Inequalities and Equationsmrpack.weebly.com/uploads/8/5/0/5/8505687/course_1_chapter_9.pdf · 9.2 Solving One-Step Equations Using Addition and Subtraction • 619 You just wrote

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603

9.1 CalltoOrderInequalities ............................................................... 605

9.2 OppositesAttracttoMaintainaBalanceSolving One-Step Equations Using

Addition and Subtraction ............................................ 615

9.3 StatementsofEqualityReduxSolving One-Step Equations Using

Multiplication and Division ..........................................625

9.4 therearemanyways...Representing Situations in Multiple Ways ....................635

9.5 MeasuringShortUsing Multiple Representations

to Solve Problems ..................................................... 643

9.6 VariablesandMoreVariablesThe Many Uses of Variables in Mathematics ................653

9.7 QuantitiesthatChangeIndependent and Dependent Variables ............................. 663

InequalitiesandEquations

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.

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604 • Chapter 9 Inequalities and Equations

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9.1 Inequalities • 605

Key Terms inequality

graph of an inequality

solution set of an inequality

ray

Learning GoalsIn this lesson, you will:

Use inequalities to order the number system.

Graph inequalities on the number line.

What 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

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!

CalltoOrderInequalities


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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.

1. For each statement, write the corresponding inequality.

a. 7 is less than or equal to 23

b. 56 is greater than 28

c. 2 is not equal to 5

d. 7.6 is less than 8.2

e. 5 3 __ 4 is greater than 4 2 __

3

Symbol Meaning Example

, less than 3 , 5 3 is less than 5

. 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

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2. Write the meaning of each inequality in words.

a. 7.8 fi 23.7

b. 8 1 __ 3 # 8.7

c. 3 __ 4

$ 0.75

d. 43,256 . 4489

e. 0.012 , 0.02

3. Write , or . to make each inequality true.

a. 12 2 b. 1.2 1.201

c. 3 1 __ 3 3.3 d. 10.25 10 1 __

5

4. Write # or $ to make each inequality true.

a. 1 2 b. 4.2 4 1 __ 4

c. 1 __ 3

0.3 d. 0.25 2 __ 5

e. 24.33 24 1 __ 3

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For any two numbers a and b, only one of the three statements is true.

● a , b

● a . b

● a 5 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

c. 3 4 __ 5

d. 4 1 __ 3

e. 4.66…

0 1 2 3 54

2. There are five points plotted on the number line shown. Identify the approximate location

of each point.

a b dc e

0 1 2 3 4 5

a.

b.

c.

d.

e.

If a fi b, then a must be less than b or greater

than b.

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3. A point at a is plotted on the number line shown.

0

a

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.

0

a

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.

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.

0

a


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Problem 3 Graphing an Inequality on a Number Line

You can use a number line to represent inequalities. The graphofaninequality 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 solutionsetoftheinequality.

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

3 4210

x $ 3

3 4210

Describe each number line representation.

c. How does the solution set of the inequality x $ 3 differ

from the solution set of x . 3?

Why does one graph show a see-through point

and the other one a black point?

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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.

3 4210

x , 3

3 4210

x # 3

Describe each number line representation.

c. How does the solution set of the inequality x # 3 differ from the solution set of x , 3?


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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 1312 16 17 18 1914 15

b. 10 11 1312 16 17 18 1914 15

c. 30 31 3332 36 37 38 3934 35

d. 20 21 2322 26 27 28 2924 25

4. Graph the solution set for each inequality.

a. x # 14

10 11 12 13 14 15

b. x , 55

50 51 52 53 54 55

c. 2 1 __ 2 # x

0 1 2 3 4 5

d. x . 3.3

0 1 2 3 4 5

e. x fi 4.2

0 1 2 3 4 5

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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,

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.

d. You must be at least 13 years old to join.

Be prepared to share your solutions and methods.

"Maximum"” means that the

weight can't go over that amount.


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9.2 Solving One-Step Equations Using Addition and Subtraction • 615

You’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”

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?

Key Terms one-step equation

Properties of Equality for Addition

and Subtraction

solution

inverse operations


Use inverse operations to solve

one-step equations.

Use models to represent one-step

equations.

OppositesAttracttoMaintainaBalanceSolving One-Step Equations Using Addition and Subtraction


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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. Then,

describe your strategies.

1.

a. Strategies:

b. What will balance one rectangle?

You might want to get your algebra

tiles out.

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2.

a. Strategies:


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


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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.

b.

