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Prentice Hall Algebra 1 • Teaching ResourcesCopyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved.

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A N S W E R S

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11-1 Practice (continued) Form GSimplifying Rational Expressions 20. A pilot packed two rectangular suitcases for her trip to Hawaii. Both hold the

same volume of clothes. Her green suitcase has a length of 2y 1 4, a width of y 1 1, and a height of 4y. Her blue suitcase has a length of 8y2 2 6y and a width of 2y. What is a simplifi ed expression for the height of the blue suitcase? Show your work.

21. Th e numerical area of a circle with radius c is equal to the numerical volume of a sphere with radius S. What is the radius of the sphere in terms of c? Show

your work. (Area circle 5 pr2. Volume sphere 5 43pr3).

Simplify each expression. State any excluded values.

22. x2 2 9

2x2 2 6x 23.

n2p2

n2p

24. 2x2 1 17x 2 9

x2 2 81 25. 4d

4 2 6d3 2 4d2

d2 2 2d

26. 11y2 1 35y 2 36

y2 2 16 27. 6a

5b 1 4ab3 1 3a4c 1 2b2c2ab 1 c

28. Your brother’s car is traveling 40mih faster than

your car. During the time it takes you to go 150 mi, your brother goes 450 mi. Make a table with the information and fi nd the speeds.

2(y 1 2)(y 1 1)y(4y 2 3) ; y u 0,

13

3"34 c 2

x 1 32x ; x u 0, 3

2x 2 1x 2 9 ; x u 9, 29

11y 2 9(y 2 4) , y u 4, 24

Your speed 20mih ; Your brother’s speed 60 mi/h

(4d2 1 2d); d u 0, 2

(3a4 1 2b2); 2ab 1 c u 0

p; n u 0, p u 0

Distance

You

Yourbrother

Rate Time

150 t

450

r

r 40 t

g

11-1 Practice Form GSimplifying Rational ExpressionsSimplify each expression. State any excluded values.

1. 6p 2 36

18 2. q 1 1

q 1 4q 1 3 3. 8b5

64b4

4. x 1 1x2 2 1

5. 56c 2 1424c 2 6 6. 3b 2 6b2 2 4

7. x2 2 144

3x2 2 36x 8. n

2 2 n 2 12n2 2 4n

9. 3x2 1 19x 2 14

x2 2 49

10. 7d3 1 14d

6d2 2 2d 11.

25y2 2 12115y 2 33 12.

99q2 2 2q 2 19q 2 1

13. Th e length of a rectangle is 3h 1 2 and the width is 9h 1 6. What is the ratio of its length to its width? Simplify your answer.

14. Th e length of a rectangle is x 2 2. Its area is 2x 2 4. What is a simplifi ed expression for the width?

15. Th e area of a rectangle is x2 2 9. Its width is x 2 3. What is a simplifi ed expression for the length?

16. Writing Why must the denominator of a rational expression not be equal to 0?

17. Th e area of a rectangle is 16a2. Th e length is 2a. What is a simplifi ed expression for the the width?

18. Are the given factors opposites? Explain. a. 3d 2 7; 7 2 3d

b. 2y 1 4; y 1 4

c. 27 1 8x; 227 2 8x

19. Th e ratio of the area of a small circle to a larger circle is p(2x)2

p(6x)2. Simplify the

expression.

p 2 63

1x 2 1 ; x u 1, 21

x 1 123x ; x u 0, 12

13; h u 2

23

2

8a

yes; 21(3d 2 7) 5 (23d 1 7) 5 (7 2 3d)

no; (21)(2y 1 4) 5 (y 2 4) u (y 1 4)

yes; (21)(27 1 8x) 5 (227 2 8x)

19; x u 0

It is undefi ned; can’t divide by 0 and can’t graph a 0 denominator

x 1 3

7(d2 1 2)2(3d 2 1); d u 0,

13

n 1 3n ; n u 0, 4

3x 2 2x 2 7 ; x u 7, 27

5y 1 113 ; y u

115

73 ; c u

14

11q 1 1, q u 19

3b 1 2 ; b u 22, 2

b8

, b u 0

1q 1 3; q u 23, 21

g

11-1 Think About a PlanSimplifying Rational Expressions a. Construction To keep heating costs down for a building, architects want

the ratio of surface area to volume to be as small as possible. What is an expression for the ratio of surface area to volume for each shape?

i. square prism ii. cylinder

b. Find the ratio for each fi gure when b 5 12 ft, h 5 18 ft, and r 5 6 ft .

Understanding the Problem

1. What is a ratio?

2. What are the other two forms of a ratio from 2 to 4?

3. What are the formulas for the surface area of a square prism and of a cylinder? What are the formulas for the volume of a square prism and of a cylinder?

Planning the Solution

4. What is the diff erence between surface area and volume?

5. What is the ratio of the surface area to the volume for the square prism?

6. What is the ratio of the surface area to the volume for the cylinder?

Getting an Answer

7. Use the ratio in Exercise 5 to fi nd the ratio for the square prism with the given measurements.

8. Use the ratio in Step 6 to fi nd the ratio for the cylinder with the given measurements. How does your answer compare to your answer for Exercise 7?

h

r

h

bb

A ratio compares two quantities measured in the same units.

SA of a square prism 5 2b2 1 4bh; SA of a cylinder 5 2πr2 1 2πrh;

V of a square prism 5 b2h; V of a cylinder 5 πr2h;

The surface area is the combined area of all of the surfaces of an object. The volume of an object is the entire amount of space inside an object.

Ratio of SA : Volume 5 2b2 1 4bhb2h

Ratio of SA : Volume 5 2πr2 1 2πrhπr2h

1152:2592 or 4:9

288:648 or 4:9; the ratios are the same.

2: 4, 24 or 12

g

11-1 ELL SupportSimplifying Rational ExpressionsA student is trying to fi nd the simplifi ed form of the expression 2x 1 8

x2 1 x 2 12.

She wrote these steps to solve the problem on note cards, but they got mixed up.

Use the note cards to complete the steps below.

1. First,

2. Second,

3. Then,

4. Next,

5. Finally,

To see if there are any common factors, factor the numerator and the denominator.

Divide out the common factor (x 1 4) from 2(x 1 4)(x 1 4)(x 2 3).

Simplify to 2x 2 3.

(

State the simplified form with any restrictions on the variable.

The denominator of the original expression is 0 when x 5 24. So the simplified form is 2(x 2 3), where x u 24.

divide out the common factor (x 1 4) from 2(x 1 4)(x 1 4)(x 2 3).

simplify to 2x 2 3.

state the simplifi ed form with any restrictions on the variable.

the denominator of the original expression is 0 when x 5 24. So the simplifi ed

form is 2(x 2 3), where x u 24.

to see if there are any common factors, factor the numerator and the denominator.


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Th e rate an object travels a distance d over time t is given by r 5 dt . Th is equation is an inverse variation since distance will be a constant and rate and time will vary inversely. Th is equation is also a rational function. One can fi nd the rate, or speed, by making calculations where d is the constant of variation. One could also fi nd the distance if given the rate and time.

1. Find the rate in ftsec when d 5 (x 2 1) miles and t 5 (x 1 20) hours by

completing the following steps. a. Write the equation that represents this problem given that 1 mile 5 5280 ft.

b. What is the simplifi ed version of this equation?

c. Solve the equation when d 5 (x 2 1) miles and t 5 (x 1 20) hours and

x 5 45.

d. How does the equation change if you know the rate and the time but want to fi nd the distance?

2. What is the rate, r, in ftsec if d 5 (x2 1 3x 1 2) ft and t 5 (x2 1 x 2 2) sec?

Simplify, then fi nd r when x 5 9.

11-1 EnrichmentSimplifying Rational Expressions

r 5 rate

d 5 distance

t 5 time

rate 5 distancetimedistance 5 rate 3 time

time 5 distancerate

r 5 d mit h 31h

3600 s 35280 ft

1 mi

r 5 22d ft15t s

r N 1fts

d 5 rt

r 5 x 1 1x 2 1 ft>s; r 5 1.25 ft>s

11-1 Standardized Test PrepSimplifying Rational ExpressionsMultiple Choice

For Exercises 1–4, choose the correct letter.

1. What is the simplifi ed form of x4 2 81x 1 3 ?

A. x3 2 3x2 1 9x 2 27 C. x3 1 3x2 1 9x 1 27

B. (x2 1 9)(x2 2 9) D. x3 1 3x2 2 9x 2 27

2. What is the simplifi ed form of (x2yz)2 (xy2z2)

(xyz)2?

F. (xyz)2 (xyz)

x G. 1

(xyz)2 H. x3y2z2 I. x4y4z

3. What is the excluded value of the rational expression 2x 1 64x 2 8?

A. 23 B. 22 C. 0 D. 2

4. What is the simplifi ed form of x2 2 25

x2 2 3x 2 10?

F. x 2 3x 2 5 G. (x 2 5)(x 1 5)(x 1 5)(x 2 3)

H. x 1 5x 1 2 I. x 2 5x 1 2

Short Response

5. When an object is free falling, the equation is d 5 16t2 where d is the distance in feet and t is the time in seconds. What happens to the distance d as t increases from 0 to 20? Draw the graph. Plot points for values of t between 0 and 20. (d 5 16t2is a quadratic equation. You can substitute y 5 16x2 to graph the equation.)

A

H

D

H

As t increases, d gets exponentially bigger. The values of d and t cannot be negative because there is no such thing as negative distance or time.

