations using the
Properties of Equality.
• Rewrite formulas by performing
operations on variable terms.
Laurie’s Notes
Exploration
Motivate
Distribute whiteboards and ask students to answer the first three questions.
• “I drove 2.1 hours at 65 miles per hour. How far did I go?” 136.5 miles
• “It took me 32 minutes to walk 4 kilometers. How fast was I walking on average?”
1
—8 kilometer per minute or 7.5 kilometers per hour
• “At 8 miles per hour, how long will it take me to bike 6 miles?” —34 hour
“How are these three equations alike?” They all involve distance, rate, and time.
• Write the three equations relating distance, rate, and time.
d
d
d = rt, r = —, t = —
t
r
• Explain that today they will be rewriting familiar formulas and solving them for different
variables, just as the familiar formula d = rt can be solved for r or t.
Exploration 1
Draw a parallelogram without a horizontal side and ask, “How do you find the
area of a parallelogram?” A = bh
“Are the two adjacent sides the base and height?” No, not unless the
parallelogram is a rectangle.
• Circulate as students begin the exploration. Note student work in part (c).
Assessing Question: “How did you solve for b (base) when you knew the area
and height?” Divide each side by 5. “Can this same process be used when you do
not know the area or height? Explain.” Yes. Divide each side of the literal equation by h.
• Have partners share their answers to part (d). You might also ask which was easier, part (b)
or part (c). The steps are the same; however, students often view working with constants as
easier than working with variables.
Exploration 2
• Have students read through the directions. Check to see that students understand that they
are solving the equation for a variable and then making a substitution, just as they did in
part (c) of Exploration 1.
• Differentiate: You might assign one of the four problems to different groups or pairs of
students in the class. Work could be recorded on a whiteboard, enabling presentation of the
work to the whole class.
• Teaching Tip: The formula for the volume of a cone involves the fraction —13 . Suggest to
students that they first work with the fraction before they consider the variables involved.
1
V = — Bh
3
3V = Bh
Int_Math1_PE_01
Communicate Your Answer
• Students should refer to solving an equation for a particular variable.
• Students should be familiar with many formulas, and not always from mathematics!
Connecting to Next Step
• If you feel students are comfortable with solving a formula for a variable, assign Exercises 28
and 29 on page 40 and Exercise 34 on page 41.
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1.5
Rewriting Equations and Formulas
Dynamic Teaching Tools
Essential Question
Dynamic Assessment & Progress Monitoring Tool
How can you use a formula for one
measurement to write a formula for a different measurement?
Lesson Planning Tool
Interactive Whiteboard Lesson Library
Using an Area Formula
Dynamic Classroom with Dynamic Investigations
Work with a partner.
a. Write a formula for the area A of
a parallelogram.
REASONING
QUANTITATIVELY
To be proficient in math,
you need to consider the
given units. For instance,
in Exploration 1, the
area A is given in square
inches and the height h
is given in inches. A unit
analysis shows that the
units for the base b are
also inches, which
makes sense.
A = 30 in.2
ANSWERS
h = 5 in.
b. Substitute the given values into the
formula. Then solve the equation
for b. Justify each step.
b
c. Solve the formula in part (a) for b without first substituting values into the formula.
Justify each step.
d. Compare how you solved the equations in parts (b) and (c). How are the processes
similar? How are they different?
Technology for the Teacher
Using Area, Circumference, and
Volume Formulas
Computer
Work with a partner. Write the indicated formula for each figure. Then write a new
Calculator
formula by solving
for the variable whose value is not given. Use the new formula to
find the value of the variable.
Online Ancillaries
a. Area A of a trapezoid
b. Circumference C of a circle
Smartboard
b1 = 8 cm
C = 24π ft
Video
h
r
A = 63 cm2
b2 = 10 cm
c. Volume V of a rectangular prism
d. Volume V of a cone
V = 75 yd3
1. a. A = bh
b. 30 = b ⋅ 5; Write the equation.
6 = b; Divide each side by 5.
c. A = bh; Write the formula.
A
— = b; Divide each side by h.
h
d. applied the same steps to
solve; used variables instead of
constants
1
2A
2. a. A = — h (b1 + b2); h = —;
2
b1 + b2
h = 7 cm
C
b. C = 2πr; r = —; r = 12 ft
2π
V
c. V = Bh; h = —; h = 5 yd
B
1
3V
d. V = —Bh; h = —; h = 6 m
3
B
3. Solve the formula for a different
variable; Sample answer: (volume of
V
a cylinder) V = πr2h; h = —2
πr
V = 24π m3
h
B = 12π m2
h
B = 15 yd2
Communicate Your Answer
3. How can you use a formula for one measurement to write a formula for a
different measurement? Give an example that is different from those given
in Explorations 1 and 2.
Section 1.5
Rewriting Equations and Formulas
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Section 1.5
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English Language Learners
Vocabulary
Write the word literal on the board.
Explain that in everyday speech, the
literal meaning of a phrase is based
on the strict definition of each word.
Compare the literal and intended
meanings of the expression It’s
raining cats and dogs. Explain that, in
mathematics, a literal equation is an
equation with two or more variables.
When you solve for one variable, the
solution includes the other variables.
1.5 Lesson
What You Will Learn
Rewrite literal equations.
Core Vocabul
Vocabulary
larry
Rewrite and use formulas for area.
Rewrite and use other common formulas.
literal equation, p. 36
formula, p. 37
Rewriting Literal Equations
Previous
surface area
An equation that has two or more variables is called a literal equation. To rewrite a
literal equation, solve for one variable in terms of the other variable(s).
Rewriting a Literal Equation
Solve the literal equation 3y + 4x = 9 for y.
SOLUTION
3y + 4x = 9
Write the equation.
3y + 4x − 4x = 9 − 4x
Extra Example 1
Subtract 4x from each side.
3y = 9 − 4x
Simplify.
3y 9 − 4x
—=—
3
3
Divide each side by 3.
4
y = 3 − —x
3
Solve the literal equation 4x − 7y = 12
4x − 12
for y. y = —
7
Simplify.
4
The rewritten literal equation is y = 3 − — x.
3
Rewriting a Literal Equation
Extra Example 2
Solve the literal equation y = 3x + 5xz for x.
Solve the literal equation 3w + 4wp = a
a
for w. w = —
3 + 4p
SOLUTION
MONITORING PROGRESS
ANSWERS
2. y = x −
Write the equation.
y = x(3 + 5z)
Distributive Property
x(3 + 5z)
y
—=—
3 + 5z
3 + 5z
y
—=x
3 + 5z
1. y = 3 + —13x
5
—2
y = 3x + 5xz
REMEMBER
Division by 0 is undefined.
3. y = 5 − 2x
4. x = y
Divide each side by 3 + 5z.
Simplify.
y
The rewritten literal equation is x = —.
3 + 5z
3
In Example 2, you must assume that z ≠ −—5 in order to divide by 3 + 5z. In general, if
you have to divide by a variable or variable expression when solving a literal equation,
you should assume that the variable or variable expression does not equal 0.
m
2+k
y−3
6. x = —
5−k
5. x = —
Monitoring Progress
Help in English and Spanish at BigIdeasMath.com
Solve the literal equation for y.
1. 3y − x = 9
2. 2x − 2y = 5
3. 20 = 8x + 4y
Solve the literal equation for x.
4. y = 5x − 4x
36
Chapter 1
5. 2x + kx = m
6. 3 + 5x − kx = y
Solving Linear Equations
Int_Math1_PE_0105.indd 36
Laurie’s Notes Teacher Actions
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• Look For and Make Use of Structure: When solving literal equations, you want students to
verbalize the operations represented. For example, when solving 3y + 4x = 9, students should
understand there are two variable terms. To isolate the 3y-term, subtract 4x. In other words,
approach solving 3y + 4x = 9 for y in the same way as you would solve 3y + 4 = 9 for y.
• Write the definition of a literal equation.
“In Example 1, can 9 and 4x be combined? Explain.” No. They are not like terms.
• In Example 2, use color to highlight each x: y = 3x + 5xz. Using the Distributive Property is not
obvious to most students because they do not think of the property in a factoring context.
Turn and Talk: “Why can’t z = −—35?” The denominator would be 0 and, therefore, undefined.
36
Chapter 1
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Rewriting and Using Formulas for Area
A formula shows how one variable is related to one or more other variables.
A formula is a type of literal equation.
Differentiated Instruction
Inclusion
Rewriting a Formula for Surface Area
The formula for the surface area S of a rectangular prism is S = 2ℓw + 2ℓh + 2wh.
Solve the formula for the lengthℓ.
SOLUTION
h
w
S = 2ℓw + 2ℓh + 2wh
Write the equation.
S − 2wh = 2ℓw + 2ℓh + 2wh − 2wh
Subtract 2wh from each side.
S − 2wh = 2ℓw + 2ℓh
Simplify.
S − 2wh =ℓ(2w + 2h)
Distributive Property
S − 2wh ℓ(2w + 2h)
— = ——
2w + 2h
2w + 2h
Divide each side by 2w + 2h.
S − 2wh
— =ℓ
2w + 2h
Simplify.
Extra Example 3
The formula for the area of a circle is
A = π r2 . Solve the formula for r.
S − 2wh
When you solve the formula forℓ, you obtainℓ= —.
2w + 2h
—
√
A
r= —
π
Using a Formula for Area
You own a rectangular lot that is 500 feet deep. It has an area of 100,000 square feet.
To pay for a new water system, you are assessed $5.50 per foot of lot frontage.
a. Find the frontage of your lot.
b. How much are you assessed for the new water system?
SOLUTION
a. In the formula for the area of a rectangle, let the width w represent the lot frontage.
frontage
500 ft
w
A =ℓw
Write the formula for area of a rectangle.