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You just wrote and solved one-stepequations. 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-stepequation. 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 inverseoperations.

Inverseoperations are pairs of operations that undo each other.

4. State the inverse operation for each stated operation.

a. addition

b. subtraction

To isolate the variable

means to get the variable by itself

on one side of the equation.


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Problem 3 Solving Equations

1. Analyze each example and the different methods used to solve each equation.

a. Describe the difference in strategy between Method 1 and Method 2 for Example 1.

b. The final step in each method shows the variable isolated. What is the coefficient

of each variable?

Example1 Example2

a 1 7 5 9 12 5 b 2 8

Method1: Method1:

a 1 7 2 7 5 9 2 7

a 5 2

12 1 8 5 b 2 8 1 8

20 5 b

Method2: Method2:

a 1 7 5 9

27 5 27

a 5 2

12 5 b 2 8

18 5 1 8

20 5 b

The answers are the same. What is different about the two methods?

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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 x 2 8

c. b 1 5.67 12.89

d. 5 3 __ 4 5 x 2 4 1 __

2


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e. 23.563 5 a 2 345.91

f. 7 ___ 12

5 y 1 1 __ 4

g. w 1 3.14 5 27

h. 13 7 __ 8

5 c 1 9 3 __ 4

Don't forget to check your

answers!

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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?

c. Is x 5 2 1 __ 3

a solution to the equation 17 2 __ 3

5 x 1 15 1 __ 3

?

d. Is x 5 4.567 a solution to the equation x 1 19.34 5 23.897?


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Talk the Talk

The PropertiesofEquality 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.

4. How do you check to see if a value is the solution to an equation?

5. Given the solution x 12, write two different equations using the

Properties of Equality.


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9.3 Solving One-Step Equations with Multiplication and Division • 625

Key Term Properties of Equality for

Multiplication and Division


Use inverse operations to solve one-step equations.

Use models to represent one-step equations.

StatementsofEqualityReduxSolving One-Step Equations with Multiplication and Division

In 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

other examples of “balancing” are there in mathematics.


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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:


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2.

a. Strategies:


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


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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.

b.

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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

c. multiplication

d. division


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Problem 3 Solving Equations

1. Analyze each example and the different methods used to solve each equation.

a. Describe the difference in strategy between Method 1 and Method 2 for Example 2.

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?

Example1 Example2

8c 5 48 2 5 d __ 4

Method1: Method1:

8c ___ 8 5 48 ___ 8

c 5 6

2 4 5 d __ 4 4

8 5 d

Method2: Method2:

8c 4 8 5 48 4 8

c 5 6

Method3:

1 __ 8 8c 5 1 __ 8 48

c 5 6

2 5 d __ 4

34 5 34

8 5 d

Looks like there is more than one

way to solve these equations. What's different about each method?

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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.

a. n__ 4

7

b. 3y 18

c. n___ 4.3

5 9.4

d. 3 ___ 10

y 5 3 1 __ 3


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e. y 4 2 __ 3

5 2 1 __ 2

f. 3.14y 81.2004

g. y 2 2 __ 3

5 2 1 __ 2

h. 514 5 81.4 1 x

Don't forget to check your solutions.

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3. Determine if each solution is true. Explain your reasoning.

a. Is p 12 a solution to the equation 9p 108?

b. Is n 4 a solution to the equation n__ 6

24?

c. Is p 18 a solution to the equation 3p 54?


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Talk the Talk

The PropertiesofEquality 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 If a 5 b, and c fi 0, then a __ c 5 b __ 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.

4. How do you check to see if a value is the solution to an equation?

5. Given the solution x 12, write two different equations using the Multiplication and

Division Properties of Equality.


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Represent two quantities that change in words, symbols, tables, and graphs.

Solve one-step equations.

People 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

mathematics would be useful?

ThereAreManyWays...Representing Situations in Multiple Ways

9.4 Representing Situations in Multiple Ways • 635


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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

b. $45.25

c. $67.13

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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

c. $7.67

Don't forget about all the estimation strategies you

have learned! Does your answer make sense in terms of the

problem situation?


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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.

3020Cost of an Item (dollars)

Tot

al C

ost (

dolla

rs)

50 60 70 80 90 100 110 120 1304010

10

0

20

30

40

50

60

70

80

90

100

110

120

130

y

x

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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?

b. 2 1 __ 2 hours?

2. Explain how you calculated your answers.

3. Define variables for the number of hours worked, and for the amount earned.



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5. Use your equation to calculate the earnings given her hours worked.

a. 6 hours

b. 5 hours and 30 minutes

c. 10 hours and 15 minutes

If I know the number of hours

worked, what do I do to that number to calculate the amount earned?

