Download Inequalities and Equations

Inequalities and
Equations
© 2011 Carnegie Learning
Tightrope
walkers often
perform at circuses.
They have trained to
keep their balance while
walking across a thin,
high rope. Some tightrope
walkers use a large
pole to help them
balance.
9.1 Call to Order
Inequalities................................................................ 605
9.2 Opposites Attract to Maintain a Balance
Solving One-Step Equations Using Addition and Subtraction............................................. 615
9.3 Statements of Equality Redux
Solving One-Step Equations Using Multiplication and Division...........................................625
9.4 there are many ways . . .
Representing Situations in Multiple Ways..................... 635
9.5 Measuring Short
Using Multiple Representations to Solve Problems...................................................... 643
9.6 Variables and More Variables
The Many Uses of Variables in Mathematics................. 653
9.7 Quantities that Change
Independent and Dependent Variables.............................. 663
603
© 2011 Carnegie Learning
604 • Chapter 9 Inequalities and Equations
Call to Order
Inequalities
Learning Goals
Key Terms
In this lesson, you will:




 Use inequalities to order the number system.
 Graph inequalities on the number line.
inequality
graph of an inequality
solution set of an inequality
ray
W
hat happens every morning in your class and usually involves your teacher
calling names? If you said roll call, you’d be right! So, does your teacher seem to
call your classmates’ names in the same order every morning? Actually, there are
a lot of ways for teachers to call roll, but one of the easiest ways is to call roll in
alphabetical order. Sometimes teachers will call roll in alphabetical order in
ascending order. This means starting at the letter A and moving to the letter Z.
Or, teachers will call roll in alphabetical order in descending order, which is the
opposite of ascending order.
Many people and items are ordered in different ways. When a photographer takes
a picture of a group of people, the photographer will usually put the shorter
© 2011 Carnegie Learning
people in the front of the group and the taller people in the back of the group.
Mechanics usually arrange their wrenches and sockets in order from smallest to largest.
What things do you order? How do you go about ordering items or people—and
this doesn’t mean ordering your brother and sister around to do your chores!
9.1 Inequalities • 605
Problem 1 Saying So Much with Just One Symbol
In the past, you probably used symbols that let you order numbers from least to greatest,
or from greatest to least. These symbols are called inequality symbols. An inequality is
any mathematical sentence that has an inequality symbol.
Symbol
Meaning
Example
,
less than
.
greater than
10 . 7 10 is greater than 7
#
less than or
equal to
3 # 9 3 is less than or equal to 9
$
greater than
or equal to
4 $ 1 4 is greater than or equal to 1
fi
not equal to
6 fi 7 6 is not equal to 7
3 , 5 3 is less than 5
1. For each statement, write the corresponding inequality.
a. 7 is less than or equal to 23
c. 2 is not equal to 5
d. 7.6 is less than 8.2
3  ​is greater than 4 __
​ 2 ​ 
e. 5 ​ __
4
3
606 • Chapter 9 Inequalities and Equations
© 2011 Carnegie Learning
b. 56 is greater than 28
2. Write the meaning of each inequality in words.
a. 7.8 fi 23.7
1  ​# 8.7
b. 8 ​ __
3
3 ​ $ 0.75
c.​ __
4
d. 43,256 . 4489
e. 0.012 , 0.02
3. Write , or . to make each inequality true.
© 2011 Carnegie Learning
a. 12 1  ​ c. 3 ​ __
3
b. 1.2 2
3.3
d. 10.25 1.201
10 __
​ 1 ​ 
5
4. Write # or $ to make each inequality true.
a. 1 2
b. 4.2 1  ​
4 ​ __
4
1  ​ c.​ __
3
0.3
d. 0.25 __
​ 2  ​
5
e. 24.33 24 __
​ 1 ​ 
3
9.1 Inequalities • 607
For any two numbers a and b, only one of the three statements is true.
● a,b
● a.b
● a5b
If a fi b,
then a must
be less than
b or greater
than b.
5. What does this statement mean in terms of the
ordering of the number system?
Problem 2 Inequalities and the Number Line
A number line is a graphic representation of all numbers.
1. Plot and label each of the numbers shown on the number line.
a. 3
b. 2.3
4  ​
c. 3 ​ __
5
__
d. 4 ​ 1 ​ 
3
e. 4.66…
2
3
4
5
2. There are five points plotted on the number line shown. Identify the approximate location
of each point.
a
0
1
b
2
a.
b.
c.
d.
e.
608 • Chapter 9 Inequalities and Equations
c
d
3
e
4
5
© 2011 Carnegie Learning
1
0
3. A point at a is plotted on the number line shown.
a
0
a. Plot a point to the right of this point and label it b. Then, write three different
inequalities that are true about a and b.
b. What can you say about all points to the right of point a on the number line?
4. A point at a is plotted on the number line shown.
a
0
a. Plot a point to the left of this point and label it b. Then, write three different
inequalities that are true about a and b.
© 2011 Carnegie Learning
b. What can you say about all the points to the left of point a on the number line?
5. Describe the position of all the points on the number line that are:
a. greater than a. b. less than a.
a
0
9.1 Inequalities • 609
Problem 3 Graphing an Inequality on a Number Line
You can use a number line to represent inequalities. The graph of an inequality in one
variable is the set of all points on a number line that make the inequality true. The set of all
points that make an inequality true is the solution set of the inequality.
1. Look at the two inequalities x . 3 and x $ 3.
a. Describe the solution sets for each.
b. Analyze the graphs of the two inequalities
shown on each number line.
x.3
0 1 2 3 4
x$3
Why does
one graph show a
see-through point
and the other one a
black point?
0 1 2 3 4
Describe each number line representation.
c. How does the solution set of the inequality x $ 3 differ
from the solution set of x . 3?
