Pre-Algebra 2

Unit 8 Rational Equations

Name________________________

Period______

Basic Skills (7B after test) Practice Name________________________________Per_______

PAP Algebra 2

NON-CALCULATOR

Simplify:

1. 1 1

3 5 2.

2 1

5 9 3.

7 4 1

8 5 9 4.

1 1 1

12 4 6

5.

2 13 5

7 3

6. 5 5

11 2

7.

8 4

15 25 8.

1 23 5

5 9 9.

3 5 1 26

4 9 6 3

10. 5 6

x x 11.

3 2

10

x y

y 12.

3

2

a c

c 13.

2 5 2

3 2 7

14. 22 2

3 3 2 3 33 3 23

15.

2 2

2 8 2 1 20( 1)

113

5

1

Solve:

16. 3 8 15x 17. 7 7 4 5x x x 18. 3

27x 19.

4 2

8 9

x

x

20. 2 7 4

7 3 7x

21. 2 3 88 0x x 22. 2 2 35x x

23. 2 5 6 0x x 24. 2 3 28 x x 25. 22 9 9 0 x x

26. 2 3 3 5 1 4 3 9 3 15 5 8 x x x x

2

Notes 8.1 Simplifying Rational Expressions Day 1 (Multiplying/Dividing) PAP Algebra II

FACTOR EACH 1. 9x2 – 4 2. x2 –5x – 24 3. x3 + 1 4. 8 – a3

SIMPLIFY EACH

5. 1260

1350 6.

2

1 x

x x

7. 2 2

2 2

6 5x xy y

y x

Multiplying/Dividing Fractions

1) 2 3

5 11 2)

15 32

24 45

3) 14 2

15 25 4)

2 2

2 3

9 4

x x

x x

3

5) 2 2

2

3 3 20

34 5

x x x x

xx x

6)

2 2

2 2

25 6 5

30 4 12

x x x

x x x x

8)

2

2

2

7

2 10

6

11 30

x

x

x x

x x

9) 2

2

2

6 15(3 5 )

4

x xx x

x

4

PAP Algebra II WS 7.1 Name: _______________________________________________ For 1 – 6, simplify each rational expression.

1. 26 9

3

c c

c

2.

2 49

7

z

z

3.

2

2 10

10 25

x

x x

4. 2

2

8 16

2 24

x x

x x

5.

2

2

2 3

6 9

x x

x x

6.

2

2

2 8 24

2 8 8

y y

y y

7. 3 15

1 2

x

x

8.

2

3

12

a

a a

÷

2

2

9

7 12

a

a a

9. 3 2

4 3

6 6

5

x x

x x

÷

2

2

3 15 12

2 2 40

x x

x x

10.

2

2

2

2 5 2

x x

x x

÷

2

2

12

2 5 3

x x

x x

11.

2

2

8

1

6

1

x y

x

xy

x

12. 3 2

3( )

5

2 2

10

f g

f g

f g

13) 2

2

5 2 32

4 16 25

x x

x x

14)

2

2

3 10 5

2 15 2

x x x

x x x

5

15) 2

2

3 2

4 208 16

4

x xx x

x x

16)

2

2

6 3x x x

x x x

17) 2 2

2

12 32 4

6 42 49

x x x x

x x

18)

22

2

8 1520

4

x xx x

x x

19) (a.) Simplify 2(2 ) 1

2 1

n

n

x

x

, where x is an integer and n is a positive integer. (b.) Use the results in 8a to show

that the value of the given expression is always an odd integer. 19) Which of the expressions below always has a positive value? Assume x ≠ 0, x ≠ 1, or x ≠ -1.

a. x1

1x

÷ (x - 1) b.

1

1x ÷ (x - 1)

1

x

c. 2

1x

x

÷ (x2 – 1)

2

1

x

d. 2 1x

x

2

2

1

( 1)

x

x

÷ 2

1

x

e. 2

1

1x

÷ x2

2

1

1x

20) Simplify:

33

4 5

2x

a b

6

PAP Algebra II Notes 7.2 Adding/Subtracting Rational Functions To add or subtract rational functions you must get a common denominator.

1. 9 11

10 15 2.