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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

c. $157.17


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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.

1510Time Worked (hours)

Ear

ning

s (d

olla

rs)

25 30 35 40 45205

25

0

50

75

100

125

150

175

200

225

y

x

9. Would it make sense to connect the points on this graph? Explain why or why not.


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9.5 Using Multiple Representations to Solve Problems • 643

How 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

same information in mathematics?

Learning GoalIn this lesson, you will:

Use multiple representations (words, symbols, tables, and graphs) to solve problems.

MeasuringShortUsing Multiple Representations to Solve Problems


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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 __ 2

inches.

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 __ 2 inches

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 __ 2

inches. What was the actual width?

4. Define variables for a measurement with the broken yardstick and the actual

measurement.


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6. Use your equation to calculate the actual measurement if the measurement taken

with the broken yardstick is:

a. 25 3 __ 4

inches.

b. 21 inches.

c. 18 5 __ 8

inches.



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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.

b. 29 1 __ 8

inches.

c. 6 7 __ 8

inches.

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8. Complete the table using your answers from Question 1 through Question 7.

Measurement with Broken Yardstick

(in.)

Actual Measurement

(in.)

11

9 1 __

2

25 3

__ 4

21

18 5

__ 8

12

29 1 __

8

6 7

__ 8

9. Use the table to complete the graph of actual measurement versus the measurement

taken with the broken yardstick.

1510Measurement with the Broken Yardstick (in.)

Act

ual M

easu

rem

ent (

in.)

25 30x

205

5

0

10

15

20

25

30

y


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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 __

4

35 1 3

__ 4

90 4 1 __

2

1. Assuming Henry bikes at the same average rate, how long

would it take him to bike:

a. 68 kilometers?

b. 42 kilometers?

2. Explain how you calculated your answers. Then, complete the last two rows of

the table.

5 kilometers is about 3.1 miles.

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3. Define variables for the distance biked and for the time.


5. Use your equation to calculate the time it would take Henry to bike:

a. 50 kilometers.

b. 60 kilometers.

c. 33.5 kilometers.



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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

c. 1 1 __ 3 hours

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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.

3020Distance (km)

Tim

e (h

ours

)

50 60 70 80 90x

4010

1

0

2

3

4

5

y

8. Would it make sense to connect the points on this graph? Explain

why or why not.



When would it not make sense to

connect points on a graph?


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9.6 The Many Uses of Variables in Mathematics • 653

In 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

other homonyms?

Key Term homonyms


Examine the many different uses of variables in mathematics.

VariablesandMoreVariablesThe Many Uses of Variables in Mathematics

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, somequantityplusoneequalstwo. 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.

a. x 1 7 5 13

b. 6 1 __ 3

5 x__ 3

c. x 2 3 1 __ 4

5 9 1 __ 3

d. 7.5b 5 189.75


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In equations, variables

represent the unknown value.

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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.

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.



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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

2. How are the variables in this Problem used differently than in Problem 1?

3. What is true for the equations in these properties and the values of the variables in

these properties that isnot true for the equations in Problem 1?

Problem 3 Variables in Formulas

1. Complete the table to calculate the perimeter of each rectangle.

Length (units)

Width (units)

Perimeter of the Rectangle (units)

6 4

12 10

25 23

34 26

2. How did you calculate the perimeter?

3. Write the formula for the perimeter of a rectangle. Define your variables.

The formula for converting a temperature in Celsius to a temperature in Fahrenheit

is F 5 9 __ 5 C 1 32, where C is the temperature in Celsius, and F is the temperature

in Fahrenheit.

4. Complete the table for the given temperatures in Celsius.

Temperature in Celsius

Temperature in Fahrenheit

100°C

0°C

25°C

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Remember, perimeter means

the distance around a

figure.


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?

b. 5 1 __ 2 hours?

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:

a. 7 3 __ 4 hours. b. 3.5 hours.

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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

5 1 __

2

10 hours 15 minutes

7 3

__ 4

3.5

100

50

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7. Use the table to complete the graph of the time, in hours, that Sherilyn bikes versus

the distance she bikes.

32Time (hours)

Dis

tanc

e (m

iles)

5 6x

41

15

0

30

45

60

75

98 11 12107

90

105

120

135

150

165

y

8. How are variables used differently in this problem than in the previous problems in

this lesson?