[4] Question answered correctly.[3] Answer includes correct graph and explanation with one

calculation error.[2] Answer includes either a correct graph or explanation but the

other is wrong.[1] Some work shown is correct.[0] No work shown is correct.

t

d

8

100020003000400050006000

16 24time (sec)

Dis

tanc

e (f

t)

O

11-1 Practice (continued) Form KSimplifying Rational Expressions 13. A mother is packing away winter clothes into two rectangular tubs. Both hold

the same volume of clothes. Th e fi rst tub has a length of 2b 1 5, a width of b 2 3, and a height of 4b. Th e second tub has a width of 4b2 1 20b and a length of b 2 3. What is a simplifi ed expression for the height of the second tub? Show your work.


14. x2 2 121

3x2 2 9x 15. v

3w3

v2w3

16. 5x2 2 41x 1 42

x2 2 49 17. 2t

4 1 t3 2 28t2

t2 1 4t

18. 9m2 2 32m 2 65

m2 2 25 19. 8a

2 2 12a 2 36a2 2 9

20. Writing Is x2 2 81x 2 9 the same as x 1 9? Explain.

21. Reasoning Is y 5 4 an acceptable value for the expression 3y2 2 10y 2 8

y2 2 16?

Explain.

x2 2 1213x2 2 9x

; x u 0, 3 v; v u 0, w u 0

5x 2 6x 1 7 ; x u w7 t(2t 2 7); t u 0, 24

9m 1 13m 1 5 ; m u w5

4(2a 1 3)a 1 3 ; a u w3

No; x2 2 81x 2 9 5

(x 1 9)(x 2 9)x 2 9 5 x 1 9, but x cannot equal 9 in the fi rst expression.

No; 42 2 16 5 0; This would make the denominator equal to 0.

(2b 1 5)(b 2 3)(4b) 5 h(4b2 1 20b)(b 2 3)

(2b 1 5)(b 2 3)(4b)(2b)(2b 1 5)(b 2 3) 5 h

2 5 h

11-1 Practice Form KSimplifying Rational ExpressionsSimplify each expression. State any excluded values.

1. 3n 2 1512 2. 12t8

36t6

3. y 1 2

y2 2 4 4. 15a 2 5010a 1 35

5. q2 2 16

7q2 1 28q 6. 5x

2 1 x 2 6x2 2 1

7. m3 1 9m

6m2 2 3m 8. 9z

2 2 3612z 1 24

9. Th e length of a rectangle is 8n 1 24 and the width is 12n 1 28. What is the ratio of its length to its width? Simplify your answer.

10. Th e area of a rectangle is x2 1 6x 2 16. Its width is x 2 2. What is a simplifi ed expression for its length?

11. Writing Describe how you determine what values should be excluded when simplifying a rational expression. Explain why this must be done.

12. Are the given factors opposites? Explain. a. 5x 2 2; 2 2 5x

b 2t 1 10; t 1 10

c. 102 1 11d; 2102 2 11d

n 2 54

t23 ; t u 0

1y 2 2; y u w4

3a 2 102a 1 7 ; a u 23.5

q 2 47q ; q u 0, 24

5x 1 6x 1 1 ; x u w1

m2 1 96m 2 3; m u 0,

12

3z 2 64 ; z u 22

2n 1 63n 1 7

x 1 8

Set each factor with a variable in the denominator equal to zero and solve for the variable. The denominator cannot equal zero.

yes, if you factor a negative out of one of the factors, the factors are equivalent.

no, if you factor a negative out of one of the factors, the factors are not equivalent.

yes, if you factor a negative out of one of the factors, the factors are equivalent.

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11-2 Think About a PlanMultiplying and Dividing Rational ExpressionsAuto Loans You want to purchase a car that costs $18,000. Th e car dealership off ers two diff erent 48-month fi nancing plans. Th e fi rst plan off ers 0% interest for 4 yr. Th e second plan off ers a $2000 discount, but you must fi nance the rest of the purchase price at an interest rate of 7.9% for 4 yr. For which fi nancing plan will your total cost be less? How much less will it be?

Know

1. Th e car costs .

2. Th ere are 2 diff erent -month plans.

3. Th e purchase price for Plan 1 is .

4. Th e interest rate for Plan 1 is .

5. Th e purchase price for Plan 2 is .

6. Th e interest rate for Plan 2 is .

Need

7. To solve the problems for the two plans, you need the .

Plan

8. Write the formula to be used by the 2 plans.

9. Set up the 2 formulas, 1 for each plan.

10. What is the total cost for Plan 1? What is the total cost for Plan 2?

11. For which fi nancing plan will the total cost be less and how much less?

$18,000

48

$18,000

0%

$16,000

7.9%

equation/formula

y 5 a(1 1 r)t

18,000(1)4; 16,000(1.079)4

Plan 1 5 $18,000; Plan 2 5 $21,687.31

Plan 1 costs $3687.31 less.

g

11-2 ELL SupportMultiplying and Dividing Rational ExpressionsTh ere are two sets of note cards below that show how Sylvia simplifi es the

expression x 1 35x 2 20 ?10x

x2 1 7x 1 12. Th e set on the left explains her thinking. Th e

set on the right shows the steps. Write the thinking and the steps in the correct order.

Think Cards Write Cards

Think Write

Divide out the common factors 5 and x 1 3.

x 1 35(x 2 4) ?

5(2x)(x 1 3)(x 1 4)

x 1 35x 2 20 ?

10xx2 1 7x 1 12

x 1 35(x 2 4) ?

5(2x)(x 1 3)(x 1 4)

2x(x 2 4)(x 1 4)

11(x 2 4) ?

2xx 1 4

Step 1

Step 2

Step 3

Step 4

Step 5

Write the problem.

Simplify.

Factor denominators.

Multiply numerators and multiply denominators. Leave the product in factored form.

x 1 35x 2 20 ?

10xx2 1 7x 1 12

x 1 35(x 2 4) ?

5(2x)(x 1 3)(x 1 4)

x 1 35(x 2 4) ?

5(2x)(x 1 3)(x 1 4)

11(x 2 4) ?

2xx 1 4

2x(x 2 4)(x 1 4)

Write the problem.

Factor denominators.

Divide out the common factors 5 and x 1 3.

Simplify.

Multiply numerators and multiply denominators. Leave the product in factored form.

g

Exercises


1. a4

a 2. 4x3

16x2 3. cd

2

3c2d

4. lml2m2n

5. 64y

16y2x 6. 2x

2 2 4xx

7. 5x3 2 15x2x 2 3 8.

x2 1 5x 1 6x 1 3

9. 2b 1 44 10. 3a 1 15

15

11. 3p 2 21

18 12. 4

4y 2 8

13. 7z 2 2814z 14. 9

18 2 81a

15. 535 2 5c 16. 2q 1 2

q2 1 4q 1 3

17. a 1 2a2 1 4a 1 4

18. 2x 2 22 2 2x

19. 9 2 x2

x 2 3 20. 2a 1 4

2

Write the opposite expression and simplify the opposite expression.

21. 10b5

40b4 22. 36 2 z

2

4z 2 24

23. x2 2 16x 2 4 24.

30 1 2z14 1 4z

11-1 Reteaching (continued)Simplifying Rational Expressions

a3, a u 0

5x2; x u 3

b 1 22

a 1 55

p 2 76

z 2 42z ; z u 0

17 2 c ; c u 7

1a 1 2 ; a u 22

2x 2 3, x u 3

210b5

40b4; 2b4, b u 0

z2 2 364z 2 24;

(z 1 6)4 , z u 6

2x2 1 16x 2 4 ; 2x 2 4, x u 4

2(30 1 2z)14 1 4z ;

215 2 z7 1 2z , z u 3.5

a 1 2

21; x u 1

12 2 9a ; a u

29

2q 1 3 ; q u 23 or 21

1y 2 2 ; y u 2

x 1 2; x u 23

1lmn; l u 0, m u 0, n u 0

x4; x u 0

4xy; x u 0, y u 0

d3c; d u 0, c u 0

2x 2 4; x u 0

g

11-1 ReteachingSimplifying Rational ExpressionsRational expressions may be in the form of monomials or polynomials.

Simplifying rational expressions is similar to simplifying numerical fractions where common factors are taken out.

Example: x 2 23x 2 6 5x 2 2

3(x 2 2)5

13

Excluded values are those that make the denominator 0. A denominator can not equal 0, so these values are not part of the solution. Consider not only the solution, but also the original expression to fi gure the excluded values.

Problem

What is the simplifi ed form of 2a3

4a2? State any excluded values.

Solve Monomials: reduce numbers; cancel out like variables

2a3

4a25

2 ? a ? a ? a2 ? 2 ? a ? a 5

a2

Th e simplifi ed form is a2 when a 2 0.

What is the simplifi ed form of x2 1 4x 1 4

x 1 2 ? State any excluded values.

Solve Polynomials: cancel out factors or groups of factors

x2 1 4x 1 4

x 1 2 5(x 1 2)(x 1 2)

x 1 2 5 x 1 2

Th e simplifi ed form is x 1 2 when x 2 22.

Recognizing Opposite Factors

You can fi nd the opposite of a number by multiplying by 21. For example, the opposite of 3 is (21)(3) 5 23.

Similarly, multiplying a polynomial by 21 results in its opposite. For example, the opposite of x 2 2 is (21)(x 2 2) 5 2x 1 2. It can also be written as 2 2 x .

Problem

Write the opposite of (20 2 x) two ways.