2 3

21 14

3) Find the least common multiple (LCM). 5x2 + 15x + 10 and 2x2 - 8 Simplify:

4. 2 3

1 2 1

x x

x x

5. 22

x

x x +

1

2x

6. 2

5

y

y +

4 5

5

y

y

7

7. 2

3

2 2 1

11

y y

yy

8. 2

5

12 4812

x

xx x

9. 2

2 2 1

2 2 4 3

x x

x x x

10. 2 2

1 6

4 4 4

x

x x x

8

WS 7.2 Name: _________________________________________

PAP Algebra II

Add or subtract the following rational functions and simplify completely.

1) 2

8 5

43 xx 2)

7 5

2

x

x

-

1x

x

3) 4 1

9

x -

2

3

x 4)

2 2

1

9 6 9

x x

x x x

5) 2 2

3 5

2 8 12 32

x x

x x x x

6)

2

3

x y -

2

2

xy

9

7) 2

4 15

24

x

xx

8)

3 1

4 6x x

9) 2

7 2

3

x y

xy

-

2

2 5x y

x y

10)

4

2x -

4

2x

11) 2

1

x

x -

3

1

x

x 12)

2

3 1 3

5 325

x x

x xx

10

13) 2

4

25

x

x -

3

5x 14)

2

4

3 6a a -

2

3 6a

15) 3

8t +

1

2t 16)

4

1x +

2

x

17) 2

3x +

2

3

( 3)x 18)

2

5

16

x

x

+

1

4

x

x

11

19) 1

1

x

x

+

1

1

x

x

20)

2 1

x

x +

2

3x

x x

21) 2

3 2

x

x +

3

2 3

x

x 22)

3

5x +

2

2

3 21 30

25

x x

x

23) 2

2

7 8

16

x x

x

+

2

4

x

x

24)

2

2

5 4x x +

2

3

16x

12

Notes 8.3 Transformational Graphing of Rationals

A rational parent function or reciprocal function has the equation: f(x) = x

1

Horizontal Asymptote_______________

Vertical Asymptote ________________

Domain and Range for the parent function:

What can x not be? What can y not be? (This is a discontinuity.)

Domain _________________ Range ___________________

The general form for translating a reciprocal function (f(x) = x

1) is:

1( )f x a k

x h

Example:

f(x) = 42

1

x

Translations: ____________________________

HA__________________

VA___________________

Domain________________

Range_________________

x -10 -1 -.5 0 0.5 1 10

y

13

Graph using transformations, explain what happens, give the domain and range, and give the asymptote equations:

3. 1

( 3)y

x

4. 2

1y

x 5.

2

2

( 5)y

x

6. 2

13y

x For each of the following, list the equation, asymptotes, domain and range.

7. 8.

Domain:

Range:

VA:

HA:

Domain:

Range:

VA:

HA:

Domain:

Range:

VA:

HA:

Domain:

Range:

VA:

HA:

14

Graphing Rationals Name:________________________________Per________ WS 8.3 Graph the following functions.

1) 1

41

yx

2) 3

3yx

VA: VA: HA: HA: domain: domain: range: range:

3) 2

32

yx

4) 2

12

( 3)y

x

VA: VA: HA: HA: domain: domain: range: range:

5. Determine if the table represents a direct or inverse variation. Find the constant of variation, and specific equation for each. 6. Architects have to consider how sound travels when designing large buildings such as theaters, auditoriums, or museums. Sound intensity, I, is inversely proportional to the square of the distance from the sound source, d.

a) Write an equation that represents this situation.

b) If a person in a theater moves to a seat twice as far from the speakers, compare the new sound intensity to that of the original.

x 4.5 3 4 1 5

y 324 144 256 16 400

x .75 2.5 6 1.5 2 y 8000 2400 1000 4000 3000

15

Simplify each expression.

4. 3 6 14 14

7 7 5 10

x x

x x

5.

2

2

3 6 6 12

4 129

x x x

xx

6.

2

5 7

2 143 28 xx x

7. 2 2

3 8 24

6 9 9

m m

m m m

8.

2

2

5 5 30

45 15

6

4 12

x x

x

x x

x

9.

1 1( )( )

1 1( )( )

x yx y

x yx y

10. 2

2

1 1

2 5

2 3

3 10

x x

x x

x x

16

PAP Algebra II Notes 8.4A Graphing Rational Functions A rational function is a function that can be written as a fraction of two polynomials:

)(

)()(

xq

xpxf

The graph has zeros (i.e roots or x-intercepts) where 0)( xp

The graph (usually) has vertical asymptotes where 0)( xq

The horizontal asymptote depends on the highest power of x in the numerator and denominator.