610 • Chapter 9 Inequalities and Equations
© 2011 Carnegie Learning
2. Look at the two inequalities x , 3 and x # 3.
a. Describe the solution sets for each.
b. Analyze the graphs of the 2 inequalities shown on each number line.
x,3
0 1 2 3 4
x#3
0 1 2 3 4
Describe each number line representation.
© 2011 Carnegie Learning
c. How does the solution set of the inequality x # 3 differ from the solution set of x , 3?
9.1 Inequalities • 611
The solution to any inequality can be represented on a number line by a ray whose starting
point is an open or closed circle. A ray begins at a starting point and goes on forever in
one direction. A closed circle means that the starting point is part of the solution set of the
inequality. An open circle means that the starting point is not part of the solution set of
the inequality.
3. Write the inequality represented by each graph.
a.
10
11
12
13
14
15
16
17
18
19
10
11
12
13
14
30
31
32
33
34
35
36
37
38
39
20
21
22
23
24
25
26
27
28
29
b. 15
16
17
18
19
c.
d. 4. Graph the solution set for each inequality.
a. x # 14
10
11
12
13
14
15
50
51
52
53
54
55
0
1
2
3
4
5
b. x , 55
© 2011 Carnegie Learning
1 ​# x
c. 2 ​ __
2
d. x . 3.3
0
1
2
3
4
5
e. x fi 4.2
0
1
612 • Chapter 9 Inequalities and Equations
2
3
4
5
Talk the Talk
1. Explain the meaning of each sentence in words.
Then, define a variable and write a mathematical
statement to represent each statement. Finally,
"Maximum"”
means that the
weight can't go over
that amount.
sketch a graph of each inequality.
a. The maximum load for an elevator is 2900 lbs.
b. A car can seat up to 8 passengers.
c. No persons under the age of 18 are permitted.
© 2011 Carnegie Learning
d. You must be at least 13 years old to join.
Be prepared to share your solutions and methods.
9.1 Inequalities • 613
© 2011 Carnegie Learning
614 • Chapter 9 Inequalities and Equations
Opposites Attract
to Maintain a
Balance
Solving One-Step Equations
Using Addition and Subtraction
Learning Goals
Key Terms
In this lesson, you will:
 one-step equation
 Properties of Equality for Addition
 Use inverse operations to solve
and Subtraction
one-step equations.
 Use models to represent one-step
equations.
 solution
 inverse operations
Y
ou’ve certainly seen parallel lines before. Railroad tracks look like parallel
lines. The opposite sides of a straight street form parallel lines. Even a very
important symbol in mathematics looks like parallel lines: the equals sign ().
Did you know there is a reason for why an equals sign looks the way it does? In 1557, mathematician Robert Recorde first used parallel line segments to
represent equality because he didn’t want to keep writing the phrase “is equal to”
© 2011 Carnegie Learning
and, as he explained, “no two things can be more equal” than parallel lines.
What does equality mean in mathematics? How can you determine whether two or
more things are equal?
9.2 Solving One-Step Equations Using Addition and Subtraction • 615
Problem 1 Maintaining Balance
Each representation shows a balance. Determine what will
balance 1 rectangle in each. Adjustments can be made in
You might
want to get
your algebra
tiles out.
each pan as long as the balance is maintained. Then,
describe your strategies.
1.
a. Strategies:
© 2011 Carnegie Learning
b. What will balance one rectangle?
616 • Chapter 9 Inequalities and Equations
2.
a. Strategies:
b. What will balance one rectangle?
© 2011 Carnegie Learning
3. Describe the general strategy you used to maintain balance in Questions 1 and 2.
4. Generalize the strategies for maintaining balance by completing each sentence.
a.To maintain balance when you subtract a quantity from one side, you must
b.To maintain balance when you add a quantity to one side, you must
9.2 Solving One-Step Equations Using Addition and Subtraction • 617
Problem 2 One Step at a Time
1. Write an equation that represents each pan balance. These are the same pan
balances that you analyzed for Question 1 and Question 2 in Problem 1. Use the
variable x to represent
, and count the
units to determine the number
they represent together. Then, describe how the strategies you used to determine what
balanced one rectangle can apply to an equation. In other words, what balances x?
a.
618 • Chapter 9 Inequalities and Equations
© 2011 Carnegie Learning
b.
You just wrote and solved one-step equations. Previously, you wrote an equation by
setting two expressions equal to each other. You solve an equation by determining what
value will replace the variable to make the equation true. If you can solve an equation
using only one operation, this equation is called a one-step equation. To determine if
your value is correct, substitute the value for the variable in the original equation. If the
equation is true, or remains balanced, then you correctly solved the equation.
2. Check each of your solutions to Question 1, part (a) and part (b), by substituting your
value for x into the original equation you wrote. Show your work.
You just determined solutions to your equations. A solution to an equation is any value for
a variable that makes the equation true.
3. State the operations in each equation you wrote for Question 1, and the operation
you used to determine the value of x. Describe how they relate to each other.
To solve an equation, you must isolate the variable by performing inverse operations.
Inverse operations are pairs of operations that undo each other.
4. State the inverse operation for each stated operation.
© 2011 Carnegie Learning
a. addition
b. subtraction
To isolate
the variable
means to get the
variable by itself
on one side of
the equation.
9.2 Solving One-Step Equations Using Addition and Subtraction • 619
Problem 3 Solving Equations
The answers
are the same. What
is different about
the two methods?
Example 1
Example 2
a1759
12 5 b 2 8
Method 1:
Method 1:
a17275927
12 1 8 5 b 2 8 1 8
a 5 2
20 5 b
Method 2:
Method 2:
a1759
12 5 b 2 8
27 5 27
a52
18 5
1 8
20 5 b
1. Analyze each example and the different methods used to solve each equation.
b. The final step in each method shows the variable isolated. What is the coefficient
of each variable?