Solve Multiply by (−1) to fi nd the opposite.

(21)(20 2 x) 5 220 1 x or x 2 20


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11-2 Practice (continued) Form KMultiplying and Dividing Rational ExpressionsDivide.

12. 6f 2 63f 2 8 4

6f 2 6f 1 9 13.

12m 2 2027m 4

3m 2 59m

14. 18c 2 279t2 2 16

42c 2 33t 1 4 15.

2x2 2 23x 1 5610x 1 6 4

x 2 85x 1 3

Simplify each complex fraction.

16. 1x 2 3

3x 2 3

17. mn 1 2mn 1 5

18. A shipping box has a base area of 4x2 1 52x 1 168 and a height of x4x 1 28. What is the volume of the box?

19. Karl drives for (x2 2 100) hours at a rate of 15x 2 50 miles per hour. How far does Karl drive?

20. Open-Ended Write two rational expressions whose product is 1.

f 1 93f 2 8

43

93t 2 4

2x 2 72

13

m 1 2nm 1 5n

x(x 1 6)

x 1 105

Answers may vary. Sample: 1x 1 3 and x 1 3

1

g

11-2 Practice Form KMultiplying and Dividing Rational ExpressionsMultiply.

1. 5n2

3n2?

3n 2.

tt 2 3 ?

t 1 1t 1 2

3. 3a 2 93a 2 6 ?a

a2 2 9 4.

18q 2 362q ?

4q2

54q 2 18

5. m2 2 m 2 20m2 2 4m

?2m2

m2 2 25 6. 8v

6v2 1 22v 2 8?

3v 2 14v2

7. z2

z2 1 5z 2 6?

2z2 2 7z 1 56z2 2 15z

8. (3x2 1 7x 1 4) ? x2 2 4x

9x2 2 16x

9. Which of the following is the reciprocal of x2 2 2x 2 63?

a. 1(x 1 7)(x 2 9)

b. (x 1 7)(x 2 9) c. 1x 2 9

Find the reciprocal of each expression.

10. x2 2 2x 2 15 11. 6p2

7p2 2 12

5n

t(t 1 1)(t 2 3)(t 1 2)

a(a 1 3)(a 2 2)

2q(q 2 2)(3q 2 1)

2m(m 1 4)(m 2 4)(m 1 5)

1v(v 1 4)

z3(z 1 6)

1(x 2 5)(x 1 3)

7p2 2 12

6p2

(x 1 1)(x 2 4)3x 2 4

g

11-2 Practice (continued) Form GMultiplying and Dividing Rational ExpressionsDivide.

14. 5y 1 7

3y 1 19 45y 1 7y 2 6 15.

25i2 2 3656i 4

5i 2 68i

16. 12j 2 36

2j 1 4 43j 2 9

4j2 2 16 17. 12x

2 1 x 2 1345x2 2 20x 2 25

4x 2 1

9x 1 5

18. (72k2 1 29k 2 21) 4 9k2 2 92k 2 77

6k 2 1

Simplify each complex fraction.

19. 1 1 1

x9

20.

ab 1 1xb 1 3

21. 1a 1

ba

1b

22. A rectangular prism has a base area of 3x2 1 21x 2 24 and a height of x

33x 2 33. What is the volume of the prism?

23. Your friend runs for (x2 2 225) seconds at a rate of 12x 2 30 meters per

second. How far does your friend run?

24. Writing How do you simplify a complex fraction?

y 2 63y 1 19

8j 2 16

(8k 2 3)(6k 2 1)k 2 11

18x

x(x 1 8)11 units

3

x 1 152 meters

Multiply the numerator by the reciprocal of the denominator. Then factor out as many like terms as possible to simplify.

a 1 bx 1 3b

b 1 b2a

5i 1 67

12x 1 135x 2 5

g

11-2 Practice Form GMultiplying and Dividing Rational ExpressionsMultiply.

1. 2 2 z4 1 5z ?3z 2.

x 2 9x 1 7 ?

xx 2 6

3. 5w 2 255w 2 10

?w

w2 2 25 4. 16u 2 322u ?

3u356u 2 24

5. j2 1 11j 2 42

26j 2 52 ?39j

j 2 3 6. 15r

18r2 1 9r 2 27?

3r 2 3r2

7. 45q2 2 3q 2 6

q2?

14q2 1 10q

35q2 1 11q 2 10 8.

4y 1 172y 2 3 ? (32y

2 2 22y 2 39)

9. (12v2 1 18v 2 84) ? v4v3 2 49v

10. (10x2 2 7x 1 2) ? 6x2 2 13x 2 63

3x 1 7

11. Which of the following is the reciprocal of x2 2 2x 2 8?

A. (x 1 2)(x 2 4) B. 1(x 1 2)(x 2 4)

C. 1x 2 8

Find the reciprocal of each expression.

12. x2 2 4x 1 18 13. 3q2

2q2 2 13

6 2 3z4z 1 5z2

ww2 1 3w 2 10

3j2 1 42j2j 2 4

18q 1 6q

64y2 1 324y 1 221

6v 2 122v 2 7

1x2 2 4x 1 18 2q2 2 13

3q2

B

x2 2 9xx2 1 x 2 42

3u3 2 6u27u 2 3

52r2 1 3r

20x3 2 104x2 1 67x 2 18


91

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g

11-2 Reteaching (continued)Multiplying and Dividing Rational ExpressionsSimplifying Complex Fractions

You can use the Outers Over Inners method to simplify complex fractions.

Th e Outers Over Inners method sets up a simplifi ed fraction that looks like this:

Product of OutersProduct of Inners m

OutersInners 5

ADBC

For example, in the fraction:

6y52

4y

6y and 4y are

the “outer” terms; 5 and 2 are the “inner” terms.

If a numerator or denominator is not a fraction, make it a fraction by rewriting it as polynomial

1 .

Problem

Simplify 6y5 2

4y

.

Solve 5(6y)(4y)

(5)(2)5

24y2

10

Check Rewrite as numerator divided by denominator. 6y5 4

24y

Rewrite as a multiplication problem. 6y5 3

4y2 5

24y2

10

Exercises

Simplify using the Outers Over Inners method.

10.

(g 1 4)2g 11.

(x 1 1)2x3

Outers Inners

ABCD

OutersInners

(g 1 4)2g

3(x 1 1)2x

g

Th ere are many types of complex fractions.

A complex fraction can be a fraction with one or more additional fractions in the numerator, or in the denominator, or in both the numerator and the denominator.

Problem

Is

5x3

6x2

x 1 1 a complex fraction? Explain.

Solve

Ask: Is the numerator a fraction? m No. 5x3 is not a fraction.

Ask: Is the denominator a fraction? m Yes. 6x2

x 1 1 is a fraction.

A fraction is in the denominator m

5x3

6x2

x 1 1 is a complex fraction.

Exercises

Tell if the following terms are complex fractions. Explain your reasoning.

1. 4y5

92y

2. 23 1 8z 3. 1x 1 2

x 2 2

4. 2x3

5x 5.

3x2

x 1 8

x3 6.

4x 1 92x 1 8

5x 2 63x 1 7

7. x 2 27

x 1 4 8.

2x x5 9.

13x2

x 1 2

4x3x 1 16

11-2 ReteachingMultiplying and Dividing Rational Expressions

yes; It has a fraction in the numerator and the denominator.

no; It does not have a fraction in the numerator or the denominator.

yes; It has a fractionin the numerator.

yes; It has a fraction in the numerator.


yes; It has a fractioin the numerator athe denominator.


yes; It has a fraction in the numerator and the denominator.

yes; It has a fraction in the numerator and thedenominator.

g

11-2 EnrichmentMultiplying and Dividing Rational ExpressionsSequences of Rational Expressions

Sometimes the terms of a sequence are numbers; sometimes the terms are algebraic expressions, including rational expressions.

Consider the sequence xx 1 1, x 1 1x 1 2,

x 1 2x 1 3,

cwhere each successive term is

determined by adding 1 to each of the numerator and denominator.

1. What are the fourth and fi fth terms of this sequence?

2. Compute the ratio of the second term to the fi rst, the ratio of the third term to the second, the ratio of the fourth term to the third, and the ratio of the fi fth term to the fourth.

3. Evaluate each of the ratios you computed in Exercise 2 for x 5 0, 10, and 100.

4. What seems to be happening to the ratios as the value of x increases? Explain.

5. Form a new sequence with the fi rst term xx 1 1, but for which the successive terms are determined by subtracting 1 from each of the numerator and denominator.

6. Compute the ratios of successive terms of the new sequence.

7. Evaluate each of the ratios you computed in Exercise 6 for x 5 0, 10, and 100.

8. What seems to be happening to the ratios as the value of x increases? Explain.

x 1 3x 1 4

, x 1 4x 1 5

(x 1 1)2

x(x 1 2), (x 1 2)2

(x 1 1)(x 1 3), (x 1 3)2

(x 1 2)(x 1 4), (x 1 4)2

(x 1 3)(x 1 5)

0; undefined, 43, 98,

1615; 10:

121120,

144143,

169168,

196195; 100:

10,20110,200,

10,40410,403,

10,60910,608,

10,81610,815

The ratios are greater than 1 but are getting closer and closer to 1.

xx 1 1,

x 2 1x ,

x 2 2x 2 1,

x 2 3x 2 2,

x 2 4x 2 3

(x 1 1)(x 2 1)x2

, x(x 2 2)(x 2 1)2

, (x 2 1)(x 2 3)(x 2 2)2

, (x 2 2)(x 2 4)(x 2 3)2

0: undefined, 0, 34, 89; 10:

99100,

8081,

6364,

4849; 100:

999910,000,

98009801,

96039604,

94089409

The ratios are less than 1 but are getting closer and closer to 1.

g

11-2 Standardized Test PrepMultiplying and Dividing Rational ExpressionsMultiple Choice


1. What is the quotient x2 2 16

2x2 2 9x 1 44

2x2 1 14x 1 244x 1 4 ?

