1 Linear Functions, Equations, and Inequalities...

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1.1 Tanks a Lot

Introduction to Linear Functions ● p. 3

1.2 Calculating Answers

Solving Linear Equations and Linear

Inequalities in One Variable ● p. 11

1.3 Running a 10K

Slope–Intercept Form of Linear

Functions ● p. 21

1.4 Pump It Up

Standard Form of Linear Functions ● p. 29

1.5 Shifts and Flips

Basic Functions and Linear

Transformations ● p. 37

1.6 Inventory and Sand

Determining the Equations of Linear

Functions ● p. 47

1.7 Absolutely!

Absolute Value Equations and

Inequalities ● p. 55

1.8 Inverses and Pieces

Functional Notation, Inverses, and

Piecewise Functions ● p. 67

Inventory is the list of items that businesses stock in stores and warehouses to supply customers.

Businesses in the United States keep about 1.5 trillion dollars worth of goods in inventory. You will

use linear functions to manage the inventory levels of a business.

1C HA PT E R

Linear Functions, Equations,and Inequalities

Chapter 1 ● Linear Functions, Equations, and Inequalities 1

1

2 Chapter 1 ● Linear Functions, Equations, and Inequalities

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Mathematical RepresentationsINTRODUCTION Mathematics is a human invention, developed as people

encountered problems that they could not solve. For instance, when people

first began to accumulate possessions, they needed to answer questions such

as: How many? How many more? How many less?

People responded by developing the concepts of numbers and counting.

Mathematics made a huge leap when people began using symbols to

represent numbers. The first “numerals” were probably tally marks used to

count weapons, livestock, or food.

As society grew more complex, people needed to answer questions such as:

Who has more? How much does each person get? If there are 5 members in

my family, 6 in your family, and 10 in another family, how can each person

receive the same amount?

During this course, we will solve problems and work with many different

representations of mathematical concepts, ideas, and processes to better

understand our world. The following processes can help you solve problems.

Discuss to Understand

• Read the problem carefully.

• What is the context of the problem? Do you understand it?

• What is the question that you are being asked? Does it make sense?

Think for Yourself

• Do I need any additional information to answer the question?

• Is this problem similar to some other problem that I know?

• How can I represent the problem using a picture, a diagram,

symbols, or some other representation?

Work with Your Partner

• How did you do the problem?

• Show me your representation.

• This is the way I thought about the problem—how did you think about it?

• What else do we need to solve the problem?

• Does our reasoning and our answer make sense to one another?

Work with Your Group

• Show me your representation.

• This is the way I thought about the problem—how did you think

about it?

• What else do we need to solve the problem?

• Does our reasoning and our answer make sense to one another?

• How can we explain our solution to one another? To the class?

Share with the Class

• Here is our solution and how we solved it.

• We could only get this far with our solution. How can we finish?

• Could we have used a different strategy to solve the problem?

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Lesson 1.1 ● Introduction to Linear Functions 3

1ObjectivesIn this lesson, you will:

● Define dependent and independent

variables.

● Write linear functions.

● Graph linear functions.

● Use multiple representations of linear

functions to model and solve problems.

Key Terms● variable

● independent variable

● dependent variable

● function

● linear function

1.1 Tanks a LotIntroduction to Linear Functions

Problem 1An oil storage tank farm has an empty tank with a capacity of 5000 gallons. It will

be filled with motor oil using a pipe that fills the tank at the rate of twelve gallons per

minute.

1. How much motor oil will be in the tank after 20 minutes? One hour? Two hours?

2. After how many minutes will there be exactly 1000 gallons of motor oil in

the tank?

3. In this problem, there are two quantities that are changing. What are they? One

quantity, the dependent quantity, depends on the other, the independent quantity.

a. Independent quantity:

b. Dependent quantity:

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4. Assign a variable to each of the quantities and write an equation that shows the

relationship between these variables. The variable assigned to the independent

quantity is called the independent variable, and the variable assigned to the

dependent quantity is called the dependent variable.

5. Use the answers to Questions 1 through 4 to complete the following table to

identify the quantities that are changing, the units that are used to measure

these quantities, and the expressions representing each of these quantities.

Then construct a graph.

Quantity Name

Unit

Expression

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6. How would you describe the graph of this situation?

7. Use your equation to determine how much oil is in the tank after two thirds

of an hour.

8. Use your equation to solve for the number of minutes it would take to have

exactly 2234 gallons of oil in the tank.

9. How much motor oil is added to the tank every minute? In 10 minutes? In one hour?

10. For what interval of time will there be less than 1000 gallons of oil in the tank?

11. Calculate the amount of time it takes to fill an empty tank. How did you get your

answer?

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Problem 2A second tank that is currently full contains 20,000 gallons of home heating oil. This

tank will be emptied through a drain pipe at the rate of twenty five gallons per minute.

1. How much heating oil will be in the tank after 10 minutes? One hour? Ten hours?

2. After how many minutes will there be exactly 10,000 gallons of heating oil in the

tank?