If the highest power of x is in the denominator, then the horizontal asymptote is 0y

If the highest power if x is the same in the numerator & denominator, then the horizontal asymptote will be the coefficients in front of those terms. For

example, if the rational function is 572

12334

4

xx

xxy then the horizontal

asymptote will be 2/3y

If the highest power of x is in the numerator, then there is no horizontal asymptote

EX1: Graph 5

2)(

x

xxf

VA: ________ HA: ________ ZEROS: __________ Domain: _______________

17

EX2: Graph 93

126)(

x

xxf

VA: ________ HA: ________ ZEROS: ________ Domain: _______________

EX3: Graph 43

5)(

2

xxxf

VA: ________ HA: ________ ZEROS: ________ Domain: _______________

18

EX4: Graph 20

2444)(

2

2

xx

xxxf

VA: ________ HA: ________ ZEROS: ________ Domain: _______________

19

PAP Algebra II WS 8.4A NONCALCULATOR! Name: ___________________________ Find the vertical asymptotes, horizontal asymptotes, x-intercepts (i.e zeros), domain and range and graph the rational function.

1) 4

3)(

x

xxf VA = ____________

HA = ____________ Zeros = ____________ Domain : _____________

2) 1

63)(

x

xxf VA = ____________

HA = ____________ Zeros = ____________ Domain: _____________

3) 102

168)(

x

xxf VA = ____________

HA = ____________ Zeros = ____________ Domain : _____________ 20

4) 16

6)(

2

2

x

xxxf VA = ____________

HA = ____________ Zeros = ____________ Domain : _____________

5) 4022

1284)(

2

2

xx

xxxf VA = ____________

HA = ____________ Zeros = ____________ Domain : ______________

6) 2

3)(

xxf VA = ____________

HA = ____________ Zeros = ____________ Domain : _____________

21

7) 107

2)(

2

xxxf VA = ____________

HA = ____________ Zeros = ____________ Domain: _______________

8) 12

633)(

2

2

xx

xxxf VA = ____________

HA = ____________ Zeros = ____________ Domain: _____________

22

PAP Algebra II Notes 8.4B

A rational function is going to have a hole (also known as a removable discontinuity) wherever a factor that cancels from the numerator & denominator equals zero.

Ex: 4

2

)4)(1(

)2)(1()(

x

x

xx

xxxf

Although a rational function can never cross a vertical asymptote, SOMETIMES it can cross the horizontal

asymptote.

EX1: 20

3093)(

2

2

xx

xxxf

V.A.: __________________ H.A.: _________________ Zeros: ________________ Holes: ____________________ Domain: ___________________ Range: __________________

23

EX2: 99

22)(

23

3

xxx

xxxf

V.A.: __________________ H.A.: _________________ Zeros: ________________ Holes: ___________________ Domain: __________________ Range: ________________ Does the graph cross the horizontal asymptote? If so, where?

EX3: 1892

1284)(

23

2

xxx

xxxf

V.A.: __________________ H.A.: _________________ Zeros: ________________ Holes: __________________ Domain: __________________ Range: ________________ Does the graph cross the horizontal asymptote? If so, where?

24

PAP Algebra II WS 8.4B NONCALCULATOR! Name: ________________________ Graph the following rational functions. Be sure to label asymptotes and holes in the graph, and find the specified limits for the function.

1) 9

62

2

x

xxy

HA = _____ VA = _________ ZEROS = ___________ HOLES: _________ Domain: __________________

2) 1

862)(

2

2

x

xxxf

HA = _____ VA = _________ ZEROS = ___________ HOLES: _________ Domain: ____________________

25

3) 50252

15123)(

23

2

xxx

xxxf

HA = _____ VA = _________ ZEROS = ___________ HOLES: _________ Domain: ____________________

4) 32162

123)(

23

3

xxx

xxxf

HA = _____ VA = _________ ZEROS = ___________ HOLES: _________ Domain: ____________________

26

Graph the following transformations to 1

yx

or 2

1y

x .

5) 1

3yx

6) 2

( 4)y

x

7) 2

3

( 2)y

x

8)

2

15y

x

9) What are the transformations from f(x) to g(x)? a) ( ) 2 ( 3)g x f x c) ( ) 2 ( 3)g x f x

b) 1

( ) ( 3)2

g x f x d) 1

( ) ( 3)2

g x f x

27

Simplify:

15.