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.



allnumbersinmathematicalproperties

Example:

Quantitiesthatvarywithinaproblemsituation

unknowns

Example:

particularquantitiesinformulas

Variables

Example: Example:

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Key Terms dependent quantity

independent quantity

independent variable

dependent variable


Identify and define independent and dependent

variables and quantities.

Sometimes 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

then determine other decisions?

QuantitiesThatChangeIndependent and Dependent Variables

9.7 Independent and Dependent Variables • 663


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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

dependentquantity. The quantity on which it depends is called the independentquantity.

The variable that represents the independent quantity is called the independentvariable,

and the variable that represents the dependent quantity is called the dependentvariable.

6. Identify the independent and dependent variables in this situation.

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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 Dependent Quantity

Distance

gallons

10

13.9

100

1000

Quantity Name

Unit of Measure

Variable



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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.

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.



Profit is the extra money for

selling items over and above the cost of

the items.

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6. Complete the table using your answers from Questions 1 through 5.


Profit

dollars

25

49.95

99.95

7. Use your table to complete the graph.

3020Cost (dollars)

Pro

�t (d

olla

rs)

50 60 70 80 90 100 120 130 140x

4010

2

0

4

6

8

10

12

14

16

18

20

22

24

26

y

Quantity Name

Unit of Measure

Variable


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Problem 3 Looking 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?

2. Name the two quantities that are changing.

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.


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6. Complete the table using your answers from Question 1.


Profit

7.50

10

19.99

7. Use this table to complete the graph.

64Pro�t ($)

Cos

t ($

)

10 12 14 16 18 20 22x

82

10

0

20

30

40

50

60

70

80

90

100

y

Quantity Name

Unit of Measure

Variable

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Talk the Talk

The situations in Problems 2 and 3 were similar, but were presented in two

different ways.

Problem2 Problem3

The profit depends on the cost of The cost of the item depends on

the item. the profit.

p 0.2c c p___

0.2

1. Solve p 0.2c for c. 2. Solve c p___

0.2 for p.

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

were labeled differently depending on how you defined the independent and dependent

quantities. They are similar because they both represent the independent and dependent on

the same axis.

Dep

ende

nt

Independentx

y


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Chapter 9 Summary

Key Terms inequality (9.1)

graph of an inequality (9.1)

solution set of an

inequality (9.1)

ray (9.1)

one-step equation (9.2)

solution (9.2)

inverse operations (9.2)

Properties of Equality

for Addition and

Subtraction (9.2)

Properties of Equality

for Multiplication and

Division (9.3)

homonyms (9.6)

dependent quantity (9.7)

independent quantity (9.7)

independent variable (9.7)

dependent variable (9.7)

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

Graph the solution set for each inequality.

x > 6

0 1 2 3 4 5 6 7 8 9

0 1 2 3 4 5 6 7 8 9

x ≤ 3.5

Your brain is a hard-working machine. A balanced lifestyle of

plenty of sleep, exercise, and healthy food will

keep it that way.

Chapter 9 Summary • 671


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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.

8 x 1 6

8 2 6 x 1 6 2 6

2 x

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 operations. Inverse operations are operations that undo each other. Addition is the

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 t 2 9

The inverse operation would be to add 9 to both sides.

13 t 2 9 Check:

13 1 9 t 2 9 1 9 13 22 2 9

22 t 13 13

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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 36.4

The inverse operation would be to divide both sides by 5.2.

5.2x 36.4 Check

5.2x____ 5.2

36.4 _____ 5.2

5.2 3 7 36.4

x 7 36.4 36.4


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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 9m

Use the equation to determine the number of gallons of water in the swimming pool after:

● 10 minutes

g 9 ∙ 10

g 90 minutes

● 20 minutes

g 9 ∙ 20

g 180 minutes

● 1 hour

g 9 ∙ 60

g 540 minutes

● 75 minutes

g 9 ∙ 75

g 675 minutes

● 125 minutes

g 9 ∙ 125

g 1125 minutes

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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

versus the number of minutes.

6040Time (minutes)

Gallons of Water in Xavier’s Pool

Am

ount

of W

ater

(gal

lons

)

100 120 1408020

200

0

400

600

800

1000

1200y

x


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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 __ 2

(12)(15)

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

Variables used as quantities that vary in a situation.

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

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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.


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Inequalities and Equationsmrpack.weebly.com/uploads/8/5/0/5/8505687/course_1_chapter_9.pdf · 9.2 Solving One-Step Equations Using Addition and Subtraction • 619 You just wrote