A. 1x 1 3 B. 2x 1 2x 1 3 C.

2x 1 22x2 1 5x 2 3

D. 2(x 1 1)

2x2 2 5x 2 3

2. What is the simplifi ed form for the product x 1 1x2 2 25

?x 1 5

x2 1 8x 1 7 ?

F. x 1 1(x 1 5)(x 1 7)

H. 1(x 2 5)(x 1 7)

G. 1(x 1 5)(x 1 7)

I. 1(x 2 5)(x 2 7)

3. What are the coordinates of the x-intercepts of the graph of y 5 2x2 1 6x 2 20?

A. (25, 0), (2, 0) B. (5, 0), (22, 0) C. (24, 0), (10, 0) D. (4, 0) (210, 0)

Short Response

4. A football is kicked with an upward velocity of 25fts from a starting height of 0.5 ft. How long does the ball stay in the air if no one catches it? Use the

formula h 5 2162 1 vt 1 c, where h is the ball’s height at time t, v is the initial upward velocity, and c is the starting height. What is the height when the time is 1 second? How does the height change as the time increases? What happens at 5 seconds?

C

H

A

9.5 ft; as time increases, the height increases, reaches its maximum, and then starts to decrease; at 5 s the height is a negative value. This is not possible, so it means the ball has already landed on the ground.

[2] Questions answered correctly.

[1] Answer is incomplete.

[0] Answer is wrong.


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11-3 Practice (continued) Form GDividing PolynomialsDivide.

24. (28n2 2 17n 2 3) 4 (4n 2 3) 25. (2t2 2 8t 1 6) 4 (t 2 3)

26. (3c2 2 5c 2 2) 4 (6c 1 2) 27. (3c2 2 5c 2 2) 4 (3c 1 1)

28. (2j2 2 3j 2 9) 4 (j 2 3) 29. (4j2 2 6j 2 18) 4 (j 2 3)

30. (23x2 1 x3 2 x 1 3) 4 (x 2 1) 31. (3x2 2 x3 1 x 2 3) 4 (2x 1 1)

32. 9d4 2 729 4 (23 1 d) 33. (23x2 1 6x3 1 x 2 40) 4 (22 1 x)

34. Find the height of a trapezoid if the area of the trapezoid is 2x2 1 11x 1 5, the length of one base is x, and the length of the other base is x 1 10. Th e formula for

the area A of a trapezoid with height h and bases b1 and b2 is A 5b1 1 b2

2 ? h.

35. Th e area of the rectangle is x4 2 9x3 2 7x2 2 8x 1 2. Th e length is given. What is the width?

36. Writing If the area of a rectangle is a polynomial and the length of one of the sides is a polynomial, can the measurement of the width of the rectangle be a polynomial quotient with a remainder? Explain.

x2 x 1

7n 1 1 2t 2 2

12 c 2 1

c 2 2

2j 1 3 4j 1 6

x2 2 2x 2 3 x2 2 2x 2 3

9d3 1 27d2 1 81d 1 243 6x2 1 9x 1 19 1 22x 2 2

2x 1 1

x2 2 10x 1 2

yes; the division of a polynomial by another polynomial can yield a polynomial with or without a remainder.

g

11-3 Practice Form GDividing PolynomialsDivide.

1. (c2 2 c 2 1) 4 c 2. (j4 2 4j3 2 8j2) 4 j2

3. (3p3 2 27p2) 4 3p2 4. (2m2 2 5m 1 2) 4 2m

5. (3b5 2 9b4 1 3b2) 4 6b2 6. (7x4 2 28x3) 4 4x3

7. (6t5 2 3t4 1 18t3 2 9t2) 4 3t 8. (2104d8 1 64d7 2 86d6 1 96d5) 4 2d4

9. (227q4 1 51q3 2 9q2) 4 3q2

10. (21040r12 2 500r11 2 620r10 1 1600r9 1 r8) 4 20r7

11. (23u6 2 105u5 1 147u4) 4 (23u3) 12. (11y26 2 132y25 1 121y24) 4 (211y24)

13. (p2 1 3p 1 2) 4 (p 1 1) 14. (x2 1 7x 1 12) 4 (x 1 4)

15. (p2 2 5p 2 36) 4 (p 1 4) 16. (2q2 2 4q 2 240) 4 (q 2 12)

17. (6x2 1 x 2 1) 4 (3x 2 1) 18. (20a2 1 2a 2 4) 4 (2a 1 1)

19. (4t2 2 64) 4 (t 1 4) 20. (z2 2 9) 4 (z 2 3)

21. (3x2 2 x3 1 x 2 3) 4 (2x 1 1) 22. (c4 2 16) 4 (c 2 2)

23. Th e area of a rectangle is x2 2 x 2 2 and the length of the rectangle is x 1 1.

a. Find the width of the rectangle.

b. Find the area of the rectangle if the width is 4 m.

c 2 1 2 1c f2 2 4f 2 8

p 2 9 m 2 52 11m

b32 2

3b22 1

12

7x4 2 7

2t4 2 t3 1 6t2 2 3t 252d4 1 32d3 2 43d2 1 48d

29q2 1 17q 2 3

252r5 2 25r4 2 31r3 1 80r2 1 r20

u3 1 35u2 2 49u 2y2 1 12y 2 11

p 1 2 x 1 3

p 2 9 2q 1 20

2x 1 1 10a 2 4

4t 2 16 z 1 3

x2 2 2x 2 3 c3 1 2c2 1 4c 1 8

x 2 2

28 m2

g

Geometry Th e volume of the rectangular prism shown at the right is m3 1 8m2 1 19m 1 12. What is the area of the base of the prism?


1. What is a rectangular prism?

2. What is the literal formula for the volume of a rectangular prism?

3. What is the formula for the volume of the rectangular prism shown?


4. Using the given information from Step 3, what is the area of the base of the rectangle equal to?

Getting an Answer

5. Divide (m3 1 8m2 1 19m 1 12) by (m 1 3) to fi nd the area of the base of the prism.

6. What does m2 1 5m 1 4 equal?

11-3 Think About a PlanDividing Polynomials

m 3

V 5 l 3 w 3 h

V 5 lw(m 1 3) 5 m3 1 8m2 1 19m 1 12

B 5 lw 5 m3 1 8m2 1 19m 1 12

m 1 3

m2 1 5m 1 4

the area of the base of the prism (B)

A rectangular prism is a solid (3-dimensional) object comprised of six faces that are

rectangles.

g

11-3 ELL SupportDividing PolynomialsWhat is the solution to (60x12 1 24x10 1 24x3) 4 12x2? Justify your steps.

(60x12 1 24x10 1 24x3) 4 12x2 Copy the problem.

(60x12 1 24x10 1 24x3) 112x2

Multiply by 112x2

, the reciprocal of 12x2.

60x12

12x21

24x10

12x21

24x3

12x2 Use the Distributive Property.

5x10 1 2x8 1 2x Subtract exponents when dividing powers with the same base.

Exercises

What is the solution to (44x5 1 55x6 1 22x3) 4 11x3? Justify your steps.

(44x5 1 55x6 1 22x3) 4 11x3 __________________________________

(44x5 1 55x6 1 22x3) 111x3

__________________________________

44x5

11x31

55x6

11x31

22x3

11x3 __________________________________

4x2 1 5x3 1 2x0 __________________________________

4x2 1 5x3 1 2 __________________________________

What is the solution to (236x4 1 18x7 1 30x5) 4 6x5? Justify your steps.

(236x4 1 18x7 1 30x5) 4 6x5 __________________________________

(236x4 1 18x7 1 30x5) 16x5

__________________________________

236x4

6x51

18x7

6x51

30x5

6x5 __________________________________

_____________________________________ _______________________________

_____________________________________ _______________________________

Copy the problem.

Multiply by 111x3


Use the Distributive Property.

Subtract exponents when dividing powers with the same base.

Simplify.

Copy the problem.

Multiply by 16x5



Subtract exponents when dividing powers with the same base.

Simplify.

2 6x21 1 3x2 1 5x0

26x 1 3x

2 1 5


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11-3 Enrichment Dividing Polynomials Dividing xn 2 1 by x 2 1

Th ere is a pattern in the quotients when polynomials of the form xn 2 1 are divided by x 2 1. Start by dividing x2 2 1 by x 2 1. Since there is no x term in x2 2 1, rewrite x2 2 1 as x2 1 0x 2 1 before performing long division.

x 1 1x 2 1qx2 1 0x 2 1

x2 2 1x2222x 2 1x 2 1

0

Th e quotient is x 1 1.

1. Divide x3 2 1 by x 2 1.

2. Divide x4 2 1 by x 2 1.

3. Based on the example and the fi rst two exercises, predict the quotient when x5 2 1 is divided by x 2 1.

4. Check your guess for Exercise 3 by performing long division for x5 2 1 by x 2 1.

5. If your guess was not correct, repeat Exercises 3 and 4 with higher exponents until you are able to predict the result of the division.