5

41 2

4

x

x x

16. 2

3 5

x y

y x

17.

35 5

6 2

3x p

p x

18.

43

2 3 4

10 10

10 10 m

19. Condense: 4ln 5lnx x 20. Expand:

4

6 8

6log

y

x

Solve. Remember to check for extraneous solutions.:

21. 6 6 6log (3 14) log 5 log (2 )x x 22. 2 2log log ( 4) 5x x

28

Notes 8.5 Direct and Inverse Variation

Direct Variation Equation Graph Example

Direct Variation Examples: 1. Julio’s wages vary directly with the number of hours that he works. If his wages for 5 hours are $29.75, how much he be paid for 30 hours? 2. If y varies directly with x, and y = 28 when x = 7, find x when y = 52. 3. A standard shower head uses 18 gallons of water in 3 minutes. Complete the table below that shows that gallons used, y, is a function of time in the shower, x.

x (minutes)

3 6 9 12 15 20 25 30

y (gallons)

a. What is the k? k = ______ b. Write the equation for the function ____________________________________ c. If 270 gallons were used, how many minutes were spent in the shower? _______________

29

Inverse Variation

Equation Graph Example

Inverse Variation Examples:

4. The volume V of gas varies inversely to the pressure P. The volume of a gas is 200 cm3 under pressure of 32 kg/cm2. What will be its volume under pressure of 40 kg/cm2? 5. The time it takes to fly from Los Angeles to New York varies inversely as the speed of the plane. If the trip takes 6 hours at 900 km/h, how long would it take at 800 km/h? 6. The number of people, P, in a theater varies inversely with the number of empty seats, s. Is this an example of a

direct variation situation, inverse variation situation, or neither type? Why? 7. Write the inverse variation equation that represents: m is inversely proportional to the square of n.

30

Mixed Examples: 8. a) Is this direct, inverse or neither? How do you know?

b) Find the constant of variation, k, and

c) Write an equation to model this data. 9. a) Is this direct, inverse or neither? How do you know?

b) Find the constant of variation, k, and

c) Write an equation to model this data. 10. a) Is this direct, inverse or neither? How do you know?

11. If each point is from a model of inverse variation, find the constant of variation. (Hint: think about the equation xy = k to remember how to find the constant)

a) ( 3, 7 ) b) ( 2.5 , 1.5 ) c) ( 15 , 3

1 )

12. Each pair of points is from a model for inverse variation. Find the missing value. (Hint: Write the equation xy = k and use it to solve for the unknown)

a) ( 3

2 ,

4

1 ) and (

2

1 , y ) b) (10, 12 ) and ( x , 5 )

Classify the following graphs as a) Direct b) Inverse c) Neither 13) 14) 15) 16)

x 2 3 4 5

y 1.25 0.833 0.625 0.5

x 10 5 2 0

y 5 2.5 1 0

x 0 2 4 5

y 0 4 16 25

31

8.5 Inverse Variation Worksheet Name:_____________________ Period:_______

1. The amount of food a dog eats varies directly with the number of days the dog is fed. If a dog is fed for 10 days, the

dog eats 25 cups of food. How much food would the dog eat in 50 days?

2. The rent for an apartment varies inversely with the number of people sharing the cost. Four people sharing an

apartment pay $120 each per month. How many people would be needed so that each would pay $80 per month?

For 3-8 state whether the problem is inverse, direct or neither. If the problem is inverse or direct variation, state the

constant of variation:

3. y = 7x 4. y + x = 4 5. 9x

y

6. xy = 12 7. x

y4

8.

9. If x and y vary inversely and x = 1 when y = 11, find x when y = 5.5

10. If x and y vary inversely and x = 2.5 when y = 100 find x when y = 25

x y

4 8

16 2

1 32

10 3.2

32

11. Heart rates and life spans of most mammals are inversely related. A cat lives for about 15.2 years on average and has

a heart rate of 126 beats per minute.

a. What is the constant of variation?

b. A hamster has a heart rate of about 634 beats per minute. About how long will a hamster live?

c. An elephant lives for about 70 years. About how many times per minute does an elephant’s heart beat?

12. Two gears are used to operate a machine. Gear A has 60 teeth and Gear B has 45 teeth. The speed at which you turn

Gear A is 5400 rpm. The number of teeth and speed in rpm are inversely related.

a. What is the constant of variation?

b. At what speed will Gear B turn?