620 • Chapter 9 Inequalities and Equations
© 2011 Carnegie Learning
a. Describe the difference in strategy between Method 1 and Method 2 for Example 1.
2. Consider the equations shown. State the inverse operation needed to isolate the
variable. Then, solve the equation. Make sure you show your work. Finally, check to
see if the value of your solution maintains balance in the original equation.
a. m 1 7  11
b. 5 5 x 2 8
© 2011 Carnegie Learning
c. b 1 5.67 5 12.89
3 ​5 x 2 4 __
​ 1 ​
d. 5 ​ __
4
2
9.2 Solving One-Step Equations Using Addition and Subtraction • 621
e. 23.563 5 a 2 345.91
7  ​ 5 y 1 __
​ 1 ​ 
f.​ ___
12
4
7  ​5 c 1 9 __
​ 3 ​ 
h. 13 ​ __
4
8
622 • Chapter 9 Inequalities and Equations
Don't forget
to check your
answers!
© 2011 Carnegie Learning
g. w 1 3.14 5 27
3. Determine if each solution is true. Explain your reasoning.
a. Is x 5 25 a solution to the equation x 1 17 5 8?
b. Is x 5 16 a solution to the equation x 2 12 5 4?
1  ​a solution to the equation 17 __
​ 2 ​ 5 x 1 15 __
​ 1 ​ ?
c. Is x 5 2 ​ __
3
3
3
© 2011 Carnegie Learning
d. Is x 5 4.567 a solution to the equation x 1 19.34 5 23.897?
9.2 Solving One-Step Equations Using Addition and Subtraction • 623
Talk the Talk
The Properties of Equality allow you to balance and solve equations involving
any number.
Properties of Equality
For all numbers a, b, and c,…
Addition Property of Equality
If a 5 b, then a 1 c 5 b 1 c.
Subtraction Property of Equality
If a 5 b, then a 2 c 5 b 2 c.
1. Describe in your own words what the Properties of Equality represent.
2. What does it mean to solve a one-step equation?
3. Describe how to solve any one-step equation.
5. Given the solution x 5 12, write two different equations using the
Properties of Equality.
Be prepared to share your solutions and methods.
624 • Chapter 9 Inequalities and Equations
© 2011 Carnegie Learning
4. How do you check to see if a value is the solution to an equation?
Statements of
Equality Redux
Solving One-Step Equations
with Multiplication and Division
Learning Goals
Key Term
In this lesson, you will:
 Properties of Equality for
 Use inverse operations to solve one-step equations.
 Use models to represent one-step equations.
Multiplication and Division
I
n 1997, Arulanantham Suresh Joachim set a world record for balancing on
one foot: 76 hours and 40 minutes. That’s slightly more than 3 days!
How long do you think you could balance on one foot? Don’t try it out now,
because you have some more to learn about balancing in mathematics. What
© 2011 Carnegie Learning
other examples of “balancing” are there in mathematics.
9.3 Solving One-Step Equations with Multiplication and Division • 625
Problem 1 Maintaining Balance
Each representation shows a balance. Determine what will balance 1 rectangle in each.
Adjustments can be made in each pan as long as the balance is maintained.
Describe your strategies.
1.
a. Strategies:
© 2011 Carnegie Learning
b. What will balance one rectangle?
626 • Chapter 9 Inequalities and Equations
2.
a. Strategies:
b. What will balance one rectangle?
© 2011 Carnegie Learning
3. Describe the general strategy you used to maintain balance in Questions 1 and 2.
4. Generalize the strategies for maintaining balance by completing each sentence.
a.To maintain balance when you multiply a quantity by one side, you must
b.To maintain balance when you divide a quantity by one side, you must
9.3 Solving One-Step Equations with Multiplication and Division • 627
Problem 2 One Step at a Time
1. Write an equation that represents each pan balance. These are the same pan
balances that you analyzed for Question 1 and Question 2 in Problem 1.
Let
represent the variable x, and let
represent one unit. Then,
describe how the strategies you used to determine what balanced one rectangle can
apply to an equation. In other words, what balances x?
a.
© 2011 Carnegie Learning
b.
628 • Chapter 9 Inequalities and Equations
2. Check each of your solutions to Question 1, parts (a) through (b) by substituting
your value for x back into the original equation you wrote. Show your work.
3. State the operations in each equation you wrote for Question 1, parts (a) through (b)
and the operation you used to determine the value of x. Describe how they relate to
each other.
As you learned previously, to solve an equation, you must isolate the variable by
performing inverse operations.
4. State the inverse operation for each stated operation.
a. addition
b. subtraction
© 2011 Carnegie Learning
c. multiplication
d. division
9.3 Solving One-Step Equations with Multiplication and Division • 629
Problem 3 Solving Equations
Example 1
8c 5 48
Example 2
d
2 5 ​ __ ​ 
4
Method 1:
8c 48
___
​   ​ 
​   ​ 5 ___
8
8
d
2  4 5 ​ __ ​  4
4
c 5 6
8 5 d
Method 2:
Method 2:
8c 4 8 5 48 4 8
d
2 5 ​ __ ​ 
4
c 5 6
Method 3:
Looks like there
is more than one
way to solve these
equations. What's
different about
each method?
Method 1:
34 5 34
8 5 d
1
1
​ __  ​  8c 5 ​ __  ​  48
8
8
c 5 6
1. Analyze each example and the different methods used to solve each equation.
b. How are Method 1 and Method 3 in Example 1 similar?
c. The final step in each method shows the variable isolated. What is the coefficient
of each variable?
630 • Chapter 9 Inequalities and Equations
© 2011 Carnegie Learning
a. Describe the difference in strategy between Method 1 and Method 2 for Example 2.