6. State the pattern in words for the quotient when xn 2 1 is divided by x 2 1.

x2 1 x 1 1

x3 1 x2 1 x 1 1

Answers may vary. Actual answer: x4 1 x3 1 x2 1 x 1 1

x4 1 x3 1 x2 1 x 1 1

Check students’ work.

The quotient is the sum of xn21 1 xn22 1c 1 x 1 1.

g

11-3 Standardized Test PrepDividing PolynomialsMultiple Choice


1. What is the remainder of (x3 2 6x2 2 9x 1 3) 4 (x 2 3)?

A. 251 B. 251x 2 3 C. 217

x D. 217

x 2 3

2. If the area of a rectangle is x2 2 9, can the length be x 2 1? F. Yes; it divides perfectly. G. No; the length would be larger. H. Yes; but the width will have a remainder. I. No; the measurements of the rectangle must multiply and divide evenly.

3. If a line passes through the points (2, 0) and (22, 23), what is the y-intercept of the line?

A. 232 B. 32 C.

34 D.

234

4. What coordinates satisfy the equation of the line y 5 278x 1 2?

F. (28, 11) G. Q24, 112 R H. both F and G I. Q4, 112 R

Short Response

5. What are the factors of the expression x2 1 12x 2 64? What numbers would you change to make it a perfect square? Explain.

A

H

A

G

(x 1 16), (x 2 4); change −64 to 36, then the factors would be (x 1 6), (x 1 6)

[2] Questions answered correctly.[1] Answer is incomplete.[0] Answer is wrong.

g

11-3 Practice (continued) Form KDividing PolynomialsDivide.

18. (9x2 1 59x 2 28) 4 (9x 2 4) 19. (11v2 1 21v 2 2) 4 (v 1 2)

20. (2p2 1 3p 2 44) 4 (p 2 4) 21. (10j2 1 93j 2 70) 4 (10j 2 7)

22. (6a2 1 7a 2 3) 4 (3a 2 1) 23. (10h2 2 15h 1 5) 4 (2h 2 1)

24. Th e area of a rectangle is x2 2 2x 2 15 and the length of the rectangle is to x 1 3.

a. Find the width of the rectangle.

b. Find the area of the rectangle if the width is 6 ft.

25. Reasoning If x 2 2 is a factor of x2 1 3x 1 k, what is the value of k?

26. Open-Ended Write a monomial and a trinomial such that the monomial is a factor of the trinomial. Explain how you know that the monomial is a factor of the trinomial.

x 1 7 11v 2 1

2p 1 11 j 1 10

2a 1 3 5h 2 5

x 2 5

84 ft2

210

Answers may vary. Sample: 3x3 1 6x2 1 9x and 3x; 3x divides evenly into 3x3 1 6x2 1 9x .

g

11-3 Practice Form KDividing PolynomialsDivide.

1. (y2 2 2y) 4 y 2. (a4 1 7a3 2 3a2) 4 a2

3. (5g5 2 15g3) 4 5g3 4. (6t2 1 8t 1 10) 4 2t

5. (2q5 2 12q3 1 9q2) 4 6q2 6. (9k4 2 12k3) 4 2k3

7. (218w4 1 48w3 2 24w2) 4 6w2 8. (24n6 2 52n4 1 36n2) 4 (24n3)

9. (n2 1 9n 1 20) 4 (n 1 4) 10. (y2 1 6y 1 5) 4 (y 1 5)

11. (h2 2 5h 2 14) 4 (h 1 2) 12. (3d2 2 12d 2 96) 4 (d 2 8)

13. (2m2 2 5m 2 3) 4 (2m 1 1) 14. (6r2 1 43r 2 40) 4 (6r 2 5)

15. (5c2 2 125) 4 (c 1 5) 16. (k2 2 36) 4 (k 2 6)

17. Find the height of a trapezoid if the area of the trapezoid is 4x3 2 2x2, the length of one base is 3x 1 1, and the length of the other base is x 2 3.

(Hint : Th e height of a trapezoid equals 2 ? AreaBase1 1 Base2.)

y 2 2 a2 1 7a 2 3

g2 2 3 3t 1 4 1 5t

q3

3 2 2q 132

9k2 2 6

23w2 1 8w 2 4 n3 1 13n 2 9n

n 1 5 y 1 1

h 2 7 3(d 1 4)

m 2 3 r 1 8

5(c 2 5) k 1 6

2x2


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g

11-4 Think About a PlanAdding and Subtracting Rational ExpressionsRowing A rowing team practices rowing 2 mi upstream and 2 mi downstream. Th e team can row downstream 25% faster than they can row upstream. a. Let u represent the team’s rate rowing upstream. Write and simplify an expression involving u for the total amount of time they spend rowing. b. Let d represent the team’s rate rowing downstream. Write and simplify an expression involving d for the total amount of time they spend rowing. c. Reasoning Do the expressions you wrote in parts (a) and (b) represent the same time? Explain.

Know

1. What is the distance the team rows upstream?

2. What is the distance the team rows downstream?

3. How much faster does the team row downstream?

Need

4. What is the formula for time written in terms of the distance d and the rate r?

5. Write expressions for the time rowing upstream and the time rowing downstream by substituting values for the distance and variables for the rate in the formula in Exercise 4.

a. Time rowing upstream 5

b. Time rowing downstream 5

Plan

6. Write an equation that relates to u and d.

7. Rewrite the expressions you wrote in Exercise 6 in terms of u.

a.

b.

8. Do the expressions you wrote in Exercise 7 represent the same time? Why or why not?

no; the distance is the same, but the trip downstream is at a faster rate, so it will take

less time. So, 21.25u R2u.

2 mi

2 mi

25% or 0.25 faster

t 5 dr

2u

2d

d 5 1.25u

2u

21.25u

g

11-4 ELL SupportAdding and Subtracting Rational ExpressionsUse the list below to complete the Venn diagram.

If a, b, and c represent polynomials (with c 2 0), then ac 1

bc 5

a 1 bc .

Add the numerators. Write the expressions with the least common denominator.

Simplify the numerator. If a, b, and c represent polynomials (with c 2 0), then ac 2

bc 5

a 2 bc .

Subtract the numerators.

Adding Rational Expressions

Subtracting Rational Expressions

If a, b, and c represent

polynomials (with c 2 0),

then ac 1bc 5

a 1 bc .

Add the numerators.

Simplify the

numerator.

Write the

expressions with

the least common

denominator.

If a, b, and c

represent polynomials

(with c 2 0), then

ac 2

bc 5

a 2 bc .

Subtract the numerators.

g

Dividing a polynomial by a binomial is similar to long division.

Problem

What is (p2 2 3p 1 2) 4 (p 2 2)?

Solve

p 2 2qp2 2 3p 1 2 Write the problem as long division.

p

p 2 2qp2 2 3p 1 2 Ask how many times p goes into p2 (p times). The variables must

align by exponent, so the p goes above 23p since both match.

pp 2 2qp2 2 3p 1 2

p2 2 2p000021p 1 2

Multiply p times (p 2 2). Subtract the product p2 2 2p. Bring down 2.

p 2 1p 2 2qp2 2 3p 1 2

p2 2 2p000021p 1 221p 1 2

Determine how many times p goes into 21p ( 21 times). Multiply (21) times (p 2 2) to get 21p 1 2. Subtract the product 21p 1 2. There is no remainder.

So p 2 2 goes into p2 2 3p 1 2 exactly p 2 1 times with no remainder.

Exercises

Divide.

5. (d2 1 4d 2 12) 4 (d 2 2) 6. (y2 2 4y 1 4) 4 (y 2 2)

7. (x2 2 2x 1 1) 4 (x 2 1) 8. (b2 2 b 2 20) 4 (b 2 5)

11-3 Reteaching(continued) Dividing Polynomials

d 1 6 y 2 2

x 2 1 b 1 4

g

A few important rules are needed for successful division of a polynomial. When dividing by a monomial, a single term, remember to divide each polynomial term by the monomial and reduce the fraction.

Problem

What is (8x3 2 3x2 1 16x) 4 2x2 ?

Solve

(8x3 2 3x2 1 16x) 4 2x2 Division equals multiplication by the reciprocal.

5 (8x3 2 3x2 1 16x) ? 12x2

Multiply by 12x2


58x3

2x22

3x2

2x21

16x2x2


5 4x1 2 32x0 1

8x Subtract exponents when dividing powers

with the same base.

5 4x 2 32 18x Simplify.

Exercises

Divide.

1. (2x2 2 9x 1 18) 4 2x 2. (16x4 2 64) 4 4x3

3. (x5 2 3x4 1 10x3 2 34x2 2 6) 4 3x2 4. (5x3 2 25x2 2 1) 4 5x

When a polynomial (many terms) is divided by a binomial (2 terms), the polynomial terms should be in order from highest to lowest exponent.

To make the polynomial 24 1 2x 2 16x2 1 3x3 division ready, put it in the correct order for division, from greatest exponent to lowest. Th e correct order for

24 1 2x 2 16x2 1 3x3 to be division ready is 3x3 2 16x2 1 2x 2 4.

For any gaps or missing exponents, the place is held with 0. For example, x2 1 1 becomes x2 1 0x 1 1, 0x being the placeholder for the x term.

11-3 Reteaching Dividing Polynomials

x 2 92 19x 4x 2

16x3

x33 2 x

2 1 10x3 214 2

2x2

x2 2 5x 2 15x


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11-4 Practice (continued) Form KAdding and Subtracting Rational ExpressionsAdd or subtract.

13. 2m 15n 14.