13. The grade you earn in math varies inversely with the number of minutes per night you watch television. If you watch

90 minutes per night, you get a 60 in math.

a. What is the constant of variation?

b. How much television can you watch if you want to make a 70?

c. You cut back on your television to only 75 minutes a night, what grade will you make in math?

d. What is the maximum amount of television you can watch and still make a 100?

14. The amount of water that has leaked from a faucet varies directly with time. In 2 hours, 10 gallons of water leak.

a. Describe what happens to the amount of water as time increases.

b. What is the constant of variation?

c. How much water leaks in 100 hours?

d. How long does it take for 100 gallons to leak?

33

Simplify:

15.

2 2

2 3 2

2 24 3 6

3 10 4

x x x x

x x x x

16.

1 1

4 4x x

17.

22 3

6 2

2

3

x m

x m

18.

43

2 3 4

x m

xy y m

Solve. Check for extraneous solutions:

19. 4

5xx 20. 6 6log 3 log 1 3x x

21. 3 1 4

12 x x 22.

2 6

6 2

x x

x x

34

Notes 8.6 Solving Rational Equation Word Problems PAP Algebra 2

1. 3 1 4

2 8 7

x

x

2. 4 3 9

1 1 5x x

3. Nancy drove from Houston to San Antonio, a distance of 250 miles. She increases her speed by 10

miles per hour for a 360 miles drive from San Antonio to Dallas. If the trip took a total of 11 hours, how fast did she drive on both legs of the trip?

4. You take your boat on a trip 36 miles down the river. On the way to your destination (ie downstream),

the 3mph current speeds you up. On the way back (ie upstream) the same current slows you down. Your total travel time is 9 hours. How fast is the boat in still water?

Timeupriver + Timedownriver = Timetotal 5. The speed of a stream is 4 km/hr. A boat travels 6 km upstream in the same time it takes to travel 12

km downstream. How fast is the boat in still water?

35

Worksheet 8.6 Name___________________________________Per_____

1) Natalie traveled 2700 miles in her private jet in the same time Allie and Stephanie flew on a jumbo jet

that traveled 3600 miles. If the jumbo jet flew 150 miles per hour faster than the private jet, what were

the speeds of both jets?

2) A speed skater travels 9 kilometers in the same amount of time it takes another speed skater to travel

8 kilometers. The first skater travels 4.38 kilometers per hour faster than the second skater. How fast is

each skater?

3) The speed of a river’s current is 3 miles per hour. You travel two miles with the current and then

return to where you started in a total of 1.25 hours. What is your speed in still water?

4) A car travels 120 miles in the same amount of time that it takes a truck to travel 100 miles. The car

travels 10 miles per hour faster than the truck. Find the speed of the truck.

36

5) David canoes upstream a distance of 8 miles and then returns. The round-trip took 1

53

hours. The

current of the stream was flowing at 2 miles per hour. What was David’s canoeing speed?

6) A boat can travel 8 miles an hour in still water. If it can travel 15 miles down a stream in the same time

that it can travel 9 miles up the stream, what is the rate of the stream?

7) Two trains starting at the same time from stations 396 miles apart, meet in 4½ hours. How far has the

faster train traveled when it meets the slower one, if the difference in their rates is 8 miles per hour?

8) Graph: 2

2

6 6 120( )

3 27 60

x xf x

x x

HA = _____

VA = _________ ZEROS = ___________

HOLES: _________

Domain: ____________________

37

Write the Equations of the following Graphs.

9)

10)

-8 -6 -4 -2 2 4 6 8 10

-10

-8

-6

-4

-2

2

4

6

8

-8 -6 -4 -2 2 4 6 8 10

-10

-8

-6

-4

-2

2

4

6

8

38

Notes 8.7 More Rational Equation Work Problems

1. Kyle can paint a room in 4 hours. If Cole helps, together they can paint the room in 1.5 hours. How long would it

take Cole to paint the room by himself?

2. Ashley and Lindsay want to start a business painting fences. The figured out that they would paint a fence in 40

minutes together. Ashley can paint the fence in 70 minutes alone. How long would it take for Lindsay to paint

the fence alone?

3. Mrs. Zurek is putting toys in a toy box at the same time her son, Noah is taking them out. Mrs. Zurek can fill the

toy box in 3 minutes and her son can empty the box in 5 min. How long will it take Mrs. Zurek to pick up the

toys if her son is emptying the box while she works?