3. In this problem, there are two quantities that are changing. What are they?

a. Independent quantity:

b. Dependent quantity:

12. Choose different values from two rows in your table in Question 5. Solve for the

change in the dependent variable and the change in the independent variable.

Calculate the quotient obtained when the change in the dependent variable is

divided by the change in the independent variable.

13. Compare your answer with the other students in your group, and then compare

your answer with another group. Explain all similarities and differences.

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4. Assign a variable to each of the quantities and write an equation that shows the

relationship between these variables.

a. What does 20,000 represent in this equation with respect to the problem

situation?

b. What does 25 represent in this equation with respect to the problem situation?

Use the answers to Questions 1 through 4 to complete the table. Then construct a

graph.

Quantity Name

Unit

Expression

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5. How would you describe the graph of this situation?

6. Use your equation to calculate the amount of oil remaining at exactly 37

minutes into emptying the tank.

7. Use your equation to calculate the number of minutes it would take for there to

be exactly 675 gallons of oil left in the tank.

8. Calculate the total amount of time it takes to empty the tank. Explain how you

got your answer.

9. For what interval of time will there be more than 10,000 gallons but less than

15,000 gallons of oil in the tank?

10. For every minute, how much heating oil is drained from the tank? In 10 minutes?

An hour?

1

11. Choose different values from two rows in your table. Solve for the change in the

dependent variable. Solve for the change in the independent variable. Calculate

the quotient obtained when the change in the dependent variable is divided by

the change in the independent variable.

12. Compare your answer with the other students in your

group, and then compare your answer with another

group. Explain all similarities and differences.

13. Problems 1 and 2 are examples of mathematical

relations called linear functions. What about their

graphs indicates this?

14. List at least three characteristics of linear functions.

Be prepared to share your work with another pair, group, or the entire class.

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RememberA function is a relation

that maps each value of the

independent variable to

one and only one value of

the dependent variable.

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Lesson 1.2 ● Solving Linear Equations and Linear Inequalities in One Variable 11

ObjectivesIn this lesson, you will:

● Solve one-step and two-step linear

equations.

● Solve and graph linear inequalities.

Key Terms● transformations

● simplification

● inequality

● number line

1.2 Calculating AnswersSolving Linear Equations and LinearInequalities in One Variable

Problem 1When you wrote an equation for the situation in Problem 1 in Tanks a Lot, you

defined variables to represent the quantities that were changing and used those

variables to write an equation that showed the relationship between the quantities.

You then used the equation to answer a number of questions about the situation.

Use that same equation to find out how long it will take for the tank to contain

1200 gallons of motor oil.

When you have an equation with two variables, and you substitute a number for one

of the variables, you are left with a linear equation to be solved for the remaining

variable. To solve an equation of this type, there are four basic transformations that

you can apply to both sides of the equation:

● Addition

● Subtraction

● Multiplication

● Division

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To solve the equation, you also have to perform a simplification. The simplifications

most commonly used are:

● Add/Subtract terms

● Perform multiplication

● Simplify fractions

● Simplify signs

● Distribute

In each of the following equations, perform the transformation and simplification,

solve the equations, and indicate which transformation and simplification you used.

Show each step separately.

Example:Transformations/Simplifications Used

Divide both sides by 4

Simplify fractions

1. x � 25 � 10 Transformations/Simplifications Used

2. 22 � m � 37 Transformations/Simplifications Used

3. Transformations/Simplifications Usedw

6.2� �3.3

y � 25

4y4

�100

4

4y � 100

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Problem 2In an oil storage tank farm, one tank that is currently full contains 20,000 gallons

of home heating oil. This tank will be emptied through a drain pipe at the rate of

twenty five gallons per minute.

In Problem 2 from the previous lesson, you defined variables to represent the

quantities that were changing and used these variables to write an equation that

showed the relationship between the quantities. You then used the equation to answer

a number of questions about the situation. Use the equation 20,000 � 25t � g to

determine how long it will take for the tank to contain 15,000 gallons of heating oil.

In this case, to solve the equation, you had to perform two transformations and two

simplifications.

In each of the following equations, perform the transformations and simplifications,

solve the equations, and indicate which transformations and simplifications you

used. Make sure to show each step separately.

1. 2w � 81 � �141 Transformations/Simplifications Used

2. �2x � 13 � �10 Transformations/Simplifications Used

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3. �22 � 7m � 37 Transformations/Simplifications Used

4. Transformations/Simplifications Used

5. �2(3k � 4) � �12 Transformations/Simplifications Used

�f3.2

� 2.4 � �3.3

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6. �2(4w) � 6 � �15 Transformations/Simplifications Used

7. �1.2s � (�9) � �1.5 Transformations/Simplifications Used

8. �3d � 4d � �12 � 5d Transformations/Simplifications Used

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10. Transformations/Simplifications Used3x4

� 7 � 11

9. 3x � 5 � 7x � 4 Transformations/Simplifications Used

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Problem 3In the first Tanks a Lot problem, you were asked: For what interval of time will there

be less than 1000 gallons of oil in the tank? This actually produces a linear inequality

rather than an equation. The five different inequality

symbols that are used in linear inequalities are listed.