3t 1 3 1 5

15. 1 1 xy 16. ab 1

2c(b 2 2)

17. 2.22n 28.83n 18.

3 1 2w

10 2 7

19. What is the perimeter of a rectangular area rug that is 3 1 p

3 ft long and 4p 2 6

5 ft wide?

20. Jennifer rode her bike to the store at a rate of 15 mi/h. She rode back home at a rate of 10 mi/h. How far is it to the store if the round trip takes 1 hour?

21. Writing Why would you change a rational expression with a denominator of x2 2 6x 1 8 to (x 2 2)(x 2 4) when adding or subtracting rational expressions?

22. Open-Ended Write a problem that uses addition of rational expressions in which you need to fi nd an LCD. Simplify the expression.

2n 1 5mmn

18 1 5tt 1 3

y 1 xy

ac(b 2 2) 1 2bbc(b 2 2)

2116n5

3w

34p 2 615 ft

6 mi

You change the denominator to make it simpler to fi nd the LCD.

Answers may vary. Sample: 12x 135 5

6x 1 510x

g

11-4 Practice Form KAdding and Subtracting Rational ExpressionsAdd or subtract.

1. 2n 12n 2.

87p 1

197p

3. x 2 3x 2x 1 3

x 4. 1

2 2 b 1b

b 2 2

5. 4d 22d2 6.

2k7 2

75k

Find the LCD of each pair of expressions.

7. 15, 2g 8.

23m2n

, 7mn2

9. 6x 2 3 , 1

x 1 4 10. 2y

y2 1 1 ,

y2

3

11. Writing Explain how you can fi nd the LCD when the denominators are diff erent.

12. What do you need to do to the numerators when using the LCD to add or subtract the rational expressions? Explain.

4n

277p

26x21 1 bb 2 2

4 2 d2

d10k2 2 49

35k

5g

(x 2 3)(x 1 4)

3m2n2

3(y2 1 1)

You can multiply the two denominators by each other.

You multiply the numerators by the same number as you multiplied the denominator by so that you do not change the value of the expression.

g

11-4 Practice (continued) Form GAdding and Subtracting Rational ExpressionsAdd or subtract.

20. 3a 14x 21.

2x 2 1 1 10 22.

2x2 1 3

23. 1 1 ab 24. mx 1

4m(x 2 1)

25. ab 1xy

26. 21.54x 25.53x 27.

1 1 2x

8 2 3 28.

1x x 2 1

4

29. Your friend bought n 1 8 outfi ts and her sister bought n 1 2n 1 3 outfi ts. How many total outfi ts did they buy?

30. What is the perimeter of a rectangular garden that is 5 1 x2 ft long and 2x 2 1

3 ft wide?

31. Your brother ran to school at a rate of 6 mi/h. He walked back home at a rate of 4 mi/h. How far is it to school if the round trip takes 1 hour?

32. Adding two rational expressions leads to a solution of 5x6 . One expression is x3.

What is the other one? Show your work.

33. Writing Explain how to use opposites to fi nd the sum 81 2 2x 1x

2x 2 1.

34. Open-Ended Write a problem that uses addition of rational expressions.

2 2 1x

3x 1 4aax

10x 2 8x 2 1

2x 1 62

b 1 ab

m2x 2 m2 1 4xmx2 2 mx

ay 1 bxby

42.5x12x

35x

2x 2 1x3 2 x2

n2 1 12n 1 26n 1 3

7x 1 133

2 25 mi

5x6 2

x3 5

5x 2 2x6 5

3x6 5

x2

You can multiply 1 2 2x by 21 to get 21(2x 2 1), so the common denominator is 21(2x 2 1).

Answers may vary. Sample: A garden is 2 1 xx ft long and 3

x 2 1 ft wide. What is the perimeter of the garden?

g

11-4 Practice Form GAdding and Subtracting Rational ExpressionsAdd or subtract.

1. 1a 11a 2.

112y 1

272y 3.

mm 1 4 1

4m 1 4

4. t 2 1t 2t 1 1

t 5. n

1 2 n 11

n 2 1 6. 1 2 mm 2 4 2

22m 1 1m 1 4

7. 2y 23y8 8.

4x3 2

34x 9.

2a 1 1a 1

a 1 22

Find the LCM of each pair of expressions.

10. 6x; 13 11. 40x2y2; 8y2 12. 3a 2 3; 3

13. z2 2 4; z 1 2 14. 4d2 2 64; 4 15. 10a2b4c4; 5ab3c2

16. Does it matter whether you use the LCD fi rst or the GCF fi rst when adding or subtracting a rational expression with diff erent denominators and simplifying? Use an example to justify your claim.

17. Is there ever a time when it is all right to add or subtract the denominators when adding or subtracting a rational expression? Explain.

Simplify. Add or subtract.

18. x 2 32(x 1 5)

11x 19.

3x2x 2 2x

2a

19y

1

22t

21 m2 2 12m 1 8m2 2 16

16 2 3y2

8y16x2 2 9

12xa2 1 6a 1 2

2a

2x 40x2 y2 3a 2 3

z2 2 4 4d2 2 64 10a2b4c4

x2 2 x 1 102x2 1 10x

3 2 4x2

You should get the same result no matter which technique you use fi rst. Check students’ work.

No. The denominator must be the same to add or subtract the numerator.


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11-4 Reteaching (continued)Adding and Subtracting Rational ExpressionsExercises

Add.

1. 4p 19p 2.

2.5x 1

1.25x

3. 2y

x2y21

x2

x2y2 4.

j2

jm2n1

3mjm2n

12n3

jm2n

5. 12a 12a 6.

73d 1

37d

7. 112m 13

4m 8. 37s 1

114s

9. 8.916t 12.19t 10.

5x 1

xx2

11. 3n 1 1 14

n 1 2 12. 2x

x2 2 91

4x 1 1x 1 3

Subtract.

13. 2y 25

3y 14. 1

2x 21

3x

15. n 2 12 22n4 16.

37y 2

63y

17. 112p 21

12p 18. 4

x 1 2 22

x 2 8

19. 10x 1 5 2x 1 1

2x 20. 3x2 2

x14x

21. 4x 1 2 2 2(x 2 2) 22. 5k2 2

6k

13p

3.75x

x2 1 2y

x2y2j 2 1 3m 1 2n2

jm2n

52a

5821d

56m

8928s

113.7144t

6x

7n 1 10(n 1 1)(n 1 2)

4x2 2 9x 2 3x2 2 9

13y

16x

12

2117y

6512p

2x 2 36x2 2 6x 2 16

2x2 1 14x 2 52x(x 1 5)

(21x 2 1)14

22x2 1 12x 1 2

5k2 2 122k

g

11-4 ReteachingAdding and Subtracting Rational Expressions Problem

What is the diff erence x 1 12x 1 1 22 2 x

2x 1 1?

x 1 1

2x 1 1 22 2 x

2x 1 1 The fractions have like denominators.

5x 1 1 2 (2 2 x)

2x 1 1 Subtract the numerators.

5x 1 1 2 2 1 x

2x 1 1 Distributive Property

52x 2 12x 1 1 Simplify the numerator.

Adding or subtracting rational expressions is similar to adding or subtracting fractions. First fi nd a common denominator, the LCD. Once the denominator is the same, the numerators can be added to or subtracted from one another. Since numerators and denominators are multiplied by the same number, it is the same as multiplying by 1. After adding or subtracting numerators, simplify, and reduce the rational expressions.

For example: 12b 13b 5

12b 1 Q

22 ?

3bR 5

12b 1

62b 5

1 1 62b 5

72b

Problem

What is the sum 58x2

12

3x2?

5

8x21

23x2

Because the fractions have different denominators, fi nd the lowest common denominator (LCD). The LCD is the smallest number that both factors have in common.

In this case, (8 ? 3)x2 or 24x2 is the LCD. The common denominator is 24x2.

533 ?

58x2

188 ?

23x2

Rewrite each fraction using the LCD. 8x2 needs to be multiplied by 3 to equal 24x2. And 3x2 needs to be multiplied by 8 to equal 24x2.

Notice that 33 5 1 and 88 5 1.

515

24x21

1624x2

Simplify numerators and denominators.

515 1 16

24x2 Add the numerators.

531

24x2 Simplify.

g

11-4 EnrichmentAdding and Subtracting Rational ExpressionsTh ere are many real-world applications for rational expressions. Rational expressions are not simply theoretical problems in mathematics, but can be used daily to solve problems of speed, time, and distance just to name a few. For instance, for problems involving rate r (which in this case is equal to speed),

distance d, and/or time t, the formula r 5 dt can be used. Depending upon which variable is being solved for, one can use this formula interchangeably to fi nd rate, distance, and time.

1. A family spent their summer vacation driving across the United States on a road trip. Th ey drove d miles before starting back home. Th eir car gets 62 miles per gallon. Because of the diff erent roadways they used on the way back, they drove 140 miles less.

a. If gas on the way up cost, on average, $3.65 per gallon and on the way back cost, on average, $3.05 per gallon, write an expression that shows how much the family spent on gas altogether.

b. Simplify the expression in part (a).

c. Th e family drove 1364 miles before starting back home. How much money did they spend on gas?

2. A student on the cross-country team ran around the school course, 5 miles, at a rate of r mi/h, and ran straight back to the school, 3 miles, at a rate of r 2 1 mi/h.

a. Write an expression to show the student’s time for the fi rst part of the run.

b. Write an expression to show the student’s time for the second part of the run.

c. Write an expression that shows the student’s time for the entire run. Th en fi nd the sum.

d. Th e student ran the fi rst part of the run at a rate of 6 mi/h. How long did it take the student to complete the entire run?