2. Consider the equations shown. State the inverse operation needed to isolate the
variable. Then, solve the equation. Make sure that you show your work. Finally, check
to see if the value of your solution maintains balance in the original equation.
n  ​5 7
a.​ __
4
b. 3y 5 18
© 2011 Carnegie Learning
n   ​ 5 9.4
c.​ ___
4.3
3  ​ y 5 3 __
​ 1 ​ 
d.​ ___
10
3
9.3 Solving One-Step Equations with Multiplication and Division • 631
2  ​5 2 __
e. y 4 ​ __
​ 1 ​ 
3
2
f. 3.14y  81.2004
h. 514 5 81.4 1 x
632 • Chapter 9 Inequalities and Equations
Don't
forget to check
your solutions.
© 2011 Carnegie Learning
g. y 2 __
​ 2 ​ 5 2 __
​ 1 ​ 
3
2
3. Determine if each solution is true. Explain your reasoning.
a. Is p 5 12 a solution to the equation 9p 5 108?
b. Is n 5 4 a solution to the equation __
​ n ​ 5 24?
6
© 2011 Carnegie Learning
c. Is p 5 18 a solution to the equation 3p 5 54?
9.3 Solving One-Step Equations with Multiplication and Division • 633
Talk the Talk
The Properties of Equality allow you to balance and solve equations involving any number.
Properties of Equality
For all numbers a, b, and c,…
Addition Property of Equality
If a 5 b, then a 1 c 5 b 1 c.
Subtraction Property of Equality
If a 5 b, then a 2 c 5 b 2 c.
Multiplication Property of Equality
If a 5 b, then ac 5 bc.
Division Property of Equality
a __
b
If a 5 b, and c fi 0, then ​ __
c ​ 5 ​ c ​.
1. Describe in your own words what the Properties of Equality represent.
2. What does it mean to solve a one-step equation?
3. Describe how to solve any one-step equation.
5. Given the solution x 5 12, write two different equations using the Multiplication and
Division Properties of Equality.
Be prepared to share your solutions and methods.
634 • Chapter 9 Inequalities and Equations
© 2011 Carnegie Learning
4. How do you check to see if a value is the solution to an equation?
There Are Many Ways...
Representing Situations
in Multiple Ways
Learning Goals
In this lesson, you will:
 Represent two quantities that change in words, symbols, tables, and graphs.
 Solve one-step equations.
P
eople are constantly confronted with problems in their lives. How many bags
of fertilizer will you need for your lawn? How much paint is needed to paint a
room? After viewing a graph of sales over the last year, what predictions can be
made for next year? After looking at a pattern of brick for a walkway, how can you
decide how many of each type of brick to order? These are all examples of
problems that could be solved more efficiently using mathematics. The real power
of mathematics is in providing people with the ability to model and solve problems
more efficiently and accurately. Can you think of other examples where
© 2011 Carnegie Learning
mathematics would be useful?
9.4 Representing Situations in Multiple Ways • 635
Problem 1 Buying on the Internet
A site on the Internet sells closeout items and charges a flat fee of $2.95 for shipping.
1. How much would the total order cost if an item costs:
a. $26.45?
b. $16.95?
2. Explain how you calculated your answers.
3. Define variables for the cost of an item and the total cost of the order.
4. Write an equation that models the relationship between these variables.
5. Use your equation to calculate the total cost of an order given the item cost.
a. $100
c. $67.13
636 • Chapter 9 Inequalities and Equations
© 2011 Carnegie Learning
b. $45.25
6. Use your equation to calculate the cost of an item given the total cost. Then, check to
see if the value of your solution maintains balance in the original equation.
a. $125
b. $37.45
Don't forget about all the
estimation strategies you
have learned! Does your answer
make sense in terms of the
problem situation?
© 2011 Carnegie Learning
c. $7.67
9.4 Representing Situations in Multiple Ways • 637
7. Complete the table with your answers from Question 1 through Question 6.
Cost of an Item
(dollars)
Total Cost with Shipping
(dollars)
26.45
16.95
100
45.25
67.13
125
37.45
7.67
8. Use the table to complete the graph of the total cost versus the cost of an item.
y
130
120
110
90
80
© 2011 Carnegie Learning
Total Cost (dollars)
100
70
60
50
40
30
20
10
0
10 20 30 40 50 60 70 80 90 100 110 120 130
Cost of an Item (dollars)
638 • Chapter 9 Inequalities and Equations
x
9. Your graph represents the cost of each item and the total cost with shipping from
your table of values. What pattern do you notice?
Would it make sense to connect the points on this graph? In this situation, the cost of an
item can be a fractional value. So, it would make sense to connect the points to show
other ordered pairs that make the equation true.
Problem 2 Working for that Paycheck!
Your friend got a job working at the local hardware store making $6.76 per hour.
1. How much would your friend earn if she worked:
a. 5 hours?
1  ​hours?
b. 2 ​ __
2
2. Explain how you calculated your answers.
© 2011 Carnegie Learning
3. Define variables for the number of hours worked, and for the amount earned.
4. Write an equation that models the relationship between these variables.
9.4 Representing Situations in Multiple Ways • 639
5. Use your equation to calculate the earnings given her hours worked.
a. 6 hours
If I know the
number of hours
worked, what do I
do to that number
to calculate the
amount earned?
b. 5 hours and 30 minutes
© 2011 Carnegie Learning
c. 10 hours and 15 minutes
640 • Chapter 9 Inequalities and Equations
6. Use your equation to calculate the number of hours she worked given her total pay.
Then, check to see if the value of your solutions maintains balance in the original equation.
a. $169
b. $226.46
© 2011 Carnegie Learning
c. $157.17
9.4 Representing Situations in Multiple Ways • 641
7. Complete the table using your answers to Question 1 through Question 6.
Time Worked
(hours)
Earnings
(dollars)
5
2.5
6
5.5
10.25
169
226.46
157.17
8. Use the table to complete the graph of the money earned versus the hours worked.
y
225
175
150
125
100
75
50
25
0
5 10 15 20 25 30 35 40 45
Time Worked (hours)
x
9. Would it make sense to connect the points on this graph? Explain why or why not.
Be prepared to share your solutions and methods.