39

4. For the student council to attend the national convention HHS will charter an airplane. The cost to charter a

plane is $5000. Each student going on the trip is going to pay an equal share of the plane and $300 for food and

lodging.

a. Write an equation that shows the cost (c) per the number of students (x) attending the convention.

b. If no more than 100 people may fly on the charter plan, what is the minimum cost per person?

c. If we needed to make sure we spend no more than $400 per student, what is the minimum number of students we need to take?

5. Dr. Waldrip is so tall he decided to try professional basketball as a potential career. He made 30 out of 50 free

throws to start. To earn a job with the Mavericks, he must have an 80% free throw average. How many

consecutive free throws must he make to get a job with the Mavs?

40

Worksheet 8.7 Name___________________________________Per_____

1. Billy can mow a lawn by himself in three hours. It takes Bobby five hours when he mows the same lawn. If the

two boys worked together, how long would it take them to mow the lawn?

2. A basketball player has made 21 of her last 30 free throws – an average of 70%. What is the number of

consecutive free throws the player needs to raise her success rate to 75%?

3) You can paint a room in 8 hours. But you and a friend can paint the same room in just 5 hours. How long

would it take for your friend to paint the room alone?

1 2 togetherR R R

4. HHS mathletes want to attend a competition. The school board has agreed to allow the mathletes to attend as

long as the cost per mathlete is less than $20. Write an expression for the cost per student if the bus costs $240,

and the entry fee per student is $12. Then, write your expression as a single fraction.

5. What is a reasonable domain and a reasonable range for #6?

41

6. The Pee-Wee soccer team has won 4 games and lost 6. The players have decided to throw a party if they can get

the winning record to 60%. How many consecutive games must they win in order to reach this winning record?

7. You scored a hole-in-one 7 out of the last 18 times while playing put-put golf. If you continued playing put-put

golf, what function would best model the percentage of hole-in-ones if the next x puts dropped in on the first

try?

8. Mrs. Elvington and Mrs. Poltl have agreed to contact every student at Heritage High School by phone to inform

them of the spring dance. Mrs. Elvington can complete the calls in six days if she works alone. Mrs. Poltl can

complete them in 4 days. How long will they take to complete the calls working together?

9. Bella can clear a lot in 5.5 hours. Rooshan can do the same job in 7.5 hours. How long would it take them to do

it together?

10. For the student council to attend the national convention HHS will charter an airplane. The cost to charter a

plane is $5000. Each student going on the trip is going to pay an equal share of the plane and $300 for food

and lodging.

a. Write an equation that shows the cost (c) per the number of students (x) attending the convention.

b. If no more than 100 people may fly on the charter plan, what is the minimum cost per person?

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11. The speed of a river’s current is 3 miles per hour. You travel two miles with the current and then return to

where you started in a total of 1.25 hours. What is your speed in still water?

12. The members of our math department are ordering Pi Day T-shirts. There is a one-time charge of $99 for

artwork on the T-shirts and a $12 charge for each T-shirt ordered. If x represents the number of T-shirts

ordered, write an equation that represents the total cost, C, in dollars per T-shirt?

a. What is the most expensive price shirts could be? (imagine only 1 person orders a shirt)

b. What is the cheapest shirts can be? (imagine lots and lots and lots of people order shirts. They are math

shirts after all.)

c. What is the range for the cost of the shirts?

13. You are organizing your high school’s sports banquet. The banquet rental hall is $350. In addition to this one

time charge, the meal will cost $8.50 per plate. Mr. Spain has decided that you can only go if the cost per

student C(x), is below $15. Write an inequalities that could be used to find the number of students, x needed to

offer the banquet?

a. What is the most expensive price the meal could cost?

b. What is the cheapest meals can be?

c. What is the range for the cost of the meals?

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16. You have subscribed to a cable television service. The cable company charges you a one-time installation fee of

$30 and a monthly fee of $50. The function gives the average cost per month as a function of the number of

months you have subscribed to the service is: m

mC

5030 . What is a reasonable range for the average cost

of cable service per month?

a. What is the most expensive price per month for cable?

b. What is the cheapest price per month for cable?

c. What is the range for the average monthly cost for cable?

17. Bob can paint a fence in 6 hours and Sam can paint a fence in 10 hours. How long would it take both men to

paint 3 fences working together?

18. Two printing machines can complete a job in 12 hours. One machine works twice as fast as the other. How long

would it take the slower machine to complete the job, working by itself?

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