A
—=w
ℓ
Divide each side byℓ to solve for w.
100,000
500
—=w
Substitute 100,000 for A and 500 forℓ.
200 = w
Some students may have difficulty
keeping track of the variable. Have
students circle the variable for which
they are solving. Remind them that
they have found the answer when that
variable is alone on one side of the
literal equation.
Extra Example 4
A patio is in the shape of a parallelogram.
Its base, which is up against the side
of the house, is 13 feet. The area of the
patio is 156 square feet. The height of
the parallelogram represents the distance
from the house to the edge of the patio.
The yard is 24 yards deep from the house
to the back fence.
a. Find the distance from the house to the
edge of the patio. 12 feet
Simplify.
b. How far is it from that edge of the
patio to the back fence? 60 feet,
or 20 yards
The frontage of your lot is 200 feet.
⋅
$5.50
b. Each foot of frontage costs $5.50, and — 200 ft = $1100.
1 ft
So, your total assessment is $1100.
Monitoring Progress
Help in English and Spanish at BigIdeasMath.com
2A
b
S − πr2
8. ℓ = —
πr
7. h = —
Solve the formula for the indicated variable.
7. Area of a triangle: A =
1
—2 bh;
Solve for h.
8. Surface area of a cone: S = πr 2 + π rℓ; Solve for ℓ.
Section 1.5
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Rewriting Equations and Formulas
Laurie’s Notes Teacher Actions
MONITORING PROGRESS
ANSWERS
37
9/23/14 7:41 AM
• Use color to highlight eachℓ: S = 2ℓw + 2ℓh + 2wh
• Look For and Make Use of Structure: Students must see the 2wh as a constant term that
can be subtracted from each side of the equation.
• Pose the problem in Example 4. Give time for partners to work the example on whiteboards. As
you circulate, observe solution methods. Determine whether there are unique and/or particularly
clear solutions that should be seen by the class. Either ask for those solutions to be shared, or
make it appear random by placing particular Popsicle sticks in the center of the can.
Section 1.5
37
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Rewriting and Using Other Common Formulas
Extra Example 5
Core Concept
Solve the distance formula d = rt for r.
d
r=—
t
Common Formulas
F = degrees Fahrenheit, C = degrees Celsius
Temperature
5
C = — (F − 32)
9
I = interest, P = principal,
r = annual interest rate (decimal form),
t = time (years)
Extra Example 6
Simple Interest
On a July day, the temperature at the
peak of Mt. Everest is −16°C. Is that
temperature warmer or colder than 0°F?
warmer
I = Prt
d = distance traveled, r = rate, t = time
d = rt
Distance
MONITORING PROGRESS
ANSWER
Rewriting the Formula for Temperature
9. no
Solve the temperature formula for F.
SOLUTION
5
C = —(F − 32)
9
9
5
— C = F − 32
Write the temperature formula.
9
Multiply each side by —.
5
— C + 32 = F − 32 + 32
9
5
Add 32 to each side.
9
5
Simplify.
— C + 32 = F
The rewritten formula is F = —95 C + 32.
Using the Formula for Temperature
Which has the greater surface temperature: Mercury or Venus?
SOLUTION
Convert the Celsius temperature of Mercury to degrees Fahrenheit.
9
F = — C + 32
5
Mercury
427°C
Venus
864°F
Write the rewritten formula from Example 5.
9
= —(427) + 32
5
Substitute 427 for C.
= 800.6
Simplify.
Because 864°F is greater than 800.6°F, Venus has the greater surface temperature.
Monitoring Progress
Help in English and Spanish at BigIdeasMath.com
9. A fever is generally considered to be a body temperature greater than 100°F. Your
friend has a temperature of 37°C. Does your friend have a fever?
38
Chapter 1
Solving Linear Equations
Laurie’s Notes Teacher Actions
Int_Math1_PE_0105.indd 38
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• Attend to Precision: In the simple interest formula, students should note that both rate and
time are in terms of years.
• Students may know the formula for converting Celsius to Fahrenheit. If so, that should be the
common formula stated and change Example 5 to rewriting the formula to solve for Celsius.
• Extension: Students have likely heard about the Kelvin temperature scale. Have students
research a formula for converting Kelvin to Fahrenheit or Celsius. Rewrite the formula.
38
Chapter 1
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Using the Formula for Simple Interest
Extra Example 7
You deposit $3500 in an account that
earns 4% simple interest. How long
will you have to leave the money in the
account to earn $350 interest? 2.5 years
You deposit $5000 in an account that earns simple interest. After 6 months, the
account earns $162.50 in interest. What is the annual interest rate?
COMMON ERROR
The unit of t is years. Be
sure to convert months
to years.
1 yr
12 mo
—
⋅ 6 mo = 0.5 yr
SOLUTION
To find the annual interest rate, solve the simple interest formula for r.
I = Prt
I
—=r
Pt
Write the simple interest formula.
Extra Example 8
Divide each side by Pt to solve for r.
162.50
(5000)(0.5)
—=r
Dan drove 330 miles at 60 miles per hour.
Ryan drove 275 miles at 55 miles per hour.
How much more time did Dan drive than
Ryan? Dan drove 0.5 hour longer than
Ryan.
Substitute 162.50 for I, 5000 for P, and 0.5 for t.
0.065 = r
Simplify.
The annual interest rate is 0.065, or 6.5%.
Solving a Real-Life Problem
MONITORING PROGRESS
ANSWERS
A truck driver averages 60 miles per hour while delivering freight to a customer. On
the return trip, the driver averages 50 miles per hour due to construction. The total
driving time is 6.6 hours. How long does each trip take?
10. $2500
11. delivery: 3h, return: 4h
SOLUTION
Step 1
Rewrite the Distance Formula to write expressions that represent the two trip
d
d
times. Solving the formula d = rt for t, you obtain t = —. So, — represents
r
60
d
the delivery time, and — represents the return trip time.
50
Step 2
Use these expressions and the total driving time to write and solve an
equation to find the distance one way.
d
60
d
50
— + — = 6.6
The sum of the two trip times is 6.6 hours.
— = 6.6
11d
300
Add the left side using the LCD.
11d = 1980
Multiply each side by 300 and simplify.
d = 180
Divide each side by 11 and simplify.
The distance one way is 180 miles.
Step 3
Use the expressions from Step 1 to find the two trip times.
60 mi
So, the delivery takes 180 mi ÷ — = 3 hours, and the return trip takes
1h
50 mi
180 mi ÷ — = 3.6 hours.
1h
Monitoring Progress
Help in English and Spanish at BigIdeasMath.com
10. How much money must you deposit in a simple interest account to earn $500 in
interest in 5 years at 4% annual interest?
11. A truck driver averages 60 miles per hour while delivering freight and 45 miles
per hour on the return trip. The total driving time is 7 hours. How long does each
trip take?
Section 1.5
Int_Math1_PE_0105.indd 39
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Rewriting Equations and Formulas
39
9/23/14 7:41 AM
Laurie’s Notes Teacher Actions
COMMON ERROR In solving the simple interest formula for r in
Example 7, students often divide by P and then divide by t. By doing so,
they end up with a compound fraction on the left side of the equation.
Turn and Talk: “Why can you divide by P and t in one step?”
Sample answer: P and t are multiplied together and can be
considered the coefficient of r.
• Whiteboarding: Pose the question in Example 8 and give
partners time to work the example. Circulate and ask advancing
questions to assist students in making progress.
• Construct Viable Arguments and Critique the Reasoning
of Others: Have a pair of students share their solution.
Closure
11 − 2x
• Exit Ticket: Solve 2x + 4y = 11 for y. y = —
4
Section 1.5
39
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1.5
Assignment Guide and
Homework Check
Exercises
Dynamic Solutions available at BigIdeasMath.com
ASSIGNMENT
Vocabulary and Core Concept Check
Basic: 1–2, 3–33 odd, 40–41, 47–54
1. VOCABULARY Is 9r + 16 = — a literal equation? Explain.
Average: 1–2, 4–22 even, 23–34,
39–41, 47–54
2. DIFFERENT WORDS, SAME QUESTION Which is different? Find “both” answers.
π
5
Advanced: 1–2, 10–22 even,
26–34 even, 35–37, 39–54
HOMEWORK CHECK
Average: 12, 22, 30, 32, 34
ANSWERS
Solve 24 − 6y = 3x for x in terms of y.
3. y − 3x = 13
4. 2x + y = 7
5. 2y − 18x = −26
6. 20x + 5y = 15
7. 9x − y = 45
1. no; It only has one variable.
2. Solve 6y = 24 − 3x for y in terms of
x; y = 4 − —12 x; x = 8 − 2y
3. y = 13 + 3x
4. y = 7 − 2x
5. y = −13 + 9x
6. y = 3 − 4x
7. y = 9x − 45
8. y = 4
9. y = x − 3
10. y = 2x + 1
11. y = 18x + 12
12. y = 16 − 12x
1
13. x = —
y
12
9. 4x − 5 = 7 + 4y
1
11. 2 + —6 y = 3x + 4
24. MODELING WITH MATHEMATICS The penny
size of a nail indicates the length of the nail.
The penny size d is given by the literal
equation d = 4n − 2, where n is the
length (in inches) of the nail.
10. 16x + 9 = 9y − 2x
1
12. 11 − —2 y = 3 + 6x
13. y = 4x + 8x
14. m = 10x − x
15. a = 2x + 6xz
16. y = 3bx − 7x
17. y = 4x + rx + 6
18. z = 8 + 6x − px
19. sx + tx = r
20. a = bx + cx + d
21. 12 − 5x − 4kx = y
22. x − 9 + 2wx = y
ERROR ANALYSIS In Exercises 25 and 26, describe and
correct the error in solving the equation for x.
25.
✗
C (in dollars) to participate in a ski club is given by
the literal equation C = 85x + 60, where x is the
number of ski trips you take.