Next to each symbol, write its meaning.

�

�

�

�

For each of the following inequalities, perform the

transformations and simplifications, solve the inequalities,

and indicate which transformations and simplifications you

used. Make sure to show each step separately. The

answers to most inequalities are an infinite set of numbers,

so we often graph these sets on a number line by using

closed or open endpoints with shading of the line.

Example:Transformations/Simplifications Used

Add 7 to both sides

Add/Subtract terms

Divide both sides by 2

Simplify fractionsx � 8

2x2

�162

2x � 16

2x � 7 � 7 � 9 � 7

2x � 7 � 9

Take NoteThere is no difference in using

transformations and

simplifications with inequalities

except when you multiply or

divide both sides by a

negative number. If x � 5,

then x is any number larger

than 5, but if we multiply or

divide both sides of this

inequality by a negative

number, for example �3, then

we get �3x � �15. If we let

x � 6, we have �3(6) � �15

but �18 is not larger than

�15. So when we multiply or

divide by a negative, we must

also reverse the direction of

the inequality sign.

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–10 –8 –6 –4 –2 0 2 4 6 8 10


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3. �22 5m � 47 Transformations/Simplifications Used

1. 3c � 8 � �13 Transformations/Simplifications Used

2. �2x � 16 � �20 Transformations/Simplifications Used

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4. Transformations/Simplifications Usedh4

� 56 � 54

5. �2(3r � 4) � �12 Transformations/Simplifications Used


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Lesson 1.3 ● Slope–Intercept Form of Linear Functions 21

Problem 1A runner is participating in a 10-kilometer road race and she is currently at the halfway

point. She looks at her watch and notices that it has taken her exactly twenty five

minutes to get to this point. Assume that she has run at a constant rate and will

complete the race at this same rate.

1. Calculate the rate she is running by first solving for the amount of time it will

take her to run 1 km. Explain how you determined her rate.

2. At which kilometer mark is she currently?

ObjectivesIn this lesson, you will

● Define and calculate the slope of a linear

function.

● Define and calculate the y-intercept of a

linear function.

● Write linear equations in slope–intercept

form.

● Use the formula to calculate the slope of

a linear equation.

● Graph linear functions using the slope

and y-intercept.

Key Terms● slope

● y-intercept

● slope-intercept form

1.3 Running a 10KSlope–Intercept Form of Linear Functions

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3. If she continues at this rate, how much of the race will she have completed in

10 more minutes? In 12 more minutes?

4. If she has currently been running for 25 minutes, how much of the race did she

complete 15 minutes ago?

5. How many minutes would it take her to complete 8 kilometers of the race?

Three fourths of the race?

6. What are the quantities that are changing after she has reached the 5 km mark?

a. Independent quantity?

b. Dependent quantity?

7. Assign a variable to each of the quantities, and write an equation that shows the

relationship between these variables.

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9. Use your graph to estimate how much of the race will be left after she has run

20 more minutes.

10. For each minute she runs, how far does she go?

This is her unit rate of change. In a linear function, this unit rate of change is

called the slope.

Quantity Name

Unit

Expression

8. Use the answers to Questions 1 through 7 to complete the following table,

making sure to identify the quantities that are changing, the units that are used

to measure these quantities, and the expressions representing each of these

quantities. Then construct a graph.

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Problem 2Linear functions are often written in the form y � mx � b where m is the _________

and b is the ______________. This form is called the slope–intercept form of a

linear function.

The slope is the unit rate of change. Each time the independent variable changes by

one unit, the dependent variable increases or decreases by the value of the slope.

The slope can be found by calculating the change in the dependent variable divided

by the change in the independent variable.

This is often written in a shorthand form as the following formula where is read

as “the change in,” (x1, y

1) and (x

2, y

2), two ordered pairs or data points from the

function, where y is the dependent variable, and x is the independent variable.

Calculate the slope and y-intercept for each of the following linear functions:

1. y � �5x � 2.3

2. A linear function that passes through the points (0, 5) and (2, �5)

m �yx

�y2 � y1

x2 � x1

m �change in dependent quantity

change in independent quantity

11. Where is she currently? Describe the location of this point on the graph. What

do we call this point on the graph?

12. In the equation you wrote for this situation in Question 7, are the slope and

y-intercept obvious? Explain. 1

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3. A linear function that passes through the points (2, 4) and (�2, 7)

4. A linear function that passes through the points (8, �5) and (�6, 9)

If the linear function is in the slope–intercept form, it enables you to graph the

function quickly and easily using the y-intercept as an initial-value starting point

and the slope as a unit rate of change.