3.65Q d62R 1 3.05Qd 2 140

62 R

6.7d 2 42762

$140.51

5r

3r 2 1

5r 1

3r 2 1;

8r 2 5r(r 2 1)

4330 h = 1

1330 h = 1 h 26 min

g

11-4 Standardized Test PrepAdding and Subtracting Rational ExpressionsMultiple Choice


1. What is the diff erence 5x 2 24x 2x 2 2

4x ?

A. 1 B. x 2 1x C. 0 D. 32

2. What is the sum 12b 1b2?

F. b 1 12b 1 2 G. 2b H. 14 I.

b2 1 12b

3. What is the sum 1g 1 2 13

g 1 1?

A. 3g 1 3 B. g 1 3

(g 1 1)(g 1 2) C.

4g 1 7(g 1 1)(g 1 2)

D. 2g 1 3

(g 1 1)(g 1 2)

4. What is the diff erence r 1 2r 1 4 23

r 1 1?

F. 21r 1 3 G. r2 2 1

(r 1 1)(r 1 4) H. r

2 2 10(r 1 1)(r 1 4)

I. r2 1 14

(r 1 1)(r 1 4)

5. What is the sum a 2 1abc3

13 2 babc3

?

A. a 2 b 2 3abc3

B. a 2 b 1 2abc3

C. a 2 4 1 babc3

D. 23c3

Short Response

6. Elena went on a 6-mile walk. She completed the fi rst half of the walk 1 mi/h faster than usual and the second half of the walk 2 mi/h slower than the fi rst half.

a. If it took her 7.2h to complete the walk, what is her usual rate? b. What is the formula necessary to solve this problem?

1.5 mi/h7.2 5 3x 1 1 1

3x 2 1

[2] Both parts answered correctly.

[1] One part answered correctly.

[2] Neither part answered correctly.

A

I

C

H

B


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11-5 Practice (continued) Form GSolving Rational Equations 19. It takes you 12 hours to paint a house, your brother 14 hours, and your sister

10 hours. If all three of you work together, how long will it take you to paint the house?

20. Maria, LaShawn, and Mike are all students. It takes Maria 8 hours to write half of her paper for history class. It takes LaShawn 2x hours to write one third of her paper, and Mike takes (x 2 2) hours to write half of his paper. If the teacher tells them they can work on the paper as a group, how long will it take them to complete it?

21. Error Analysis Edward solved the rational equation 3x(x 2 2)

x 2 x Q963xR 5 3x Q13R and got an answer of x 5 219. What was his

mistake?

22. Writing Write a rational equation that has n 5 10 for the answer. Include at least 3 terms in your equation, one of which should be a quadratic equation or a perfect square.

23. A pool has 2 pipes, one to fi ll it and one to empty it. Ms. Simon wants to fi ll the pool, but she mistakenly turns on both pipes at the same time. Th e pipe that fi lls the pool can fi ll it in 6 hours and the one that drains it can do that job in 10 hours. How long will it take to fi ll the pool now that both pipes are fi lling and emptying it at the same time?

24. What is the LCD of the equation t(t 2 2)2t 2 3 2 4Q

1t R 5 5t 2

3(t 1 4)t 1 1 ?

Solve each equation. Check your solutions.

25. cc 1 4 13

c 2 3 516

c2 1 c 2 12 26. 12y 1 1 2

(y 1 4)(y 2 4)y 2 2 5 21

3.93 h

13x2 1 26x 2 16

h

15 hours

t(2t 2 3)(t 1 1)

2 or 22 5

When he solved to the point that 22x 5 238, he didn’t cancel both negative signs that were on both sides of the equation. He should have gotten x 5 19.

Answers will vary. Sample: (n2 2 4)>(n 1 2) 1 2>3 (3n) 5 2(n 1 4)

g

11-5 Practice Form GSolving Rational EquationsSolve each equation. Check your solutions.

1. 12 2 j 1 2 54

2 2 j 2. 8

c 1 2 2 6 54

c 1 2 3. 3

2p 2 2 2 1 54

p 2 1 1 2

4. 2x 2 2 134 5

2x 2 2 5.

5d 1 2 1

d5 5

d 1 55 6. 2

23a 2

3a 2 3 5

32

7. 4n 2 1 52

n 1 2 2 1 8. x

x 2 3 12

x 1 3 5 1 9. p 1 7p 1 2 2 2 5

2 2 pp 1 4

10. 2p 1 3 57

28p 11. a

a 1 6 52

a 1 6 12. 26

4 2 d 52d

d 2 2

13. It takes you about an hour to make one batch of cookie dough and your brother about 42 minutes to make one batch. How much time does it take you to make a batch of cookie dough together?

14. Your dad can clean the house in 2 hours and 10 minutes. Your mom can clean it in an hour and 45 minutes. How many hours does it take them to clean the house if they work together?

Solve each equation. Check your solutions. If there is no solution, write no solution.

15. x 2 1x 1 2 14x

2x2 2 2x 2 125 2 16. t 2 1

3t2 2 t 2 22

2t 2 33t 1 2 5

242t 2 2

17. 2 2 2p

p2 2 6p 1 81

3pp 2 4 5

pp 2 2 18.

d 2 4d 1 4 5

4 1 dd 2 2 2

d 1 8d2 1 2d 2 8

12 2

43

16

2 3 22

24 2358

37

2 1 or 6

about 24.7 minutes

about 0.97 hours

2!15, !15 0, 6

no solution 0

g

11-5 Think About a PlanSolving Rational EquationsRunning You take 94 min to complete a 10-mi race. Your average speed during the fi rst half of the race is 2 mi/h greater than your average speed during the second half of the race. What is your average speed during the fi rst half of the race?


1. What is the distance of the fi rst half of the race?

2. What is the distance of the second half of the race?

3. What is the total time it takes to complete the race?


4. Rewrite the distance formula d 5 rt for time t in terms of distance d

and rate r.

5. Use your answer to Exercise 4 to write an expression for the time it takes to run each part of the race.

a. Time for fi rst half of race = ___________________

b. Time for second half of race = __________________

6. Compare the units of the given time it takes to complete the race with the units of the description of the average speed. Write the time so that it matches the units in the rates.

Getting an Answer

7. Write an equation for the total time it takes to complete the race. Solve the equation.What is the rate for the fi rst half of the race?

5 mi

5 mi

94 min

t 5 dt

5r 1 2

5r

the average rates are in mi/h and the given time is in min; 9460 h

5r 1 2 1

5r 5

9460; about 5.5 mi/h; about 7.5 mi/h

g

Concept List

rational equation Distributive Property true statement

Cross Products Property factor extraneous solution

LCD quadratic expression Zero-Product Property

Choose the concept from the list above that best represents the item in each box.

1. 79 22

3x 55

3x 2. 4

x 1 3 5x 1 2x 1 3

4(x 1 3) 5 (x 1 2)(x 1 3)

3. 25 525 3

4. 4(x 1 3) 5 (x 1 2)(x 1 3)

4x 1 12 5 x2 1 5x 1 6

5. (x 2 4)(x 1 5) 5 0

x 2 4 5 0 or x 1 5 5 0

x 5 4 or x 5 25

6. x2 2 x 2 6

7. 1x 2 1 01

x2 2 1

x 5 1

8. 0 5 x2 2 4

0 5 (x 2 2)(x 1 2)

9. 5x 1 3 52

x 1 2

(x 1 3)(x 1 2)

11-5 ELL SupportSolving Rational Equations

rational equation Cross Products Property

LCD

Distributive Property

factor

quadratic expression

true statement

extraneous solution

Zero-Product Property


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11-5 Practice (continued) Form KSolving Rational Equations 15. It takes Jeremy 4 hours to clean an auditorium, Cheryl 5 hours, and Gerald

8 hours. If all three of them work together, how long will it take them to clean the auditorium?

16. Shannon and Bruce are library workers. It takes Shannon 75 minutes to shelve a cart of books. It takes Bruce 3x min to shelve a cart of books. If they work on shelving a cart of books together, it will take them 30 min to complete it.

a. Write an equation for this problem.

b. Solve for x.

c. How long does it take Bruce to shelve a cart of books?

17. Error Analysis A student solved the rational equation 9xx(x 1 3)

212x6x 5 6Q

12R

and found x 5 1.2. What was her mistake?

18. Open-Ended Write a rational equation that has x 5 2 as a solution.


19. 2t 2 1 53

t 1 1 20. 1

x2 2 252

3x 1 5 5

5x 2 5

4023 hr, or about 1 hr 44 min

3075 1

303x 5 1

503

50 min

She omitted the negative.

5 21.125

Answers may vary. Sample: x2 1 x 1 2x(x 1 2) 5 1

g

11-5 EnrichmentSolving Rational EquationsGoing “green” is an expression that is used for products, techniques, and even buildings that are being presented into society or are being changed in order to be more environmentally friendly—to help save the environment and make the world a better and safer place for future generations. Th ere are various ways to go “green” and the use of organic products is one of them.

1. Th e Environmental Club is planting an organic garden for their school. First, the members must build a fence surrounding the garden. Alone, it would take Maria, Julie, and Gerry x, 3x, and 6x hours to fi nish the fence, respectively. Together, they fi nish the fence in 2 hours. How long would it take each member to build the fence if working alone?