642 • Chapter 9 Inequalities and Equations
© 2011 Carnegie Learning
Earnings (dollars)
200
Measuring Short
Using Multiple Representations
to Solve Problems
Learning Goal
In this lesson, you will:
 Use multiple representations (words, symbols, tables, and graphs) to solve problems.
H
ow do you get your news? Do you listen to the radio? Watch TV? Read the
newspaper? Check sites online? There are many ways to get the same news story.
The information you might hear on the radio in a 30 second news blurb could also
be talked about on an hour long television special. That same story may also be
mentioned in a brief article in the newspaper or perhaps there is a whole web site
devoted to it online. Even though this news story is presented in different ways,
the basic facts are still the same. Can you think of different ways we represent the
© 2011 Carnegie Learning
same information in mathematics?
9.5 Using Multiple Representations to Solve Problems • 643
Problem 1 Broken Yardstick
Jayme and Liliana need to measure some pictures so they can buy picture frames. They
looked for something to use to measure the pictures, but they could only find a broken
yardstick. The yardstick was missing the first 2 __
​ 1 ​ inches.
2
They both thought about how to use this yardstick.
Liliana said that all they had to do was measure the pictures and then subtract 2 __
​ 1  ​inches
2
from each measurement.
1. Is Liliana correct? Explain your reasoning.
2. They measured the first picture’s length to be 11 inches. What was the
actual length?
3. They measured the first picture’s width to be 9 __
​ 1 ​ inches. What was the actual width?
2
4. Define variables for a measurement with the broken yardstick and the actual
5. Write an equation that models the relationship between these variables.
644 • Chapter 9 Inequalities and Equations
© 2011 Carnegie Learning
measurement.
6. Use your equation to calculate the actual measurement if the measurement taken
with the broken yardstick is:
3  ​inches.
a. 25 ​ __
4
b. 21 inches.
© 2011 Carnegie Learning
5  ​inches.
c. 18 ​ __
8
9.5 Using Multiple Representations to Solve Problems • 645
7. Use your equation to calculate the measurement taken with the broken yardstick
given the actual measurement. Then, check your solution using the original equation.
a. 12 inches.
1  ​inches.
b. 29 ​ __
8
© 2011 Carnegie Learning
7  ​inches.
c. 6 ​ __
8
646 • Chapter 9 Inequalities and Equations
8. Complete the table using your answers from Question 1 through Question 7.
Measurement with
Broken Yardstick
(in.)
Actual
Measurement
(in.)
11
1
9 ​ __  ​ 
2
3
25 ​ __  ​
4
21
5
18 ​ __  ​
8
12
1
29 ​ __  ​ 
8
7
6 ​ __  ​
8
9. Use the table to complete the graph of actual measurement versus the measurement
taken with the broken yardstick.
y
30
Actual Measurement (in.)
© 2011 Carnegie Learning
25
20
15
10
5
0
5
30
25
10
15
20
Measurement with the Broken Yardstick (in.)
x
9.5 Using Multiple Representations to Solve Problems • 647
10. Would it make sense to connect the points on this graph? Explain why or why not.
Problem 2 Biking
Henry is biking to get ready for football season. He records his distances and times
after each ride in a table.
Distance Biked
(km)
Time
(hours)
5
1
__
5 kilometers is
about 3.1 miles.
​    ​ 
4
35
3
1 __
​    ​
4
90
1
4 __
​    ​ 
2
1. Assuming Henry bikes at the same average rate, how long
would it take him to bike:
b. 42 kilometers?
2. Explain how you calculated your answers. Then, complete the last two rows of
the table.
648 • Chapter 9 Inequalities and Equations
© 2011 Carnegie Learning
a. 68 kilometers?
3. Define variables for the distance biked and for the time.
4. Write an equation that models the relationship between these variables.
5. Use your equation to calculate the time it would take Henry to bike:
a. 50 kilometers.
© 2011 Carnegie Learning
b. 60 kilometers.
c. 33.5 kilometers.
9.5 Using Multiple Representations to Solve Problems • 649
6. Use your equation to calculate how far Henry could bike given each amount of time.
Then, check your solution using the original equation.
a. 45 minutes
b. 2 hours
© 2011 Carnegie Learning
1  ​hours
c. 1 ​ __
3
650 • Chapter 9 Inequalities and Equations
7. Use the table and your calculations in Question 1
through Question 6 to complete the graph of the
time in minutes Henry bikes versus the distance
he bikes in kilometers.
When would
it not make sense to
connect points on
a graph?
y
5
Time (hours)
4
3
2
1
0
10 20 30 40 50 60 70 80 90
Distance (km)
x
8. Would it make sense to connect the points on this graph? Explain
why or why not.
© 2011 Carnegie Learning
Be prepared to share your solutions and methods.
9.5 Using Multiple Representations to Solve Problems • 651
© 2011 Carnegie Learning
652 • Chapter 9 Inequalities and Equations
Variables and More Variables
The Many Uses of Variables
in Mathematics
Learning Goal
Key Term
In this lesson, you will:
 homonyms
 Examine the many different uses of variables in mathematics.
I
n English there are many words that have the same spelling and the same
pronunciation, but have different meanings. For example, the word left can mean a
direction, as in “goes to the left.” “Left” can also be the past tense of the word
leave, as in “he just left.” Words like this are called homonyms. Can you think of
© 2011 Carnegie Learning
other homonyms?