12 − 2x = −2(y − x)
−2x = −2(y − x) − 12
x = (y − x) + 6
26.
✗
10 = ax − 3b
10 = x(a − 3b)
10
—=x
a − 3b
23. MODELING WITH MATHEMATICS The total cost
1
—9 m
n
a. Solve the equation for n.
b. Use the equation from part (a) to find
the lengths of nails with the following
penny sizes: 3, 6, and 10.
8. 6 − 3y = −6
In Exercises 13–22, solve the literal equation for x.
(See Example 2.)
In Exercises 27–30, solve the formula for the indicated
variable. (See Examples 3 and 5.)
a. Solve the equation for x.
27. Profit: P = R − C; Solve for C.
b. How many ski trips do
you take if you spend
a total of $315? $485?
28. Surface area of a cylinder: S = 2π r 2 + 2π rh;
Solve for h.
1
29. Area of a trapezoid: A = —2 h(b1 + b2); Solve for b2.
v −v
t
1
0;
30. Average acceleration of an object: a = —
40
Chapter 1
Solving Linear Equations
Solve for v1.
40
25.Int_Math1_PE_0105.indd
The equation
is not solved for x because
9/23/14 7:41 AM
Int_Math1_PE_010
there is still a term with x on both sides;
y+6
x = y − x + 6; 2x = y + 6; x = —
2
26. There is no x in 3b to factor out;
27.
28.
29.
30.
40
Solve 6y = 24 − 3x for y in terms of x.
In Exercises 3–12, solve the literal equation for y.
(See Example 1.)
Advanced: 12, 22, 35, 37, 40
a
15. x = —
2 + 6z
y
16. x = —
3b − 7
y−6
17. x = —
4+r
z−8
18. x = —
6−p
r
19. x = —
s+t
a−d
20. x = —
b+c
y − 12
21. x = —
−5 − 4k
y+9
22. x = —
1 + 2w
C − 60
23. a. x = —
85
b. 3 trips; 5 trips
d+2
24. a. n = —
4
1
b. 1— in.; 2 in.; 3 in.
4
Solve 24 − 3x = 6y for x.
Monitoring Progress and Modeling with Mathematics
Basic: 7, 17, 23, 27, 31
14. x =
Solve 3x + 6y = 24 for x.
10 = ax − 3b; 10 + 3b = ax;
10 + 3b
x=—
a
C=R−P
S − 2πr2
h=—
2π r
2A
b2 = — − b1
h
v1 = at + v0
Chapter 1
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31. REWRITING A FORMULA A common statistic used in
35. PROBLEM SOLVING You deposit $2000 in an account
professional football is the quarterback rating. This
rating is made up of four major factors. One factor is
the completion rating given by the formula
that earns simple interest at an annual rate of 4%. How
long must you leave the money in the account to earn
$500 in interest? (See Example 7.)
(
C
R = 5 — − 0.3
A
36. PROBLEM SOLVING A flight averages 460 miles per
)
hour. The return flight averages 500 miles per hour
due to a tailwind. The total flying time is 4.8 hours.
How long is each flight? Explain. (See Example 8.)
where C is the number of completed passes and A is
the number of attempted passes. Solve the formula
for C.
m1m2
F=G —
d2
( )
37. USING STRUCTURE An athletic facility is building an
indoor track. The track is composed of a rectangle and
two semicircles, as shown.
x
33. MODELING WITH MATHEMATICS The sale price
r
S (in dollars) of an item is given by the formula
S = L − rL, where L is the list price (in dollars)
and r is the discount rate (in decimal form).
(See Examples 4 and 6.)
r
a. Solve the formula for r.
a. Write a formula for the perimeter of the
indoor track.
b. Solve the formula for x.
Sale price:
$18
34. MODELING WITH MATHEMATICS The density d of a
c. The perimeter of the track is 660 feet, and r is
50 feet. Find x. Round your answer to the
nearest foot.
Dynamic Classroom with Dynamic Investigations
)
R
31. C = A — + 0.3
5
Fd2
32. m1 = —
m2G
L−S
33. a. r = —
L
b. 0.4
34. a. m = dV
b. 6.012 g
35. 6.25 yr
d
36. 2.5 h; 2.3 h; — represents the
460
d
original trip time, and —
500
represents the return trip time. Add
these expressions and solve for the
one-way distance. Substituting the
distance into each of the expressions
gives the time for each flight.
37. a. P = 2x + 2πr
b. x = —12 P − πr
38. MODELING WITH MATHEMATICS The distance
m
substance is given by the formula d = —, where m is
V
its mass and V is its volume.
c. 173 ft
38. a. 55t = 20g
11t
b. g = —
4
c. 16.5 gal; 330 mi; The amount of
gasoline used can be found using
the formula from part (b). Either
of the original formulas can be
used to find the distance.
d (in miles) you travel in a car is given by the two
equations shown, where t is the time (in hours) and
g is the number of gallons of gasoline the car uses.
Pyrite
Density: 5.01g/cm3
Interactive Whiteboard Lesson Library
(
is given by the formula
b. The list price of the shirt
is $30. What is the
discount rate?
Dynamic Assessment & Progress Monitoring Tool
ANSWERS
32. REWRITING A FORMULA Newton’s law of gravitation
where F is the force between two objects of masses
m1 and m2, G is the gravitational constant, and d is
the distance between the two objects. Solve the
formula for m1.
Dynamic Teaching Tools
Volume: 1.2 cm3
d = 55t
d = 20g
a. Write an equation that relates g and t.
b. Solve the equation for g.
c. You travel for 6 hours. How many gallons of
gasoline does the car use? How far do you travel?
Explain.
a. Solve the formula for m.
b. Find the mass of the pyrite sample.
Section 1.5
Rewriting Equations and Formulas
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Section 1.5
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39. MODELING WITH MATHEMATICS One type of stone
ANSWERS
39.
40.
41.
42.
41. MAKING AN ARGUMENT Your friend claims that
formation found in Carlsbad Caverns in New Mexico
is called a column. This cylindrical stone formation
connects to the ceiling and the floor of a cave.
C
a. r = —
2π
b. 1.1 ft; 1.3 ft; 1.4 ft
c. First find the radius using the
formula from part (a), then
substitute this into the formula for
the area of a circle.
a. S = 4ℓb + 2b2
b. Sample answer: ℓ; The formula
contains terms with both b and b2,
but only one term withℓ.
no; 70°F is about 21.1°C, which is
greater than 20°C.
Sample answer: h = 8; missing
base = 2; A = —12 h (b1 + b2),
40 = —12 8 (2 + 8)
Thermometer A displays a greater temperature than
Thermometer B. Is your friend correct? Explain
your
y
reasoning.
g
100
90
80
70
60
50
40
30
20
10
0
−10
column
°F
Thermometer A
Thermometer B
stalagmite
42. THOUGHT PROVOKING Give a possible value for h.
a. Rewrite the formula for the circumference of a
circle, so that you can easily calculate the radius
of a column given its circumference.
Justify your answer. Draw and label the figure using
your chosen value of h.
b. What is the radius (to the nearest tenth of a foot)
of a column that has a circumference of 7 feet?
8 feet? 9 feet?
⋅ ⋅
h
A = 40 cm2
c. Explain how you can find the area of a
cross section of a column when you know its
circumference.
2 cm
8 cm
40. HOW DO YOU SEE IT? The rectangular prism shown
MATHEMATICAL CONNECTIONS In Exercises 43 and 44,
has bases with equal side lengths.
8 cm
write a formula for the area of the regular polygon.
Solve the formula for the height h.
A = 40 cm
2
43.
44.
8 cm
center
center
2A
5
43. A = — bh; h = —
2
5b
A
44. A = 4bh; h = —
4b
b+c
45. a = —
bx − 1
yb
46. a = —
y − xb
47–54. See Additional Answers.
b
h
b
h
b
b
a. Use the figure to write a formula for the surface
area S of the rectangular prism.
REASONING In Exercises 45 and 46, solve the literal
equation for a.
b. Your teacher asks you to rewrite the formula
by solving for one of the side lengths, b orℓ.
Which side length would you choose? Explain
your reasoning.
a+b+c
ab
45. x = —
( a ab− b )
46. y = x —
Mini-Assessment
Maintaining Mathematical Proficiency
1. Solve 9 − c = 17d for c.
c = 9 − 17d
2. Solve nt + 6 − xt = s for t.
s−6
t=—
n−x
3. Solve the formula for the volume of
a rectangular prism V =ℓhw for w.
V
w=—
ℓh
4. A circular trampoline has a padded
rope that goes around the outside
of the frame. The rope is sold for
$2.75 per foot and costs $104.50
before tax for this trampoline. What
is the diameter of the trampoline, to
the nearest foot? 12 feet
5. Kayla drives 144 miles in 3 hours.
Kendra drives 224 miles in 4 hours.
How much faster does Kendra drive
on average? Kendra drives an
average of 8 miles per hour faster
than Kayla.
42
(Skills Review Handbook)
Evaluate the expression.
47. 15 − 5 + 52
⋅
48. 18 2 − 42 ÷ 8
⋅
49. 33 + 12 ÷ 3 5
Solve the equation. Graph the solutions, if possible.
51.
42
∣x − 3∣ + 4 = 9
Chapter 1
Reviewing what you learned in previous grades and lessons
52.
∣ 3y − 12 ∣ − 7 = 2
50. 25(5 − 6) + 9 ÷ 3
(Section 1.4)
53. 2∣ 2r + 4 ∣ = −16
54. −4∣ s + 9 ∣ = −24
Solving Linear Equations
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If students need help...
If students got it...