5. For instance, in the equation y � 3x � 5, identify the slope and y-intercept.

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7. y � �2x � 4 Slope � y-intercept �

For each of the following linear functions in slope–intercept form, identify the slope

and the y-intercept, and then construct their graphs using the slope and

y-intercept.

6. Using the equation y � 3x � 5, plot the y-intercept on the graph. Then plot the

next point by moving one unit to the right and then moving up, if the slope

is positive, or down, if the slope is negative, by the value of the slope. From this

second point, repeat the process to plot as many points as you wish, and then

draw your line through these points.1

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8. y � �4x Slope � y-intercept �

9. y � �3x � 5 Slope � y-intercept �

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11. y � 6 Slope � y-intercept �

10. Slope � y-intercept �y � �23

x � 2

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Lesson 1.4 ● Standard Form of Linear Functions 29

Problem 1After a water main break, a large building’s basement was flooded to the ceiling.

The local fire department provided two pump trucks to pump the water out of the

basement. The first truck, No. 175, can pump 25 cubic feet of water per minute,

and the second truck, No. 236, can pump 32 cubic feet of water per minute.

The building’s basement is a large rectangular prism, measuring 150 feet long,

120 feet wide, and 10 feet deep.

1. How many cubic feet of water are there in the basement? Explain how you

determined this answer.

2. If both pump trucks are used for different amounts of time to pump out the

basement, define variables for the time that each truck pumps, and then write

an equation that represents this situation.


● Write linear equations in standard form.

● Graph linear functions in standard form

using intercepts.

● Transform linear equations in standard

form to slope–intercept form.

Key Term● standard form of a linear equation

1.4 Pump It UpStandard Form of Linear Functions

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3. If Truck No. 175 works for an hour, how many minutes must the other truck

work to completely empty the basement?

4. If Truck No. 236 works for 90 minutes, how many minutes must the other truck

work to completely empty the basement?

5. If only Truck No. 175 is used, how long would it take to empty the basement?

Truck No. 236?

6. If Truck No. 236 empties 20,000 cubic feet of water, how many minutes must

the other truck work to completely empty the basement?

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10. What kind of function does this situation produce? Explain how you know.

7. If Truck No. 175 empties 20,000 cubic feet of water, how many minutes must

the other truck work to completely empty the basement?

8. If Truck No. 175 works for three and one half hours, how many cubic feet of

water are left for the other truck to pump?

9. Complete the following table and graph the times for each truck.

Quantity Name

Unit

Expression

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11. Using two data points/ordered pairs from your table, calculate the slope. What

is the y-intercept?

12. Rewrite this equation in slope–intercept form.

x-intercept y-intercept

Problem 2A linear function of the form Ax � By � C where A, B, and C are integer constants

with no common factor larger than 1 is said to be the standard form of a linear

equation. One advantage of the standard form is that it enables you to calculate

both the x- and y-intercepts easily. For each of the following linear equations written

in standard form, calculate both the x- and y-intercepts, and then use these inter-

cepts to graph the function.

1. �4x � 9y � 144

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3. 8x � 7y � 56

2. �5x � 7y � 35


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When a linear equation is written in standard form, it can be transformed into

the slope–intercept form by solving for y. For each of the following linear

equations in standard form, transform it into slope–intercept form, calculate the

slope and y-intercept, and construct its graph.

4. 3x � 6y � 15

slope � y-intercept �

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6. 8x � 7y � 56


5. �5x � y � �11

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7. y � 3x � 4 8.

9. y � 3.4x � 5.6 10.


y �7

4x �

9

8

y � �2

3x � 6

Sometimes you may be asked to transform a linear function that is in

slope–intercept form into standard form. For each of the following linear

functions in slope–intercept form, transform it into standard form.

1

Use the table to graph the functions, and indicate the

transformations, both in terms of transforming the equation

and the graph, which were performed on the basic function

to arrive at the transformed function.

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Lesson 1.5 ● Basic Functions and Linear Transformations 37

Problem 1We have now worked with two different forms of linear functions, slope–intercept

form and standard form. You should remember that a function is defined as a

relation for which every input value has one and only one output value. We are going

to look at linear functions from the view of a family of functions. The most basic

form for a linear function is

y � x

which is called the basic function. Any linear function can

be constructed through a series of transformations to the

basic function.


● Define basic functions.

● Use translations, dilations, and

reflections to transform linear functions.

● Graph parallel lines.

● Graph perpendicular lines.

Key Terms● basic function

● dilation

● reflection

● line of reflection

1.5 Shifts and FlipsBasic Functions and Linear Transformations

RememberA dilation is a transformation

of a figure in which the figure

stretches or shrinks with

respect to a fixed point.

RememberA reflection is a

transformation in which a

figure is reflected, or flipped,

in a given line called the

line of reflection.