9.6 The Many Uses of Variables in Mathematics • 653
Problem 1 A Little History –The Unknown
One of the first artifacts showing the use of mathematics is a cuneiform tablet.
This tablet, from about 1800 b.c., in Babylonia, illustrates mathematical relationships that
are still being studied and learned today. The Babylonians were the first to write equations
that were full sentences, for example, some quantity plus one equals two. The Babylonians
were also the first to use a symbol or word to represent an unknown quantity. This early
algebra was the dominant form of algebra up through 1600 a.d. Thus, we have one
meaning, or use, for a variable as an unknown quantity.
One way variables are used is in solving equations.
1. Solve each equation for the unknown quantity. Then, check the value of your solution.
1  ​5 __
​ x  ​
b. 6 ​ __
3 3
1  ​5 9 __
​ 1 ​ 
c. x 2 3 ​ __
4
3
d. 7.5b 5 189.75
654 • Chapter 9 Inequalities and Equations
In equations,
variables
represent the
unknown value.
© 2011 Carnegie Learning
a. x 1 7 5 13
2. Omar owns 345 more chickens than Henry. If Omar owns 467 chickens, how many
does Henry own?
a. Write an equation for this problem situation.
b. Solve your equation to answer the original question. Then, check the value of
your solution.
c. If possible, write another equation that can be used for this problem.
© 2011 Carnegie Learning
d. Does writing an equation help you solve this problem? Explain your reasoning.
3. There is an unknown number such that when it is multiplied by 5 and the product is
added to 200, the answer is 265. What is the number? Hint: Remember, you can undo
the process. Explain how you calculated your answer.
9.6 The Many Uses of Variables in Mathematics • 655
Problem 2 Variables That Represent All Numbers
1. Use the variables a, b, and c to state each property.
a. Commutative Property of Addition
b. Associative Property of Multiplication
c. Distributive Property of Multiplication over Addition
3. What is true for the equations in these properties and the values of the variables in
these properties that is not true for the equations in Problem 1?
656 • Chapter 9 Inequalities and Equations
© 2011 Carnegie Learning
2. How are the variables in this Problem used differently than in Problem 1?
Problem 3 Variables in Formulas
1. Complete the table to calculate the perimeter of each rectangle.
Length
(units)
Width
(units)
6
4
12
10
25
23
34
26
Perimeter of the
Rectangle (units)
Remember,
perimeter means
the distance
around a
figure.
2. How did you calculate the perimeter?
3. Write the formula for the perimeter of a rectangle. Define your variables.
© 2011 Carnegie Learning
The formula for converting a temperature in Celsius to a temperature in Fahrenheit
is F 5 __
​ 9 ​ C 1 32, where C is the temperature in Celsius, and F is the temperature
5
in Fahrenheit.
4. Complete the table for the given temperatures in Celsius.
Temperature in
Celsius
Temperature in
Fahrenheit
100°C
0°C
25°C
9.6 The Many Uses of Variables in Mathematics • 657
5. How are variables used differently in these formulas than in the previous
two problems?
Problem 4 Variables That Vary
Sherilyn is a bicyclist training for a long-distance bike race. She usually rides her bike at the
rate of 16 miles per hour.
1. If Sherilyn maintains her rate, how far would she cycle in:
a. 4 hours?
1  ​hours?
b. 5 ​ __
2
c. 10 hours and 15 minutes?
2. Define variables for the time she cycles and her distance.
4. Use your equation to calculate the distance Sherilyn would bike in:
b. 3.5 hours.
a. 7 __
​ 3 ​  hours.
4
658 • Chapter 9 Inequalities and Equations
© 2011 Carnegie Learning
3. Write an equation that models the relationship between these variables.
5. If Sherilyn maintains her rate, write and solve your equation to calculate how long it
would take Sherilyn to bike:
a. 100 miles.
b. 50 miles.
6. Complete the table using your answers from Question 1 through Question 5.
Time
(hours)
Distance
(miles)
4
© 2011 Carnegie Learning
1
5 ​ __  ​ 
2
10 hours 15 minutes
3
7 ​ __  ​
4
3.5
100
50
9.6 The Many Uses of Variables in Mathematics • 659
7. Use the table to complete the graph of the time, in hours, that Sherilyn bikes versus
the distance she bikes.
y
165
150
135
Distance (miles)
120
105
90
75
60
45
30
15
0
1
2
3
4
5 6 7 8
Time (hours)
9 10 11 12
x
8. How are variables used differently in this problem than in the previous problems in
Talk the Talk
The concept of variable is a foundation concept in the study of mathematics.
1. Complete the graphic organizer. Show examples of the different ways variables
are used in the study of mathematics.
Be prepared to share your solutions and methods.
660 • Chapter 9 Inequalities and Equations
© 2011 Carnegie Learning
this lesson?
unknowns
Example:
all numbers in mathematical properties
Example:
Variables
Example:
© 2011 Carnegie Learning
Example:
particular quantities
in formulas
Quantities that vary within a problem situation
9.6 The Many Uses of Variables in Mathematics • 661
© 2011 Carnegie Learning
662 • Chapter 9 Inequalities and Equations
Quantities That Change
Independent and
Dependent Variables
Learning Goal
Key Terms
In this lesson, you will:




 Identify and define independent and dependent
variables and quantities.
dependent quantity
independent quantity
independent variable
dependent variable
S
ometimes you get to make choices and other times you do not. Sometimes
making one decision depends on an earlier decision. If you go to a carnival and
decide to pay the ride-all-day price versus paying the admission price and then
paying for each ride, what decisions do you still need to make and what decisions
are already made for you? Can you think of other decisions that you make that
© 2011 Carnegie Learning
then determine other decisions?