Resources by Chapter
• Practice A and Practice B
• Puzzle Time
Resources by Chapter
• Enrichment and Extension
• Cumulative Review
Student Journal
• Practice
Start the next Section
Differentiating the Lesson
Skills Review Handbook
Chapter 1
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1.4–1.5
What Did You Learn?
Dynamic Teaching Tools
Dynamic Assessment & Progress Monitoring Tool
Core Vocabulary
absolute value equation, p. 28
extraneous solution, p. 31
Interactive Whiteboard Lesson Library
literal equation, p. 36
formula, p. 37
Dynamic Classroom with Dynamic Investigations
ANSWERS
1. Sample answer: An absolute value
equation must be set equal to a
positive value to have a solution.
Isolate the absolute value on one side
of the equation to determine whether
it has a solution.
2. Sample answer: The absolute values
are the same, so treat them as a single
variable.
3. Sample answer: the center and two
adjacent vertices to form congruent
triangles
Core Concepts
Section 1.4
Properties of Absolute Value, p. 28
Solving Absolute Value Equations, p. 28
Solving Equations with Two Absolute Values, p. 30
Special Solutions of Absolute Value Equations, p. 31
Section 1.5
Rewriting Literal Equations, p. 36
Common Formulas, p. 38
Mathematical Practices
1.
How did you decide whether your friend’s argument in Exercise 46 on page 33 made sense?
2.
How did you use the structure of the equation in Exercise 59 on page 34 to rewrite the equation?
3.
What entry points did you use to answer Exercises 43 and 44 on page 42?
Performance Task:
Dead Reckoning
Have you ever wondered how sailors navigated the oceans
before the Global Positioning System (GPS)? One method
sailors used is called dead reckoning. How does dead
reckoning use mathematics to track locations? Could you
use this method today?
To explore the answers to these questions and
more, check out the Performance Task and
Real-Life STEM video at BigIdeasMath.com.
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1
ANSWERS
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
z = −9; Subtract 3 from each side.
t = −13; Divide each side by −0.2.
n = 10; Multiply each side by −5.
y = −9
b = −5
n=6
z = −5
x = 18
25
w=—
4
x = 10; 110°, 50°, 20°
x = 126; 126°, 96°, 126°, 96°, 96°
Chapter Review
1.1
Dynamic Solutions available at BigIdeasMath.com
Solving Simple Equations (pp. 3–10)
a. Solve x − 5 = −9. Justify each step.
x − 5 = −9
Addition Property of Equality
+5
Write the equation.
+5
Add 5 to each side.
x = −4
Simplify.
The solution is x = −4.
b. Solve 4x = 12. Justify each step.
Division Property of Equality
4x = 12
Write the equation.
4x 12
—=—
4
4
Divide each side by 4.
x=3
Simplify.
The solution is x = 3.
Solve the equation. Justify each step. Check your solution.
1. z + 3 = − 6
3.2
3
.2
2
1.2
n
5
3. −— = −2
2. 2.6 = −0.2t
Solving Multi-Step Equations (pp. 11–18)
Solve −6x + 23 + 2x = 15.
−6x + 23 + 2x = 15
Write the equation.
−4x + 23 = 15
Combine like terms.
−4x = −8
Subtract 23 from each side.
x=2
Divide each side by −4.
The solution is x = 2.
Solve the equation. Check your solution.
4. 3y + 11 = −16
5.
7. −4(2z + 6) − 12 = 4
8. —32 (x − 2) − 5 = 19
6=1−b
6. n + 5n + 7 = 43
1
7
9. 6 = —5 w + —5 w − 4
Find the value of x. Then find the angle measures of the polygon.
10.
11.
110°
5x°
2x°
Sum of angle measures: 180°
44
Chapter 1
(x − 30)°
(x − 30)°
x°
x°
(x − 30)°
Sum of angle measures: 540°
Solving Linear Equations
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ANSWERS
1.3
12.
13.
14.
15.
16.
17.
18.
Solving Equations with Variables on Both Sides (pp. 19–24)
Solve 2( y − 4) = −4( y + 8).
2( y − 4) = −4( y + 8)
Write the equation.
2y − 8 = −4y − 32
Distributive Property
6y − 8 = −32
Add 4y to each side.
6y = −24
n = −4
infinitely many solutions
no solution
y = 14, y = −20
1
w = 3, w = −—5
x = −1
∣ v − 84.5 ∣ = 10.5
Add 8 to each side.
y = −4
Divide each side by 6.
The solution is y = −4.
Solve the equation.
12. 3n − 3 = 4n + 1
1.4
1
13. 5(1 + x) = 5x + 5
14. 3(n + 4) = —2 (6n + 4)
Solving Absolute Value Equations (pp. 27–34)
a. Solve ∣ x − 5 ∣ = 3.
x−5=
+5
3
x − 5 = −3
or
+5
+5
x=8
+5
x=2
Write related linear equations.
Add 5 to each side.
Simplify.
The solutions are x = 8 and x = 2.
Check
b. Solve ∣ 2x + 6 ∣ = 4x. Check your solutions.
2x + 6 = 4x
−2x
−2x
or
2x + 6 = −4x
−2x
−2x
Write related linear equations.
Subtract 2x from each side.
6 = 2x
6 = −6x
—=—
2x
2
—=—
Solve for x.
3=x
−1 = x
Simplify.
6
2
6
−6
−6x
−6
Simplify.
∣ 2x + 6 ∣ = 4x
?
∣ 2(3) + 6 ∣ =
4(3)
∣ 12 ∣ =? 12
12 = 12
∣ 2x + 6 ∣ = 4x
?
∣ 2(−1) + 6 ∣ =
4(−1)
∣ 4 ∣ =? −4
Check the apparent solutions to see if either is extraneous.
The solution is x = 3. Reject x = −1 because it is extraneous.
∕ −4
4=
Solve the equation. Check your solutions.
15.
∣ y + 3 ∣ = 17
16. −2∣ 5w − 7 ∣ + 9 = − 7
17. ∣ x − 2 ∣ = ∣ 4 + x ∣
18. The minimum sustained wind speed of a Category 1 hurricane is 74 miles per hour. The maximum
sustained wind speed is 95 miles per hour. Write an absolute value equation that represents the
minimum and maximum speeds.
Chapter 1
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Chapter Review
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ANSWERS
19. y = —12 x − 5
20. y = 2x − 2
a
21. y = —
9 + 3x
3V
22. a. h = —
B
b. 18 cm
23. a. K = —59 (F − 32) + 273.15
b. about 355.37 K
1.5
Rewriting Equations and Formulas (pp. 35–42)
a. The slope-intercept form of a linear equation is y = mx + b. Solve the equation for m.
y = mx + b
Write the equation.
y − b = mx + b − b
Subtract b from each side.
y − b = mx
Simplify.
y − b mx
—=—
x
x
y−b
—=m
x
Divide each side by x.
Simplify.
y−b
When you solve the equation for m, you obtain m = —.
x
b. The formula for the surface area S of a cylinder is S = 2𝛑 r 2 + 2𝛑 rh. Solve the formula
for the height h.
2π r 2 + 2πrh
S=
−
2πr 2
S−
2πr 2
− 2πr 2
= 2πrh
S − 2πr 2 2πrh
—=—
2πr
2πr
2πr 2
S−
2πr
—=h
Write the equation.
Subtract 2πr 2 from each side.
Simplify.
Divide each side by 2πr.
Simplify.
S − 2π r 2
When you solve the formula for h, you obtain h = —.
2πr
Solve the literal equation for y.
19. 2x − 4y = 20
20. 8x − 3 = 5 + 4y
21. a = 9y + 3yx
1
22. The volume V of a pyramid is given by the formula V = —3 Bh, where B is the area of the
base and h is the height.
a. Solve the formula for h.
b. Find the height h of the pyramid.
V = 216 cm3
B = 36 cm2
9
23. The formula F = —5 (K − 273.15) + 32 converts a temperature from kelvin K to degrees
Fahrenheit F.
a. Solve the formula for K.
b. Convert 180°F to kelvin K. Round your answer to the nearest hundredth.
46
Chapter 1
Solving Linear Equations
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1
Chapter Test
ANSWERS
1. x = 22; Add 7 to each side.
2. x = −3; Subtract 5 from each side;
Multiply each side by —32 .
1
3. x = −—5 ; Subtract x and 1 from each
side; Divide each side by 10.
4. x = −3, x = 9
5. x = 3
6. infinitely many solutions
7. x = −6
8. x = −4, x = 8
9. no solution
10. c ≠ 5; If c is 5, then the equation is
an identity. For all other values of c,
subtracting 3x from each side will
give a statement that is always false.
11. c < 0; An absolute value cannot be
negative.
12. ∣ h − 34 ∣ = 4
P − 2ℓ
13. a. w = —
2
b. 65 yd
c. about 4.8%
14. a. 2.1 h; The cost at the dealership is
24 + 99t and the cost at the local
mechanic is 45 + 89t. Set these
two expressions equal and solve
for the time.
b. time is less than 2.1 h; time is
greater than 2.1 h; Because
the expressions are equal for
2.1 hours, that is the cutoff point
from the dealership being less
expensive to the local mechanic
being less expensive.
15. It will give a negative value on the
right and absolute value cannot be
negative.
16. It is an identity, meaning that all
real numbers are solutions of the
equation.
Solve the equation. Justify each step. Check your solution.
2. —23 x + 5 = 3
1. x − 7 = 15
3. 11x + 1 = −1 + x
Solve the equation.
4. 2∣ x − 3 ∣ − 5 = 7
5.
∣ 2x − 19 ∣ = 4x + 1
6. −2 + 5x − 7 = 3x − 9 + 2x
7. 3(x + 4) − 1 = −7
8.
∣ 20 + 2x ∣ = ∣ 4x + 4 ∣
9. —13 (6x + 12) − 2(x − 7) = 19
Describe the values of c for which the equation has no solution. Explain your reasoning.