1

Algebraic Graphical

Transformations Transformations

Add a constant Shift up

Subtract a constant Shift down

Multiply or divide by a

positive constantDilation

Multiply by �1 Reflection

1

Algebraic transformation:

Graphical transformation:


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1. Basic function y � x



2. y � x � 3

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3. y � x � 4



4. y � 2x



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5. y � 2x � 1



6. y � �3x



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7. y � �4x � 1



8. y � 3x � 5



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9. y �23

x � 1



10. y � �12

x � 3



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For each of the following equations of linear functions, describe the transformations

you would need to perform to the graph of the basic function in order to transform it

into the given function.

11. y � 4x

12. y � x � 7

13. y � �2x � 7

14. y � �7x � 11

Problem 2Graph the following equations on the same grid.

1. y � 2x and y � 2x � 5

2. Describe how the graphs are related geometrically.

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3. Graph the following equations on the same grid.

y � �3x and y � �3x � 5

4. Describe how the graphs are related geometrically.

5. What conclusion might you make about equations with the same slope?

6. The graphs of y � 2x and are shown on the graph.y � �1

2x

From the x-axis, draw a line segment vertically from (2, 0) to the line y � 2x to

form a right triangle. From the x-axis, draw a line segment vertically from (�4, 0)

to the line y � to form a second right triangle.�12

x

1

4

2

6

8

–4

6 8 4 –6 –4 –8 –2

y

x

–8

–6

y = 2x

y = x– 1 2

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7. Using what you know from geometry, why are the two triangles congruent?

What can you conclude about the angles formed by the intersecting lines

y � 2x and y � ?�1

2x

8. The graphs of y � 3x and are on the grid.y � �13

x

From the x-axis, draw a line segment vertically from (2, 0) to the line y � 3x to

form a right triangle. From the x-axis, draw a line segment vertically from (�4, 0)

to the line y � to form a second right triangle.

9. Using what you know from geometry, why are the two right triangles congruent?

What can you conclude about the angles formed by the intersecting lines y � 3x

and y � ?

10. What conclusion can you draw about linear functions with related slopes?


�13

x

�1

3x

11

4

2

6

8

6 8 4 –6 –4 –8 –2

y

x

–8

–6

y = 3x

y = x– 1 3


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Lesson 1.6 ● Determining the Equations of Linear Functions 47

Problem 1A company discovers that the person who was to keep track of the inventory of a

vital component left without leaving any records. Assume that the company uses the

same number of these components every day, and they had 724 on hand on the

10th of the month. Three days later they have 688 on hand.

1. How many components do they use per day on average?

2. How many components did they have at the beginning of the month?

3. Define variables for both the independent and dependent variables, and write a

linear function that represents the number of components on hand based on the

10th day of the month.


● Determine the equation of linear

functions when

● given the slope and the y-intercept.

● given the slope and one point on

the line.

● given two points on the line.

● given the equation of a line parallel to

the line and a point on the line.

● given the equation of a line

perpendicular to the line and a point

on the line.

Key Terms● point-slope form

● two-point form

● parallel lines

● perpendicular lines

1.6 Inventory and Sand Determining the Equationsof Linear Functions 11


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4. Use the function to determine how long before the inventory is gone.

Problem 2A company that sells river sand starts the week with 15,000 tons of sand, and they

sell about 15 tons of sand per day.


linear function that represents the amount of sand on hand based on the day.

2. Use the function to calculate how long before there are only 8000 tons of sand.

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Problem 3A gasoline station sells approximately 1250 gallons of gas per day. Assume that they

sell the same amount per day and they receive a shipment at the beginning of each

month. On the fifth day after a delivery, they have 26,450 gallons on hand.

1. How many gallons of gas do they sell per day on average?

2. How many gallons of gas did they have at the beginning of the month?


linear function that represents the amount of gas on hand based on the number

of days since the delivery.

4. Use the function to calculate how long before the amount of gas will reach

5000 gallons.

In each case, you defined a linear function in order to model a situation and

then used the function to answer important questions. The ability to write a

linear function to model a situation is very useful. In Problem 1, you were

given two data points or ordered pairs. In Problem 2, you were given an initial

value (y-intercept) and the unit rate of change (slope). In Problem 3, you were

given one data point and the unit rate of change. The following is a list of

different situations that you may be presented with for which you will need to

write the equation of a linear function.

A. Given the slope and the y-intercept (Problem 2)

B. Given the slope and one point on the line that is not the y-intercept (Problem 3)

C. Given two points on the line (Problem 1)

D. Given a point that the line passes through and the equation of a parallel line

E. Given a point that the line passes through and the equation of a perpendicular line

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For each of these situations, there are several ways to determine the equations of

the linear functions.

A. Given the slope and the y-intercept (Problem 2)

The equation can be written directly using the slope–intercept form

y � mx � b

Example: Slope � 4 and y-intercept � �5

y � 4x � 5

B. Given the slope and one point on the line that is not the y-intercept (Problem 3)

You can determine the equation using the slope–intercept form by substituting

in the value of m and then substituting the x and y coordinates from the

ordered pair in y � mx � b and solving for b.