9.7 Independent and Dependent Variables • 663
Problem 1 Quantities That Change
In this chapter, you have solved and analyzed problems where quantities changed or
varied. Let’s consider another example, but this time let’s think about how one quantity
depends on another quantity.
Dawson just purchased a new diesel-powered car that averages 41 miles to the gallon.
1. How far does this car travel on:
a. 10 gallons of fuel?
b. a full tank of fuel, 13.9 gallons?
2. There are two quantities that are changing in this problem situation. Name the
quantities that are changing.
3. Does the value of one quantity depend on the value of the other?
4. Define variables for each quantity.
5. Write an equation for the relationship between these variables.
When one quantity depends on another in a problem situation, it is said to be the
The variable that represents the independent quantity is called the independent variable,
and the variable that represents the dependent quantity is called the dependent variable.
6. Identify the independent and dependent variables in this situation.
664 • Chapter 9 Inequalities and Equations
© 2011 Carnegie Learning
dependent quantity. The quantity on which it depends is called the independent quantity.
7. Use your equation to calculate the fuel needed to travel:
a. 1000 miles.
b. 100 miles.
8. Complete the table using your values from Question 1 and Question 7.
Independent Quantity
Distance
Quantity Name
Unit of Measure
Dependent Quantity
gallons
© 2011 Carnegie Learning
Variable
10
13.9
100
1000
9.7 Independent and Dependent Variables • 665
Problem 2 Sometimes One, Sometimes the Other
A store makes 20% profit on each item they sell.
1. Determine the store’s profit in selling an item for:
a. $25.00.
b. $49.95.
Profit is the
extra money for
selling items over and
above the cost of
the items.
c. $99.95.
2. Name the two quantities that are changing.
3. Describe which value depends on the other.
Let c represent the cost of the items in dollars, and let p represent the profit in dollars.
5. Identify the independent and dependent variables in this situation.
666 • Chapter 9 Inequalities and Equations
© 2011 Carnegie Learning
4. Write an equation for the relationship between these variables.
6. Complete the table using your answers from Questions 1 through 5.
Independent Quantity
Dependent Quantity
Profit
Quantity Name
dollars
Unit of Measure
Variable
25
49.95
99.95
7. Use your table to complete the graph.
y
26
24
22
20
© 2011 Carnegie Learning
Profit (dollars)
18
16
14
12
10
8
6
4
2
0
10 20 30 40 50 60 70 80 90 100 120 130 140
Cost (dollars)
x
9.7 Independent and Dependent Variables • 667
Problem 3Looking at Problem Situations
in a Different Way
Let’s think about the problem situation in a different way.
Suppose you are operating this business and you know how much profit you want to
make on each item.
1. How much should an item cost if you want to make:
a. $7.50 profit?
b. $10 profit?
c. $19.99 profit?
3. Describe which value depends on the other.
Let p be equal to the profit, and let c be equal to the cost of the item.
4. Write an equation for the relationship between these variables.
668 • Chapter 9 Inequalities and Equations
© 2011 Carnegie Learning
2. Name the two quantities that are changing.
5. Identify the independent and dependent variables in this situation.
6. Complete the table using your answers from Question 1.
Independent Quantity
Dependent Quantity
Profit
Quantity Name
Unit of Measure
Variable
7.50
10
19.99
7. Use this table to complete the graph.
y
100
90
80
Cost ($)
© 2011 Carnegie Learning
70
60
50
40
30
20
10
0
2
4
6
8 10 12 14 16 18 20 22
Profit ($)
x
9.7 Independent and Dependent Variables • 669
Talk the Talk
The situations in Problems 2 and 3 were similar, but were presented in two
different ways.
Problem 2
Problem 3
The profit depends on the cost of
The cost of the item depends on
the item.
the profit.
p 5 0.2c
p
c 5 ___
​    ​ 
0.2
p
2. Solve c 5 ___
​    ​ for p.
0.2
1. Solve p 5 0.2c for c.
3. What do you notice about the two solutions?
4. How does examining this same situation from different perspectives affect the
independent and dependent variables?
5. What can you conclude about the designation of a variable as independent
or dependent?
Go back and look at the two graphs in Question 7 for both Problem 2 and 3. The graphs
quantities. They are similar because they both represent the independent and dependent on
the same axis.
Dependent
y
x
Independent
Be prepared to share your solutions and methods.
670 • Chapter 9 Inequalities and Equations
© 2011 Carnegie Learning
were labeled differently depending on how you defined the independent and dependent
Chapter 9 Summary
Key Terms
 inequality (9.1)
 graph of an inequality (9.1)
 solution set of an
 inverse operations (9.2)
 Properties of Equality





for Addition and Subtraction (9.2)
inequality (9.1)
 ray (9.1)
 one-step equation (9.2)
 solution (9.2)
 Properties of Equality
for Multiplication and
homonyms (9.6)
dependent quantity (9.7)
independent quantity (9.7)
independent variable (9.7)
dependent variable (9.7)
Division (9.3)
Graphing Inequalities on the Number Line
An inequality is any mathematical sentence that has an inequality symbol. The solution to
any inequality can be represented on a number line by a ray whose starting point is an
open or closed circle. A ray begins at a starting point and goes on forever in one direction.
A closed circle means that the starting point is part of the solution set of the inequality. An
open circle means that the starting point is not a part of the solution set of the inequality.
The graph of an inequality in one variable is the set of all points on a number line that
make the inequality true. This set of points is the solution set of the inequality.
Example
© 2011 Carnegie Learning
Graph the solution set for each inequality.
Your brain is a
hard-working machine.
A balanced lifestyle of
plenty of sleep, exercise,
and healthy food will
keep it that way.
x>6
0
1
2
3
4
5
6
7
8
9
x ≤ 3.5
0
1
2
3
4
5
6
7
8
9
Chapter 9 Summary • 671
Using Models to Represent One-Step Equations
Equations are mathematical statements that declare that two expressions are equal.