10. 3x − 5 = 3x − c
11.
∣x − 7∣ = c
12. A safety regulation states that the minimum height of a handrail is 30 inches. The
maximum height is 38 inches. Write an absolute value equation that represents the
minimum and maximum heights.
13. The perimeter P (in yards) of a soccer field is represented by the formula P = 2ℓ+ 2w,
whereℓ is the length (in yards) and w is the width (in yards).
a. Solve the formula for w.
b. Find the width of the field.
c. About what percent of the field
is inside the circle?
P = 330 yd
10 yd
= 100 yd
14. Your car needs new brakes. You call a dealership and a local
mechanic for prices.
Cost of parts
Labor cost per hour
Dealership
$24
$99
Local Mechanic
$45
$89
a. After how many hours are the total costs the same at both places? Justify your answer.
b. When do the repairs cost less at the dealership? at the local mechanic? Explain.
15. Consider the equation ∣ 4x + 20 ∣ = 6x. Without calculating, how do you know that x = −2 is an
extraneous solution?
16. Your friend was solving the equation shown and was confused by the result
“−8 = −8.” Explain what this result means.
4(y − 2) − 2y = 6y − 8 − 4y
4y − 8 − 2y = 6y − 8 − 4y
2y − 8 = 2y − 8
−8 = −8
Chapter 1
Chapter Test
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If students need help...
If students got it...
Lesson Tutorials
Resources by Chapter
• Enrichment and Extension
• Cumulative Review
Skills Review Handbook
Performance Task
BigIdeasMath.com
Start the next Section
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1
ANSWERS
1. B
a+b
2. cx − a + b = 2b, x = —,
c
b + a = cx
3. a. <
b. <
c. >
d. <
e. =
f. =
4. a. 24x + 28(5 − x) = 132 or
24(5 − x) + 28x = 132
b. $4; Sample answer: Switching
gives a total cost of $128, which
is $4 less than $132.
Cumulative Assessment
1. A mountain biking park has 48 trails, 37.5% of which are beginner trails. The rest are
divided evenly between intermediate and expert trails. How many of each kind of trail
are there?
A 12 beginner, 18 intermediate, 18 expert
○
B 18 beginner, 15 intermediate, 15 expert
○
C 18 beginner, 12 intermediate, 18 expert
○
D 30 beginner, 9 intermediate, 9 expert
○
2. Which of the equations are equivalent to cx − a = b?
cx − a + b = 2b
0 = cx − a + b
b
2cx − 2a = —
2
b
x−a=—
c
a+b
x=—
c
b + a = cx
3. Let N represent the number of solutions of the equation 3(x − a) = 3x − 6. Complete
each statement with the symbol <, >, or =.
a. When a = 3, N ____ 1.
b. When a = −3, N ____ 1.
c. When a = 2, N ____ 1.
d. When a = −2, N ____ 1.
e. When a = x, N ____ 1.
f. When a = −x, N ____ 1.
4. You are painting your dining room white and your living room blue. You spend
$132 on 5 cans of paint. The white paint costs $24 per can, and the blue paint costs
$28 per can.
a. Use the numbers and symbols to write an equation that represents how many cans
of each color you bought.
x
132
5
24
28
=
(
)
+
−
×
÷
b. How much would you have saved by switching the colors of the dining room and
living room? Explain.
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Solving Linear Equations
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ANSWERS
5. 8x + 6 = −2x − 14 and 5x + 3 = −7
6. B
7. a. 45m = 99 + 36m; month 11
b. yes; Because the expenses are
equal for month 11, any time
after that satellite TV will be less
expensive.
8. no solution: 12x − 2x = 10x − 8,
0 = ∣ x + 13 ∣ + 2,
3x − 12 = 3(x − 4) + 1
one solution: ∣ 8x + 3 ∣ = 0,
−2x + 4 = 2x + 4, −6 = 5x − 9
two solutions: 9 = 3∣ 2x − 11 ∣
infinitely many solutions:
−4(x + 4) = −4x − 16,
7 − 2x = 3 − 2(x − 2)
sec sec
9. 80 —, —
ft 80 ft
5. Which of the equations are equivalent?
6x + 6 = −14
8x + 6 = −2x − 14
5x + 3 = −7
7x + 3 = 2x − 13
6. The perimeter of the triangle is 13 inches. What is the length of the shortest
side?
(x − 5) in.
A 2 in.
○
B 3 in.
○
x
2
C 4 in.
○
in.
6 in.
D 8 in.
○
7. You pay $45 per month for cable TV. Your friend buys a satellite TV receiver for $99 and
pays $36 per month for satellite TV. Your friend claims that the expenses for a year of
satellite TV are less than the expenses for a year of cable.
a. Write and solve an equation to determine when you and your friend will have paid the
same amount for TV services.
b. Is your friend correct? Explain.
8. Place each equation into one of the four categories.
No solution
One solution
Two solutions
Infinitely many solutions
∣ 8x + 3 ∣ = 0
−6 = 5x − 9
3x − 12 = 3(x − 4) + 1
−2x + 4 = 2x + 4
0 = ∣ x + 13 ∣ + 2
−4(x + 4) = −4x − 16
12x − 2x = 10x − 8
9 = 3∣ 2x − 11 ∣
7 − 2x = 3 − 2(x − 2)
9. A car travels 1000 feet in 12.5 seconds. Which of the expressions do not represent the
average speed of the car?
second
80 —
feet
feet
80 —
second
80 feet
second
—
—
second
80 feet
Chapter 1
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Cumulative Assessment
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Integrated
Mathematics I
Ancillaries
Assessment Book .....................................................A2
Resources by Chapter ..............................................A5
Student Journal .......................................................A8
Differentiating the Lesson ....................................A10
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Assessment Book
The Assessment Book contains formative and summative assessment options, providing teachers
with the ability to assess students on the same content in a variety of ways. It is available in print
and online in an editable format.
Name _________________________________________________________ Date _________
Skills
Test
Prerequisite Skills Test
Add or subtract.
u Prerequisite Skills Test with
Item Analysis
Answers
1. − 9 + ( −15)
2. 2 + ( − 3)
3. 6 − 9
1.
4. − 6 − 11
5. 13 + 8
6. −12 − ( −10)
2.
8. − 8 • 2
9. 9 ÷ 3
4.
12. −1( − 7)
5.
3.
Multiply or divide.
7. 2( − 7)
The Prerequisite Skills Test checks students’
understanding of previously learned
mathematical skills they will need to be
successful in their math class. You can use
the Item Analysis to diagnose topics your
students need to review to prepare
them for the school year.
10. 25 ÷ ( − 5)
11. − 30 ÷ ( − 6)
6.
Solve the problem and specify the units of measure.
13. The length of a rectangle is 6 feet and the width is 3 feet. Find the perimeter
of the rectangle.
8.
9.
14. One side of a square measures 9 centimeters. Find the area of the square.
10.
Graph the number.
15. 4
16.
−5 −4 −3 −2 −1
7.
0
1
2
3
4
−3
11.
−5 −4 −3 −2 −1
5
0
1
2
3
4
5
12.
13.
17. − 6 + 5
18. 1 − − 3
−5 −4 −3 −2 −1
0
1
2
3
4
14.
−5 −4 −3 −2 −1
5
0
1
2
3
4
5
Complete the statement with <, > , or = .
19. 3____7
20. −1____4
21. − 4____ −10
22.
− 6 ____ − 3
15.
See left.
16.
See left.
17.
See left.
18.
See left.
19.
Evaluate the expression for the given value of x.
23. 2 x − 6; x = 9
24. − 7 + 9 x; x = 3
20.
25. 12 x + 13; x = 5
21.
22.
23.
24.
25.
Name _________________________________________________________ Date _________
Post
Course
u Pre-Course Test with Item
Analysis/Post-Course Test
with Item Analysis
Post Course Test
Solve the equation. Check your solution.
Answers
2. 3x − 2 = 4 − x
1. x + 14 = − 5
1.
Solve the equation.
2.
4. − 23 = − 3( − 4b − 3) − 8(1 + b)
3. − 3 + 4 8n + 10 = 53
Describe the values of c for which the equation has no solution.
The Pre-Course Test and PostCourse Test cover key concepts that
students will learn in their math
class. You can gauge how much
your students learned throughout
the year by comparing their PreTest score against their Post-Test
score. These tests also can be given
as practice for state
assessments.
4.
5. − 3 x + c = − 3 x + 8
6. − 3 + x = c
Name _________________________________________________________ Date _________
7. The quotient
of n and 3 is less than 5.
Pre-Course
Test
7.
8. 10 more than y is greater than or equal to 17.
Solve the equation. Check your solution.
Solve the inequality.
2 x + 7the
= solution.
−5 + x
1. x − 12 = 9
2. Graph
9. −3
− 3 ≤ − 3(1 − x)
Solve the equation.
8.
Answers
9.
1.
10. − 3 + 2 x + 1 ≤ −2
See left.
2.
(− 61 x
3. 8 2 − 9 p − 2 = 14
−5 −4 −3 −24.−1 − 30
+2 63) + 6( 4 x − 3) −6
= −−478−2
0
3.
2
4
6
8
Write and graph a compound inequality that represents the numbers that are
4.
Describe the values of c for which the equation has no solution.
nott solutions of the inequality represented by the graph shown.
5. 5 x + 4 = 5 x − c
6. 2 x + 6 = c
5.
11.
−2 −1
0
1
2
Write the sentence as an inequality.
12.
7. A number m increased by 12−2
is less0 than248.
3
4
5
y
2
3
4
5
6
7
8
−3
−4
−5
−2 −1
12.
−2
0
1
0
2
3
−6
0
5
2
4
6
8
10.
Determine whether the relation is a function. If the relation is a function,
determine whether the function is linear or nonlinear.