Example: Slope � �2 and the line passes through the point (3, 4)

y � �2x � b substitute for m

4 � �2(3) � b substitute in x and y coordinates

10 � b

y � �2x � 10

You can use another form called the point–slope form, ( y � y1) � m(x � x

1)

where m is the slope and (x1, y

1) is a point on the line.

Example: Slope � �2 and the line passes through the point (3, 4)

C. Given two points on the line (Problem 1)

You can determine the equation using the slope–intercept form by calculating the

slope by using the slope formula and then substituting the x

and y coordinates from one of the ordered pairs in y � mx � b and solving for b.

Example: Determine the equation of the line that passes through (2, 4) and (�3, 14).

y � �2x � 8

8 � b

4 � �2(2) � b

y � �2x � b

m �yx

�y2 � y1

x2 � x1�

14 � 4�3 � 2

�10�5

� �2

m �yx

�y2 � y1

x2 � x1

y � �2x � 10

y � 4 � �2x � 6

( y � 4) � �2(x � 3)

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You can use another form called the two–point form:

Example: Determine the equation of the line that passes through (2, 4) and (�3, 14).

D. Given a point that the line passes through and the equation of a parallel line

You can determine the equation of the line by identifying the slope of the given line.

Parallel lines must have equal slopes. Then calculate the y-intercept by substituting

the x and y coordinates from the ordered pair in y � mx � b and solving for b.

Example: Determine the equation of a line parallel to the line y � 4x � 7 and

passing through the point (�2, �3).

The slope of the given line is 4.

E. Given a point that the line passes through and the equation of a perpendicular line

You can determine the equation of the line by identifying the slope of the given

line. Perpendicular lines have slopes that are negative reciprocals. Then

calculate the y-intercept by substituting the x and y coordinates from the ordered

pair in y � mx � b and solving for b.

Example: Determine the equation of the line perpendicular to the line

y � 3x � 6 and passing through the point (�4, 5).

y � 4x � 5

5 � b

�3 � 4(�2) � b

y � 4x � b

y � �2x � 8

�2x � 4 � y � 4

�2(x � 2) � y � 4

10

�5�

y � 4

x � 2

14 � 4�3 � 2

�y � 4x � 2

y2 � y1

x2 � x1�

y � y1

x � x1

11

The slope of the given line is 3, and the slope of the perpendicular line is .

For each of the following situations, determine the equation of the line.

1. The line has a slope of �4 that passes through the point (4, 0).

2. The line passes through the points (3, �7) and (�5, 9).

3. The line is parallel to the line 2x � 4y � 9 and passes through the point (1, 1).

y � �13

x �113

113

� b

5 � �13

(�4) � b

y � �13

x � b

�13


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4. The line is perpendicular to the line and passes through the point

(0, �2).

5. The line has a slope of and y-intercept of 3.4.

6. The line is perpendicular to the line y � �x � 7 and passes through the point

(10, �5).

7. The line has a slope of �9 and a y-intercept of �50.

8. The line passes through the points (�9, �5) and (�2, 4).

�23

y � �15

x �65

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9. The line is parallel to the line y � x � 9 and passes through the point (�9, 5).

10. The line has a slope of 2.1 and passes through the point (�3, 7).


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Lesson 1.7 ● Absolute Value Equations and Inequalities 55

Problem 1 Solving Absolute Value EquationsAbsolute value is a concept that has wide application in higher mathematics, and

you may remember that the absolute value of a number is the distance from the

number to zero on the number line. Distance is always positive. The more technical

definition is

1. Calculate the absolute value of each of the following:

a. |7| �

b. |�17| �

c. |�101| �

d. |0| �

e. �

2. Calculate the value(s) for each of the following variables that have the given

absolute values:

a. |a| � 9

b. |c| � 3.4

c. |y| � 0

d. |d| � �5

� 7

3 �

|x| � x if x � 0

|x| � �x if x � 0


● Write absolute value equations and

inequalities in one and two variables.

● Solve absolute value equations and


● Graph absolute value equations and


Key Terms● absolute value

● absolute value equation

● absolute value inequality

● compound inequality

1.7 Absolutely!Absolute Value Equations and Inequalities

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When we solve a linear equation with one variable, we can perform the various

transformations and simplifications to isolate the variable and calculate our solution.

An absolute value equation contains an absolute value and may have more than

one solution. For instance:

Add �5 to both sides

Combine like terms

or

Add �5 to both sides

Combine like terms

3. Solve each of the following equations:

a. |x � 5| � 10

b. |2x � 5| � 17

x � 3 or x � �13

x � 5 � �5 � �8 � �5

x � 5 � �8

|x � 5| � 8

x � 3 or x � �13

x � 5 � �5 � 8 � �5 or x � 5 � �5 � �8 � �5

x � 5 � 8 or x � 5 � �8

|x � 5| � 8

c. 12.4 � |4x � 6.5|

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d. |�5x � 2| � 18

e. |7x � 5| � �9

4. What do you notice about the solution(s) to the equations in Question 3? How do

the solutions differ from those of other equations with one variable that you have

solved before?