Similarly, a balanced scale shows that two quantities are equivalent. Scales can be used
as models to represent equations.
Example
Write an equation that represents the given pan balance. Use the variable x to represent
and use numbers to represent each group of
units. Then, solve the equation.
85x16
8265x1626
25x
In the pan balance, one
is equivalent to 2
units.
Using Inverse Operations to Solve One-Step Equations
To solve an equation means to determine what value or values will replace the variable to
make the equation true. If you can solve an equation using only one operation, the
equation is called a one-step equation. A solution to an equation is any value for a variable
that makes the equation true. To solve an equation, you must isolate the variable using
inverse operation of subtraction, and subtraction is the inverse operation of addition.
Example
In the given equation, state the inverse operation needed to isolate the variable. Then,
solve the equation. Check to see if the value of your solution maintains balance in the
original equation.
13 5 t 2 9
The inverse operation would be to add 9 to both sides.
13 5 t 2 9
13 1 9 5 t 2 9 1 9
22 5 t
672 • Chapter 9 Inequalities and Equations
Check:
13  22 2 9
13 5 13
© 2011 Carnegie Learning
inverse operations. Inverse operations are operations that undo each other. Addition is the
Using Inverse Operations to Solve One-Step Equations
As you learned in Lesson 9.2, to solve an equation means to determine what value or
values will replace the variable to make the equation true. To solve an equation, you must
isolate the variable using inverse operations. Inverse operations are operations that undo
each other. Multiplication is the inverse operation of division, and division is the inverse
operation of multiplication.
Example
In the given equation, state the inverse operation needed to isolate the variable. Then,
solve the equation. Check to see if the value of your solution maintains balance in the
original equation.
5.2x 5 36.4
The inverse operation would be to divide both sides by 5.2.
© 2011 Carnegie Learning
5.2x 5 36.4
Check
36.4
5.2x
____
_____
 
 ​ 5 ​ 
 ​
5.2 3 7  36.4
​ 
5.2
5.2
x 5 7 36.4 5 36.4
Chapter 9 Summary • 673
Representing Situations in Multiple Ways
It is common to use equations, tables, and graphs in order to solve and describe certain
problem situations.
Example
Xavier is preparing to fill his empty swimming pool. The water hose he uses produces
9 gallons of water per minute. Define variables to represent the number of gallons of water
in the swimming pool and the number of minutes Xavier uses the water hose.
Let g represent the number of gallons of water in the swimming pool, and let m represent
the number of minutes Xavier uses the water hose.
Write an equation that models the relationship between these variables.
g 5 9m
Use the equation to determine the number of gallons of water in the swimming pool after:
10 minutes
g 5 9 ∙ 10
g 5 90 minutes
●
20 minutes
g 5 9 ∙ 20
g 5 180 minutes
●
1 hour
g 5 9 ∙ 60
g 5 540 minutes
●
75 minutes
g 5 9 ∙ 75
g 5 675 minutes
●
125 minutes
g 5 9 ∙ 125
g 5 1125 minutes
674 • Chapter 9 Inequalities and Equations
© 2011 Carnegie Learning
●
Using Multiple Representations to Solve Problems
Problem situations can be represented in different ways. Tables and graphs can be
created using information calculated from solving an equation.
Complete the table with your answers from Xavier’s water hose situation.
Time
(minutes)
Amount of Water
(gallons)
10
90
20
180
60
540
75
675
125
1125
Use the table to complete the graph of the number of gallons of water in the pool
Amount of Water (gallons)
© 2011 Carnegie Learning
versus the number of minutes.
Gallons of Water in Xavier’s Pool
y
1200
1000
800
600
400
200
0
20 40 60 80 100 120 140
Time (minutes)
x
Chapter 9 Summary • 675
Examining the Many Different Uses of Variables in Mathematics
Variables are often used to represent numbers in order to describe certain properties.
Variables are also used to represent unknown quantities in equations and formulas.
Example
Variables used in the formula to calculate area of a triangle:
A = area
b = length of the triangle’s base
h = height of the triangle
Calculate the area of a triangle with a base of 12 inches and a height of 15 inches.
A = __
​ 1 ​ (12)(15)
2
A = 90 square inches
Example
Variables used to represent an unknown quantity:
8.5y = 102
8.5y ÷ 8.5 = 102 ÷ 8.5
y = 12
Example
Variables used to describe properties:
Associative Property of Addition for any numbers a, b, and c.
a + (b + c) = (a + b) + c
Example
A car is traveling at a constant speed of 40 miles per hour. How many miles would the car
travel in h hours?
40h = m
676 • Chapter 9 Inequalities and Equations
© 2011 Carnegie Learning
Variables used as quantities that vary in a situation.
Identifying and Defining Independent and Dependent
Variables and Quantities
When one quantity depends on another in a problem situation, it is said to be the
dependent quantity. The quantity on which it depends is called the independent quantity.
The variable that represents the independent quantity is called the independent variable.
The variable that represents the dependent quantity is called the dependent variable.
Example
Ramona works in a crayon factory. The machine she operates produces 200 crayons
per minute.
●
Name the two quantities that are changing in this problem situation.
The two quantities that are changing are the time and the number of crayons produced.
●
Describe which quantity depends on the other.
The number of crayons is the dependent quantity, because the number of crayons
produced depends on the amount of time Ramona operates her machine.
●
Let c represent the number of crayons produced, and let t represent the time (in
minutes) that Ramona operates the machine. Write an equation to represent the
problem situation.
c = 200t
●
Identify the independent and dependent variables in the equation.
The dependent variable is c and the independent variable is t.
© 2011 Carnegie Learning
Chapter 9 Summary • 677
© 2011 Carnegie Learning
678 • Chapter 9 Inequalities and Equations