13.
x
1
2
3
4
5
y
1
4
9
16
25
15.
16.
See left.
6
16. through: ( − 3, − 4), perpendicular to y = − 3 x + 5
4
4
6
8
2
See left.
13.
14.
−7
−2
15. through: ( − 3, 1), slope = 2
4
See left.
12.
7.
8
See left.
Write an
equation
in slope-intercept
form of the
line
Write and graph a compound
inequality
that
represents the numbers
that
arewith the given
characteristics.
not solutions of the inequality
represented by the graph shown.
11.
11.
See left.
11.
6
6
−4
9 10
10.
6.
4
Determine
whether
the
relation
is a function,
8. The product of x and
10 is greater
than or
equal
to 23.is a function. If the relation 8.
determine whether the function is linearr or nonlinear.
r
9.
Solve the inequality. Graph the solution.
2
13.
14. y = − ( x + 1)
010. 32 2 −6x + 94 ≤ 16
x −3
9. − 6 x − ( − 7 x − 1) ≤ 6
See left.
1
5.
6.
Write the sentence as an inequality.
PreCourse
3.
12.
See left.
13.
14. y = 3 x + 1
Write an equation in slope-intercept form of the line with the given
characteristics.
15. through: (3, − 3) and ( − 3, 4)
14.
15.
16.
16. through: ( − 3, 5), parallel to y = 2 x + 2
A2
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Name _________________________________________________________ Date _________
Chapter
1
Quiz
For use after Section 1.3
Solve the equation.
Answers
1. x + 7 = 3
2. 5.3 = z − 2.7
3. 72 = − 6r
4.
2.
5. 4m − 6 = 14
6. 8 = 11 − v
7. 9 = − 7 w + 12 w − 6
8. 40 = −
3.
11. − 3( g − 4) = 2 − 3 g
The Quiz provides ongoing assessment of
student understanding. You can use this
as a gradable quiz or as practice.
4.
1
(9 x + 30) + 2
3
5.
6.
Solve the equation. Determine whether the equation has one solution, no
solution, or infinitely many solutions.
9. 6c + 3 = c − 12
u Quiz
1.
1
p = 15
5
7.
10. 4( 2q − 3) = 8q − 12
8.
12. 8( y − 1) − 3 y = 6( 2 y − 6)
9.
13. To estimate your average monthly salary, divide your yearly salary by the
number of months in a year. Write and solve an equation to determine your
yearly salary when your average monthly salary is $4560.
10.
14. You are a contractor and charge $45 for a site visit plus an additional
11.
$24 per hour for each hour you spend working at the site. Write and solve
an equation to determine how many total hours you have to work to earn
$810 working at two separate work sights.
12.
15. You and a friend are both traveling from the Seattle area. You start 38 miles
east of Seattle and travel east on Interstate 90 at 62 miles per hour. Your
friend starts 20 miles south of Seattle and travels south on Interstate 5 at
65 miles per hour.
13.
a. Who would be farther from Seattle in two hours and by how much?
14.
b. How many hours will it take for you and your friend to be the same
distance from Seattle?
15. a.
b.
Name_________________________________________________________
Chapter
1
Solve the equation. Justify each step.
1. x +
Date __________
Test A
Answers
1
3
=
2
4
2.
z
= 12
4
Solve the equation. Determine whether the equation has one solution,
no solution, or infinitely many solutions.
2.
3. 5n = − 20
4. g + 5 = 17
5. 13 + 3 p + 10 = 23 + 3 p
6. 7 + 4 y = 39
3.
7. 3 = t + 11.5 − t
8. 4 x + 8 + 6 x − 5 = 33
4.
9. 5( 2c + 7) − 3c = 7(c + 5)
10.
5.
3
1
b + 6 + b = 15 + 2b
2
2
6.
Describe the values of c for which the equation has no solution.
11. 2 x − 6 = 2 x − c
12.
7.
x +8 = c
8.
Find the value of the variable. Then find the angle measures of the polygon.
13. 5x°
14.
u Chapter Tests
1.
5b°
4b°
9.
10.
The Chapter Tests provide assessment
of student understanding of key
concepts taught in the chapter. There
are two tests for each chapter. You
can use these as gradable tests or as
practice for your students to master
an upcoming test.
11.
x°
12.
4b°
Sum of angle measures: 180°
Sum of angle measures: 360°
Solve the equation.
15. 7 n + 3 = 2n + 23
17.
19.
5b°
3
1
(d + 12) = (2d − 6)
2
2
16.
1
(6 x + 4) = 5(2 x − 8)
2
18.
b − 12 = 15
20.
2k + 6 = k
13.
14.
15.
16.
2 r + 5 = 3r
17.
18.
19.
Name_________________________________________________________
Chapter
1
Date __________
20.
Alternative Assessment
1. Which of the following equations have only one solution? Which have two
solutions? Which have no solution? Which have infinitely many solutions?
a.
3
x −8 =1
5
b. 5 x − 2( x − 2) = 7 x + 4 − 4 x
c.
u Alternative Assessment with
Scoring Rubric
3x + 2 − 2 = 6
d. 3 x − 2 + 4 = 2
1
e.
(6 x − 3) − x = 2 x + 1
2
f.
8 x − 5 = 3x + 5
2. A doormat is in the shape of a trapezoid. The area A of the doormat is represented
by the formula A =
1
h(b1 + b2 ).
2
a. Solve the formula for h.
b. Show that solving the formula for b1 by first multiplying both sides by 2 and
then dividing both sides by h leads to b1 =
2A
− b2 .
h
c. Show that solving the formula for b1 by first multiplying both sides by 2 and
then using the Distributive Property to distribute the h leads to b1 =
2 A − b2h
.
h
d. Show that the final formula in part (b) is equivalent to the final formula in part
(c) by showing the steps to transform one formula to the other one.
e. Explain how you could solve the original formula for b2.
Copyright © Big Ideas Learning, LLC. All rights reserved.
Integrated Math I PE_TE sampler.indd A3
Each Alternative Assessment includes at
least one multi-step problem that combines
a variety of concepts from the chapter.
Students are asked to explain their
solutions, write about the mathematics, or
compare and analyze different situations.
You can use this as an alternative to
traditional tests. It can be graded, assigned
as a project, or given as practice.
A3
10/20/14 1:42 PM
Name_________________________________________________________
Chapter
Performance Task (continued)
1
u Performance Task
The Performance Task presents an
assessment in a real-world situation.
Every chapter has a task that allows
students to work with multiple
standards and apply their knowledge to
realistic scenarios. You can use this task
as an in-class project, a take-home
assignment or as a graded assessment.
Date __________
Magic of Mathematics
Have you ever watched a magician perform a number trick? You can use algebra to
explain how these types of tricks work.
Work through problems 1 and 2 with a partner. Take turns having one of you read the
steps aloud as the other calculates and records the results. Then repeat the steps with a
variable in place of a specific number. Work together using algebra to explain why the
magic trick works.
1. Step 1: Think of any number.
Step 2: Double the number.
Step 3: Add 5.
Step 4: Double your new number.
Step 5: Subtract 6.
Step 6: Divide by 4.
Step 7: Subtract your original number.
Did you get 1? Use algebra to show why this works.
2. Step 1: Select a two-digit number less than 100.
Step 2: Add 3 to your number.
Step 3: Double the result.
Step 4: Subtract 9.
Step 5: Multiply by 5.
Step 6: Add 15.
Step 7: Divide by 10.
Did you get your original number? Use algebra to show why this works. How is
this problem different from the previous problem?
3. Create your own magic number trick and use algebra to explain why it works.
u Quarterly Standards Based Test
The Quarterly Standards Based Test
provides students practice answering
questions in standardized test format.
The assessments are cumulative and
cover material from multiple chapters of
the textbook. The questions represent
problem types and reasoning patterns
frequently found on standardized tests.
You can give this test to your students as
a cumulative assessment for the quarter
or as practice for state assessment.
Name_________________________________________________________
Chapters
1–3
Date __________
Quarterly Standards Based Test (continued)
8. Translate the equation 4 x − 5 = 3( x + 2) into a verbal sentence.
A. Four times the difference of x and five equals three times the sum of x and two.
B. Four times x less than five is the same as three times x plus two.
C. The sum of four times x and five is three times the sum of x and two.
D. The difference of four times x and five is three times the sum of x and two.
9. A boat travels upstream on the Allegheny River for 2 hours. The return trip only
takes 1.7 hours because the boat travels 2.5 miles per hour faster downstream due
to the current. How far does the boat travel one way?
A. 14 1 mi
6
B. 14 mi
C. 28 1 mi
3
D. 28 mi
10. Solve 1 (6 x + 30) = 4 x + 2( x − 7).
3
A. x = 6
B. x = 11
C. x = 4 1
4
D. x = −1
11. Solve 3n + 6 = 21.
A. n = − 5 and n = 5
B. n = − 9 and n = 9
C. n = − 9 and n = 5
D. no solution
12. Translate 5 x + 7 ≥ 2 into a verbal sentence.
A. Five times the sum of x and seven is greater than two.
B. Five times x, increased by seven, is less than or equal to two.
C. The product of five and x, increased by seven, is greater than or equal to two.
D. Seven more than the product of five and x is greater than two.
A4
Integrated Math I PE_TE sampler.indd A4
Copyright © Big Ideas Learning, LLC. All rights reserved.
10/20/14 1:42 PM
Resources by Chapter
The Resources by Chapter ancillary includes a number of supplemental resources for
every chapter in the program. It is available in print and online in an editable format.