Problem 2 Graphing Absolute Value FunctionsWhen we solve linear equations in two variables, the solution is a set of ordered

pairs that satisfies the equation.

1. Graph the following equation on the grid:

y � x � 2

a. For what values of x are the values of y � 0?

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b. Graph on the same grid by plotting the points in the table below:y � | x � 2 |

x

0

3

�2

�4

�5

y

c. What do you notice about this graph?

d. What is the least value of y that is a solution of ? For what value of

x does this occur? What is this ordered pair called?

2. Graph the solutions for each of the following equations:

a. y � | x � 3 |

y � | x � 2 |

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Problem 3 Solving Linear Inequalities withAbsolute Value in One Variable

Linear inequalities that contain absolute value, an absolute value inequality, may

have infinite solutions, and the solutions are represented by segments or rays on the

number line.

1. To solve the inequality , first solve the equation.

a. |x � 2| � 4

|x � 2| � 4

b. Graph these two points on the number line below.

c. Will these endpoints be included in this inequality? How do you know?

d. These two points divide the number line into three distinct portions; choose a

number from each of the regions and substitute them into the inequality to

determine which of these portions satisfy the inequality. Shade the portion(s)

that satisfy the inequality.

e. We can also solve the inequality algebraically by rewriting it as a compound

inequality.

Can be rewritten as

Why?

x � 2 � 4 and x � 2 � �4

| x � 2| � 4

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f. Solve the compound inequality. How does this solution correspond to the

solution you calculated in part (a)?

When a solution is confined in a region, both the lower and upper bounds must

be specified.

2. Solve the following inequality graphically and then algebraically.

| x � 1| � 2

When a solution is confined outside a region, x is below the lower bound or above

the upper bound of the region.

Solve each of the following inequalities and graph their solutions on the number

lines provided.

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3. |2x � 3| � 5

4. |2x � 3| � 7

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Problem 4 Solving Linear Inequalities withAbsolute Value in Two Variables

1. Graph the following equation on the grid:

y � |x � 3|

a. Which portions of the coordinate plane has ordered pairs so that the values of

y � |x � 3|? y � |x � 3|? y � |x � 3|?

The solution set of a linear inequality with an absolute value is the portion of

the coordinate plane that satisfies the inequality. To indicate the solution set,

we shade the portion. If the line segments are included in the solution, we use

a solid line, and if they are not included, we use a dotted or dashed line.

b. Shade the solution set of y � |x � 3| on the grid in Question 1.

11

2. Graph the solutions for each of the following inequalities:

a. y � |x � 2|

b. y � |3x � 1|


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Lesson 1.8 ● Functional Notation, Inverses, and Piecewise Functions 67

Problem 1A relation is any correspondence between a set of input values and output values.

The set of all inputs is the domain of the relation. The set of all outputs is the range of

the relation. A function is a special type of relation in which for every member of its

domain is associated with exactly one member of its range. In other words, a function

is a relationship in which each input value has a unique output value. For example,

● If you are selling candy bars for a fundraiser, the relationship between the number

of candy bars sold and the resulting income is a function. For any amount of

candy bars sold (the input or domain), there is a unique value for the income (the

output or range).

● If you are counting how much money you have based on the total number of

coins in your pocket, the relationship is not a function. The amount of money may

vary depending on the types of coins you have. While this is not an example of a

function, it is still a relation.

1. Every holder of a social security card in the United States is assigned a nine-digit

social security number.

a. Let the domain be all assigned social security numbers and the range be

the names of all social security cardholders. Is this relationship a function?

Why or why not?


● Write linear functions using functional

notation.

● Write compositions of functions.

● Determine inverses of linear functions.

● Define piecewise linear functions.

● Graph piecewise linear functions.

Key Terms● relation

● domain

● range

● function

● inverse operation

● functional notation

● identity function

● inverse function

● composition of functions

● piecewise functions

1.8 Inverses and PiecesFunctional Notation, Inverses, and Piecewise Functions 11


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b. Consider reversing the situation. Let the domain be the names of all social

security cardholders and the range be all assigned social security numbers. Is

this reverse relationship a function? Why or why not?

2. Every member on the school football team is assigned a number.

a. Let the domain be all the football players on the school team and the range be

all of the assigned player numbers. Is this relationship a function? Why or why

not?

b. Reverse the situation. Let the domain be all assigned player numbers and the

range be all of the football players on the school team. Is this reverse relation-

ship a function? Why or why not?

3. Each person has a favorite color.

a. Let the domain be all people in the world and let the range be all the colors. Is

this relationship a function? Why or why not?

b. Reverse the situation so that the domain is all of the colors and the range is all

people in the world. Is the reverse relationship a function? Why or why not?

Problem 2In Questions 1 through 3, you considered general relationships and their reverse

relationships and decided whether or not they were functions. Now you will look

at more specific situations and determine their reverse. In other words, you will

determine how to “undo” the situation. “Undoing,” working backwards, or retracing

steps to return to an original value or position is referred to as the inverse operation.