Nombre _______________________________________________________
Capítulo
1
Fecha _________
Resolver ecuaciones lineales
Estimada familia:
Resolver ecuaciones es una destreza importante en la clase de matemáticas,
pero ¿qué pasa en la vida cotidiana? ¿Alguna vez han considerado cómo
pueden usar esta destreza en la vida real?
u Family Communication Letters
(English and Spanish)
The Family Communication Letters
provide a way to quickly communicate
to family members how they can
help their student with the
material of the chapter. You can
send this home with your students
to help make the mathematics less
intimidating and provide
suggestions for families to help
their students see mathematical
concepts in common activities.
Consideren la siguiente situación:
•
Usted y su familia quieren comprar un nuevo sistema de videojuegos.
El sistema cuesta $400. Ya han ahorrado $250 para el sistema.
¿Cuánto dinero aún necesitan ahorrar para comprar el sistema?
Podrían simplemente restar $250 de $400 para obtener la respuesta.
En cambio, consideren escribir una ecuación. ¿Cómo sería? En este caso,
la incógnita, la cantidad de dinero que falta ahorrar, puede representarse
con una variable. Un ejemplo de la ecuación sería y + 250 = 400, donde
y es el valor desconocido. Comenten cómo hallarían el valor desconocido.
Ahora, en familia, propongan maneras en que podrían ganar el dinero.
¿Ganarán el dinero trabajando juntos? ¿O dividirán la Date
restante
cantidad
Name _________________________________________________________
_________
que deben ahorrar entre los integrantes de su familia y trabajarán por
Chapter
separado?
1
Solving Linear Equations
Dear Family,
Luego, escriban una ecuación para cada situación. ¿En qué se parecen las
ecuaciones? ¿En qué se diferencian? ¿El valor de la variable es igual para
ambas situaciones?
A medida que su hijo avanza en el capítulo 1, aprenderá cómo resolver
Solving equations is an important skill in the math classroom, but how about in
tipos de ecuaciones semejantes. Compartan otras maneras en que su familia
everyday life? Have you ever considered how you may use this skill in real life?
usa ecuaciones en la vida cotidiana, quizás sin ni siquiera darse cuenta.
Consider the following scenario:
•
You and your family want to purchase a new video game system. The
system costs $400. You already have $250 saved to put toward the
system. How much money do you still need to save to buy the system?
You could simply subtract $250 from $400 to get your answer. Consider
writing an equation instead. What would that look like? The unknown, in this
case, the amount of money left to save, can be represented by a variable. One
example of an equation is y + 250 = 400, where y is the unknown value.
Discuss how you would find the unknown value.
Now, as a family, brainstorm ways you could earn the money. Will you earn the
money working together? Or will you divide the remaining amount needed to be
earned among the members of your family and work independently?
Copyrightfor
© Big
Ideasscenario.
Learning, LLC
Next, write an equation
each
How are the equations the same?
All rights reserved.
How are they different?
Is the value of the variable the same for both
scenarios?
Algebra 1
Resources by Chapter
3
As your child works through Chapter 1, he or she will learn how to solve similar
types of equations. Share together other ways you as family use equations in
everyday life, maybe without even realizing it.
Copyright © Big Ideas Learning, LLC. All rights reserved.
Integrated Math I PE_TE sampler.indd A5
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u Start Thinking/Warm Up/
Cumulative Review Warm Up
Each Start Thinking/Warm Up/
Cumulative Review Warm Up includes
two options for getting the class started.
The Start Thinking questions provide
students with an opportunity to
discuss thought-provoking questions and
analyze real-world situations. You can
use this to begin a class discussion. The
Warm Up questions review prerequisite
skills needed for the lesson. You can give
this to students at the beginning of class
to prepare them for the upcoming
lesson. The Cumulative Review Warm Up
questions review material from earlier
lessons or courses. You can give these
to students to continue their mastery of
concepts they have been taught.
1.1
Start Thinking
The angle measures of any four-sided figure sum to 360 degrees.
How can you find the fourth angle measure when you know the
sum of the other three angle measures?
1.1
Warm Up
Simplify the expression.
1. 5 + ( −15)
2. 6 − 7
3. 10 • ( −1)
− 30
4.
2
5. −1 × 0
6. 4 − ( − 5)
1.1
Cumulative Review Warm Up
Tell which property the statement illustrates.
1. 2 + 4 = 4 + 2
2. (3 • 7) 4 = 3(7 • 4)
3. 8 + 0 = 8
4. 7 • ¨ ¸ = 1
5. 4 • 0 = 0
6. 12(8 + 3) = 12 • 8 + 12 • 3
§1·
©7¹
Name_________________________________________________________
u Practice
The Practice exercises provide additional
practice on the key concepts taught
in the lesson. There are two levels of
practice provided for each lesson:
A (basic) and B (average). You can assign
these exercises for students that need
the extra work or as a gradable
assignment.
1.1
Date __________
Practice A
In Exercises 1–6, solve the equation. Justify each step. Check your solution.
1. x + 2 = 5
2. g − 4 = 3
4. d + 4 = − 2
5.
3. m − 1 = 8
p + 7 = 5
6. k − 6 = − 4
The sum of the angle measures of a quadrilateral is 360°. In Exercises 7 and 8,
write and solve an equation to find the value of x. Use a protractor to check the
reasonableness of your answer.
7.
8.
x°
125°
132°
110°
72°
x°
90°
80°
In Exercises 9–14, solve the equation. Justify each step. Check your solutions.
9. 3t = 24
12.
j ÷5 = 2
10. 7 p = 28
11. s ÷ 4 = 3
13. − 6q = 54
14. c ÷ ( − 9) = 2
In Exercises 15–20, solve the equation. Check your solution.
15. h + 1 = 5
3
3
16. w − 7 = 2
9
9
17. 3 f = 9
5
18. u + 2.7 = 1.5
19. 32π t = 64π
20. m ÷ ( − 7) = 2.1
In Exercises 21–23, write and solve an equation to answer the question.
21. The width of a laptop is 11.25 inches. The width is 0.75 times the length. What is
the length of the laptop?
22. The temperature at 10 A.M. is 12 degrees Fahrenheit. The temperature at 6:00 A.M.
was − 7 degrees Fahrenheit. How many degrees did the temperature rise?
23. The population of a city is 645 people less than it was 5 years ago. The current
population is 13,500. What was the population 5 years ago?
24. Identify the property of equality that makes Equation 1 and Equation 2 equivalent.
A6
Integrated Math I PE_TE sampler.indd A6
Equation 1
4.2 x − 1.5 = 1.7 x + 8.3
Equation 2
42 x − 15 = 17 x + 83
Copyright © Big Ideas Learning, LLC. All rights reserved.
10/20/14 1:42 PM
Name_________________________________________________________
1.1
Date __________
Enrichment and Extension
u Enrichment and Extension
Solving Simple Equations
Solve the equation. Justify each step. Check your solution.
1. x +
4.
4
5
=
5
2
2.
m
1
=1
−7
4
5.
7. 3.2t = 6.4
9
3
= t
16
4
π
2
t =
5π
6
8. 150 = 7.5x
3. w −
1
2
= 2
2
3
6. x ÷
4
7
= −
5
8
Each Enrichment and Extension extends
the lesson and provides a challenging
application of the key concepts. These
rigorous exercises can be assigned to
challenge your students to use higher
order thinking skills or to build on a
concept via an extension of the lesson.
9. w ÷ 2.4 = 6.52
The sum of the angle measures of a polygon follows the general rule of ( n − 2) • 180°,
where the variable n represents the number of sides. In Exercises 10–15, write and solve
an equation to find the value of x. Use a protractor to check the reasonableness of
your answer.
10.
11.
96°
72°
x°
35°
x°
114°
48°
12.
x°
131°
13.
x°
47°
x°
x°
x°
x°
x°
14.
x°
15.
90°
x°
2x + 20°
x°
2x − 10°
x°
16. It takes a plane 4 hours and 15 minutes to fly from Orlando, Florida, to Boston,
ance between th
iti iis 1114 miles.
il
Massachusetts. The distance
the ttwo cities
a. What is the average sspeed of the plane in miles per hour?
_________________________________________________________
Date _________
b. If every mile is approximately
o
1.6Name
kilometers,
what is the speed of the airplane
in kilometers per hour?
u
1.1 Puzzle Time
u Puzzle Time
Did You Hear About The Tree's Birthday?
Circle the letter of each correct answer in the boxes below. The circled letters will spell
out the answer to the riddle.
Each Puzzle Time provides additional
practice in a fun format in which
students use their mathematical
knowledge to solve a riddle. This
format allows students to self-check
their work. Your students can learn
the lesson concepts by finding the
answers to silly jokes and riddles.
Find the value of the variable of the equation.
2. x + 11 = 4
1. m + 7 = 9
3. n −
3
2
=
5
5
4. −18 = r − 12
5. s − ( −10) = 2
6. 6.3 = b − 1.5
8. y ÷ 9 = − 3
7. 1.4h = 5.6
9. − 7c = − 63
10.
6
11. − a = 18
7
x
= −3
8
12. −144π = −12π k
Solve an equation to answer the question.
13. The students on a decorating committee create a banner. The length of the
banner is 2.5 times its width. The length of the banner is 20 feet. What is the
width (in feet) of the banner?
14. The student council consists of 32 members. There are 27 members decorating
for the dance. How many members are not decorating?
B
L
I
B
M
15
−18
4
2.1
13
A
Q
G
U
S
T
T
−16 −21
A
C
N
W
D
A
O
E
S
P
4.6
17
9
−13
5
9.1
8.2
7.8
11.6
P
R
I
P
Y
O
J
N
E
−7
8
−20
12
−6
H
−8
18
1
−14 −27
20 −17 16
2 −24 14
Name_________________________________________________________ Date __________
Chapter
1
Cumulative Review
In Exercises 1–8, add or subtract.
1. 4 − 6
2. 0 + ( −17)
3. − 5 + 7
4. 10 + ( − 2)
5. 4 − ( −13)
6. −11 − ( − 23)
7. 17 + 8
8. −19 + 21
In Exercises 9 –24, multiply or divide.