Write a phrase, expression, or sentence for the inverse of each given action.

1. Open a door.

2. Turn on a light.

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3. Add 6 to a number.

4. Walk 2 blocks east and then 3 blocks south.

5. Get into the deep end of a pool and swim to the shallow end.

6. Multiply a number by 3 and subtract 5.

7. For Question 6, Jan gave the inverse as “Divide by 3 and then add 5.” Marcus

gave the inverse as “Add 5 and then divide by 3.” Who is correct? Why?

Problem 3In this chapter, you have been working with linear relations in slope–intercept form,

y � mx � b, and standard form, Ax � By � C.

1. Are the linear relations functions? Why or why not?

2. For each of the following functions written in functional

notation, calculate its value for the given values of the

independent variable.

a. f (x) � �3x, for x � 2, �7

b. g(x) � 7x � 8, calculate g(0), g(5)

c. h(x) � �x2 � 8x, calculate h(�1), h(3)

d. f(x) � x, calculate f(�1), f(3)

In Question 2d, the function f assigns every value of x to itself; this is called the

identity function.

RememberFunctional notation is often

used to represent functions.

f(x) is read f of x or the value

of the function f at x.

For example, if f(x) � 3x � 5

then f(�2) � 3(�2) � 5 � �11.

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3. To determine the inverse of a linear function, you need

to derive the function that “undoes” the original function.

For each of the following functions determine its inverse

function.

a.

b.

c.

4. For each of the functions and inverses from Question 3,

calculate each of the following:

a.

b.

c.

5. In each case, what happens when an inverse of a function is composed with the

function?

A function f is the set of all ordered pairs (x, y) or (x, f(x)), where for every value

of x there is one and only one value of y or f(x). To represent any linear equation

in two variables using functional notation, first solve for y and replace y with f(x).

The inverse of this function f�1(x) is the set of all ordered pairs (y, x) or (f(x), x). To

derive the inverse function, solve for the dependent variable and then reverse the

variables.

For each of the following, first write the linear equation in two variables in functional

notation and then determine its inverse.

6. f�1 (x) �f( x) �y � 4x � 8

h�1 (h(x) ) �

h�1 (h(�9.6) ) �

h�1 (h(6) ) �

g�1 (g(x) ) �

g�1 (g(6) ) �

g�1 (g(�1) ) �

f�1 (f(x) ) �

f�1 (f(�2) ) �

f�1 (f(3) ) �

h�1 (x) �h( x) �x

1.2

g�1 (x) �g( x) � x � 8

f�1 (x) �f( x) � �3x

Take NoteApplying one function to the

answer of another function is

called the composition of

functions. f g(x) or f(g(x)) is

read f of g of x or the value of

the function f at the value of

the function g at x.

For example, if f(x) � 3x � 5

and g(x) � 2x then f(g(�2)) �

f (2(�2)) � f (�4) �

3(�4) � 5 � �17.

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7.

8.

9.

Problem 4You have been offered a new job selling medical supplies. As part of your

compensation package, you are to receive a commission on your total sales each

year with a rate that changes as you sell more. You will receive

● 1% of your total sales for the first $20,000 sold

● 2% of your total sales from over $20,000 to $40,000 sold

● 5% of your total sales over $40,000

1. Define variables for your total sales and your commission.

a. Use these variables to write a function for your total commissions up to $20,000.

b. What would be your commission if you sold $5000 of medical supplies?

$10,000? $20,000?

c. Use these variables to write a function for your total commissions from over

$20,000 to $40,000.

d. What would be your commission if you sold $25,000 of medical supplies?

$30,000? $40,000?

e. Use these variables to write a function for your total commissions over $40,000.

f�1 (x) �f(x) �3y � 7x � 11

f�1 (x) �f(x) ��4x � �2y � 10

f�1 (x) �f(x) �3x � 7y � 9

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f. What would be your commission if you sold $45,000 of medical supplies?

$50,000? $60,000?

g. Use the values to complete the table, making sure to find the three

different expressions for the different commissions. Construct a graph of your

commission with total sales from $0 to $60,000 using the values from parts (b),

(d), and (f ) on the grid. Using what you know about linear graphs, connect

these points.

Quantity Name

Unit

Expression

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2. Using this graph, answer the following questions:

a. How is this graph different from the others you have constructed?

b. Is this graph a function? Why or why not?

c. A relation like this is called a piecewise relation. Why?

Piecewise functions are functions which are defined with different functional

relationships between the independent and dependent variables over different

domains. This function could be written as a single function as

3. Graph each of the following piecewise functions.

a. f(x) � � 2x � 1 x � 4�2x � 17 x � 4

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f(x) �

0.01x x 20,000

200 � 0.02 (x � 20,000) 20,000 x 40,000

600 � 0.05 (x � 40,000) x 40,000�

��

�

�

1 Linear Functions, Equations, and Inequalities...

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