10. − 5 • ( − 6)
11. 3 • ( −1)
13. 19 • 2
9. − 4(5)
14. − 7 ( − 6)
15. 9 • ( − 8)
12. 7( − 2)
16. 4( 2)
17. − 20 ÷ 5
18. 48 ÷ ( − 3)
19. − 38 ÷ ( − 2)
20. 8 ÷ ( − 2)
21. −18 ÷ ( − 3)
22. 32 ÷ 8
23. − 48 ÷ 6
24. 63 ÷ ( − 9)
In Exercises 25–28, solve the problem and specify the units of measure.
25. You mow your neighbor’s lawn in 5 hours and earn $45. What is your hourly wage?
26. How many packages of diapers can you buy with $36 when one package costs $9?
27. At a gas station you buy 15 gallons of gas. The total cost is $60. What is the cost per
gallon?
28. On Saturday, you run 4 miles more than your friend. Your friend ran 3 miles. How
many miles did you run?
29. A flower bed is in the shape of a rectangular prism. Its dimensions are 3 feet wide by 16
u Cumulative Review
The Cumulative Review includes exercises
covering concepts and skills from the
current chapter and previous chapters.
Students can work on their mastery of
previously learned material.
feet long by 6 inches deep.
a. How many cubic feet of topsoil do you need to fill the flower bed?
b. You can spread topsoil at a rate of 4 cubic feet per minute. How long will it take you
to spread all the topsoil?
In Exercises 30 –37, solve the equation, justify each step, and check your answer.
30. x + 5 = 7
31.
32. x + ( − 3) = 5
33. 10 − m = − 3
y −4 = 2
34. 4 g = 36
35. 3b = 39
36. c ÷ 7 = 14
37.
y
= 15
2
Copyright © Big Ideas Learning, LLC. All rights reserved.
Integrated Math I PE_TE sampler.indd A7
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Student Journal
The Student Journal serves as a valuable component where students may work extra problems,
take notes about new concepts and classroom lessons, and internalize new concepts by expressing
their findings in their own words. Available in English and Spanish
Name_________________________________________________________
Chapter
Date __________
Maintaining Mathematical Proficiency
1
Add or subtract.
1. −1 + ( − 3)
u Maintaining Mathematical
Proficiency
The Maintaining Mathematical Proficiency
corresponds to the Pupil Edition Chapter
Opener. Your students have the opportunity
to practice prior skills necessary to move
forward.
2. 0 + ( −12)
3. 5 − ( − 2)
4. − 4 − 7
5. Find two pairs of integers whose sum is − 6.
6. In a city, the record monthly high temperature for March is 56°F. The record
monthly low temperature for March is −4°F. What is the range of temperatures
for the month of March?
Multiply or divide.
7. − 2(13)
8. − 8 • ( − 5)
10. − 30 ÷ ( − 3)
9. −14 ÷ 2
11. Find two pairs of integers whose product is − 20.
12. A football team loses 3 yards in 3 consecutive plays. What is the total yardage
gained?
u Exploration Journal
The Exploration pages correspond to the
Explorations and accompanying exercises in
the Pupil Edition. Your students can use the
room given on these pages to show their
work and record their answers.
Name _________________________________________________________ Date _________
Solving Simple Equations
1.1
For use with Exploration 1.1
Essential Question How can you use simple equations to solve real-life
problems?
1
EXPLORATION: Measuring Angles
Go to BigIdeasMath.com for an interactive tool to investigate this exploration.
Work with a partner. Use a protractor to measure the angles of each quadrilateral.
Complete the table to organize your results. (The notation m∠ A denotes the measure
of angle A.) How precise are your measurements?
a.
A
B
b.
c.
A
D
B
A
B
C
D
C
D
C
Quadrilateral
m∠ A
(degrees)
m∠B
(degrees)
m∠ C
(degrees)
m∠D
(degrees)
m∠ A + m∠B
+ m∠C + m∠D
a.
b.
c.
2
EXPLORATION: Making a Conjecture
Go to BigIdeasMath.com for an interactive tool to investigate this exploration.
Work with a partner. Use the completed table in Exploration 1 to write a conjecture
about the sum of the angle measures of a quadrilateral. Draw three quadrilaterals that
are different from those in Exploration 1 and use them to justify your conjecture.
A8
Integrated Math I PE_TE sampler.indd A8
Copyright © Big Ideas Learning, LLC. All rights reserved.
10/20/14 2:31 PM
Name _________________________________________________________ Date _________
Notetaking with Vocabulary
1.1
For use after Lesson 1.1
u Notetaking with Vocabulary
In your own words, write the meaning of each vocabulary term.
conjecture
This student-friendly notetaking
component is designed to be a reference
for key vocabulary, properties, and core
concepts from the lesson. Students can
write the definitions of the vocabulary
terms in their own words and take notes
about the core concepts.
rule
theorem
equation
linear equation in one variable
solution
inverse operations
equivalent equations
Core Concepts
Addition Property of Equality
Words Adding the same number to each side of an equation produces an equivalent
equation.
Algebra If a = b, then a + c = b + c.
Notes:
Name _________________________________________________________ Date _________
1.1
Notetaking with Vocabulary (continued)
u Extra Practice
Common Problem-Solving Strategies
Use a verbal model.
Guess, check, and revise.
Draw a diagram.
Sketch a graph or number line.
Write an equation.
Make a table.
Look for a pattern.
Make a list.
Work backward.
Break the problem into parts.
Each section of the Pupil Edition has
an Extra Practice in the Student Journal
with room for students to show their
work.
Notes:
Extra Practice
In Exercises 1–9, solve the equation. Justify each step. Check your solution.
1. w + 4 = 16
2. x + 7 = −12
3. −15 + w = 6
4. z − 5 = 8
5. − 2 = y − 9
6. 7 q = 35
7. 4b = − 52
8. 3 =
q
11
9.
n
= −15
−2
10. A coupon subtracts $17.95 from the price p of a pair of headphones. You pay $71.80 for the
headphones after using the coupon. Write and solve an equation to find the original price of
the headphones.
11. After a party, you have 2 of the brownies you made left over. There are 16 brownies left.
5
How many brownies did you make for the party?
Copyright © Big Ideas Learning, LLC. All rights reserved.
Integrated Math I PE_TE sampler.indd A9
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Differentiating the Lesson
The Differentiating the Lesson online ancillary provides complete teaching notes and worksheets
that address the needs of diverse learners. Lessons engage students in activities that often
incorporate visual and kinesthetic learning. Some lessons present an alternative approach to
teaching the content, while other lessons extend the concepts of the text in a challenging way
for advanced students. Each chapter also begins with an overview of the differentiated lessons
in the chapter and describes the students who would most benefit from the approach used in
each lesson.
Lesson
1A
Solving Simple Linear Equations Using the
Addition and Subtraction Properties
For use with Section 1.1
Lesson Preparation
Materials: math tiles*, Resource Sheet 1A: Math Tile Mat for Solving Equations,
Resource Sheet 1B: Tape Diagram Mat (optional)
*If math tiles are not available, laminate copies of Resource Sheet 1A and have students
draw the math tiles with dry erase markers as they complete the lesson.
Beforehand: Pre-assess students’ understanding of integer operations using two-color
counters. Students will need to understand the concept of a zero pair.
Classroom Management: Use the pre-assessment information to group students
into pairs.
Lesson Procedure
Distribute math tiles to each student, along with Resource Sheet 1A.
Explain that a yellow square (or a square with + ) represents +1, and a red square
(or a square with − ) represents −1. Make connections to two-color counters used in
previous years. Introduce that a green rod (or a rod with + ) represents 1x, and a red rod
(or a rod with − ) represents −1x. Remind students that placing one yellow square and
one red square together equals 0, which is called a zero pair. Ask students what is being
represented when placing one green rod and one red rod together. Be sure to have
students recognize that this is also a zero pair.
= +1
= −1
= +1x
= −1x
Present students with the equation x + 4 = 7. Explain that an equation must always be
balanced, which is why students will represent each side of the equation on the balance
provided in Resource Sheet 1A. Have students represent the equation on their resource
sheets using the math tiles. Reference the example below.
=
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Integrated Math I PE_TE sampler.indd A10
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Credits
Cover
© Abidal | Dreamstime.com, Jason Winter/Shutterstock.com
Front Matter
vi Goodluz/Shutterstock.com
Chapter 1
0 top left Doug Meek/Shutterstock.com; top right aceshot1/
Shutterstock.com; center left Stephanie Lupoli/Shutterstock.com;
bottom right Stefan Schurr/Shutterstock.com; bottom left Ververidis
Vasilis/Shutterstock.com; 6 Ververidis Vasilis/Shutterstock.com;
9 cowardlion/Shutterstock.com; 14 Stefan Schurr/Shutterstock.com;
15 Eric Isselee/Shutterstock.com; 22 Stephanie Lupoli/
Shutterstock.com; 25 Nattika/Shutterstock.com; 26 Invisible Studio/
Shutterstock.com, avNY/Shutterstock.com, barragan/
Shutterstock.com; 29 aceshot1/Shutterstock.com; 32 Jagodka/
Shutterstock.com; 38 Petr84/Shutterstock.com; 40 Maxim Tupikov/
Shutterstock.com; 41 bottom left Vladimir Dokovski/Shutterstock.com;
top right Potapov Alexander/Shutterstock.com; 42 John Blanton/
Shutterstock.com; 43 tuulijumala/Shutterstock.com, Kokhanchikov/
Shutterstock.com
Copyright © Big Ideas Learning, LLC. All rights reserved.
Integrated Math I PE_TE sampler.indd A11
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Integrated Math I PE_TE sampler.indd A12
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