2. Th e Environmental Club decides to plant carrots, tomatoes, and lettuce. Maria can plant a carrot patch in 3 hours, a tomato patch in 7 hours, and a lettuce patch in 2 hours. Julie can plant a carrot patch in 4 hours, a tomato patch in 5 hours, and a lettuce patch in 4 hours. Gerry can plant a carrot patch in 2 hours, a tomato patch in 14 hours, and a lettuce patch in 1 hour. Maria and Julie plant the carrots together and Maria and Gerry plant the tomatoes together. How long does it take to plant each vegetable?

3. All three members plant the lettuce patch together. How long will it take to plant the lettuce?

4. Th e Environmental Club is dismayed to fi nd that fl ea beetles and aphids have infested their new tomato patch. Alone, the beetles could seriously damage the crops in two weeks and the aphids could seriously damage the crops in three days. With fl ea beetles and aphids simultaneously attacking their crops, how long before the garden is destroyed?

5. To rid the garden of the pests, the members of the Environmental Club make an organic insecticidal soap. Th e soap is 95% water, 4% cayenne pepper, and 1% organic dish soap.

a. Write an equation for making the soap using those ingredients and proportions.

b. How many liters of each ingredient will they need if they start with 15 liters of water?

Maria 3 h, Julie 9 h, and Gerry 18 h

Carrots: 1.7 h; Tomatoes: 4.67 h

Lettuce: 47 N 0.57

4217 N 2.47 days

0.95s 1 0.04s 1 0.01s 5 s (soap)

15.78 (0.04) 5 0.63 L cayenne pepper15.78 (0.01) 5 0.15 L organic dish soap

g

11-5 Standardized Test PrepSolving Rational EquationsMultiple Choice


1. What is the excluded value of the equation y 5 2x 2 1 1 1?

A. 21 B. 0 C. 1 D. 2

2. A bus trip along the coast takes one route going, for a total of 1024 miles, and another route returning, for a total of 896 miles. If the bus travels at a constant speed of 65 mi/h, how far did the bus travel per second on the return trip?

F. 0.07 mi/s G. 13.8 mi/s H. 896 mi/s I. 0.0181 mi/s

3. What is the LCD for the equation 6 2 x2x2y

22x

3xy25

xx2y2

?

A. 6x2y2 B. x2y2 C. 12x2y2 D. 6xy

4. What is the excluded value of the rational expression? Include all possible solutions.

y 5 1x2 1 2x 2 24

F. 0, 24 G. 6, 24 H. 4, 6 I. 4, 26

Short Response

5. Every morning Diane runs 6 miles in about an hour. What is her rate in feet per second? What equations would you use to solve? Explain and show your work.

C

I

A

I

Use the conversion factors 5280 ft1 mi and 1 h

3600 s; 8.8 ft/sec

[2] Both parts answered correctly.[1] One part answered correctly.[0] Neither part answered correctly.

g

11-5 Practice Form KSolving Rational EquationsSolve each equation. Check your solutions. If there is no solution, write no solution.

1. 33 1 a 1 5 58

3 1 a 2. 6

x 2 4 2 2 58

x 2 4

3. 7h 2 5 112 5

6h 2 5 4.

9f 1 9 1 4 5

ff 1 9

5. 8x 1 1 2 10 54x 2 10 6.

xx 2 1 1

2x 1 1 5 1

7. 5z 2 2 53

2 1 z 8. t

t 1 1 54

t 1 1

9. It takes Frank about 45 minutes to mow a yard and Paul about 38 minutes to mow a yard. How much time does it take them to mow a yard together?

10. Sam can wash 8 cars in 1 hour. Shelly can wash 6 cars in 1 hour. How much time does it take them to wash 70 cars if they work together?


11. 1v 52v21

19 12.

1n 2 1 1

nn 1 3 5

4n2 1 2n 2 3

13. 1 1 43k 1 12 52k

k 1 4 14. 2x

3x2 2 4x 2 41

3 1 x2x 2 4 5

2x3x 1 2

22 3

3 215

1 no solution

28 4

20.6 min

5 hr

3, 6 21

163 21, 21

15


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11-6 Think About a PlanInverse VariationWriting Explain how the variable y changes in each situation. a. y varies directly with x. Th e value of x is doubled. b. y varies inversely with x. Th e value of x is doubled.


1. What is the formula for a direct variation involving the values x and y, and what is the constant of variation in that formula?

2. How does the value of y change when the value of x increases in a direct variation?

3. What is the formula for an inverse variation involving the values for x and y and what is the constant of variation?

4. How does the value of y change when the value of x increases in an inverse variation?


5. Choose a constant of variation and write the direct variation formula and the inverse variation formula using this constant of variation. Th en fi nd values of x and y that satisfy each equation. Be sure to double the values of x that you choose.

Getting an Answer

6. Explain how the variable y changes. a. For a direct variation, when x is doubled, y .

b. For an inverse variation, when x is doubled, y .

yx 5 k or y 5 kx ; k is the constant of variation

as x increases, y increases

xy 5 k or y 5 kx ; k is the constant of variation

as x increases, y decreases

Answers may vary. Sample: direct variation: y 5 2x ; inverse variation: y 5 12x ;

doubles.

decreases by 12

x 2x

2 4

2x1

4 818

14

8 16 116

g

11-6 ELL SupportInverse VariationUse the list below to complete the Venn diagram.

y varies directly with x. y varies inversely with x. Th e ratio xy is constant.

y is inversely proportional to x. y is directly proportional to x.

Th e product xy is constant.

One quantity changes in relation to another quantity.

Th e graph is linear. Th e graph is not linear.

Inverse Variation

Variation

Direct Variation

y varies directly with x.

y is directly proportional to x.

The ratio yx is constant.

The graph is linear.

One quantity changes in relation to another quantity.

y is inversely proportional

to x.

The product xy is constant.

The graph is not linear.

y varies inversely with x.

g

11-5 Reteaching (continued)Solving Rational EquationsExercises


1. 1x 12x 5 3 2.

42x 1

62x 5 10

3. 69b 25

3b 5 9 4. 1

20c 25

4c 5 8

5. 1 1 2x 521x2

6. 4y21

6y25 2

7. 22x 2 1 5 1x2

8. 2z 1 1 24

z 2 1 5 22

9. Your sister can paint a room in 5 hours. Your dad can paint the same room in

312 hours. How long will it take your sister and your dad to paint the room if

they work together?


10. 8x 1 1 54

x 2 1 11. 6

x2 2 x 2 125

22x 2 4

12. 512 21

4x 52

3x 13. 4x 1 1x 1 2 5

6 1 xx 2 2

14. 57x 114 5

5 1 xx 15.

69(x 1 6)

1 1 5 4xx 1 6

1 12

219 23

20

21 21, 3

21 21.56, 2.56

15 1

27 5

1t

3517 h or about 2.06 h

3 26

2.2 5.8 or 20.8

2407 or about 25.7209

g

11-5 ReteachingSolving Rational EquationsSolving rational equations uses the properties of simplifying rational expressions in an equation that solves for an unknown variable.

Problem

What is the the solution of 2x 13

2x 514 ?

Solve 2x 13

2x 514 The denominators are x, 2x, and 4. The LCD is 4x.

4xQ2x 1 32xR 5 4xQ14R Multiply each side by 4x.

4xQ2xR 1 4xQ 32xR 5 4xQ14R Distributive Property

8 1 6 5 x Simplify.

14 5 x Add.

Check your solution.

For some rational equations, you may have to solve a quadratic equation after multiplying each side of the equation by the LCD. Th ere may be more than one solution.

Problem

What is the the solution of 2 1 2x 54x2

?

Solve x2Q2 1 2xR 5 x2 Q 4x2R Multiply each side by the LCD, x2.

x2 (2) 1 x2 Q2xR 5 x2 Q 4x2R Distributive Property

2x2 1 2x 5 4 Simplify.

x2 1 x 5 2 Divide each side by 2.

x2 1 x 2 2 5 0 Collect terms on one side.

(x 1 2)(x 2 1) 5 0 Factor the quadratic expression.

x 1 2 5 0 x 2 1 5 0 Zero-Product Property

x 5 22 or x 5 1 Solve for x.

Check your solutions.


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11-6 Practice (continued) Form KInverse Variation 12. Graph the equations xy 5 24 and xy 5 4. How are the graphs alike? How are

they diff erent?

Do the data in each table represent a direct variation or an inverse variation? Write an equation to model the data in each table.

13. 14. 15.

Tell whether each situation represents a direct variation or an inverse variation.

16. You pay $0.10 for each minute you talk long distance.

17. $100 is split up by a club to buy lunch for each person.

18. You get paid $20 each time you mow the lawn.

19. Writing Describe how you can determine if a relationship represents a direct variation without graphing it.

20. Open-Ended Write an equation modeling direct variation and an equation modeling inverse variation in which the graphs will never intersect.

x y

3

22

5

9

15

26

x y

2

22

4

1

0.5

21

x y

2

22

4

21

22

1

The graphs are identical in shape but rotated 90° about the origin.

direct; y 5 3x

direct

direct

The relationship represents a direct variation if the rate of change is constant throughout.

Answers may vary. Sample: y 5 22x and y 5 2x

inverse

inverse; y 5 2x direct; y 5 2 12x

xO

y

2

4

4

2

24 42

g

11-6 Practice Form KInverse VariationSuppose y varies inversely with x. Write an equation for the inverse variation.

1. y 5 8 when x 5 2 2. y 5 10 when x 5 23

3. y 5 2.5 when x 5 25 4. y 5 12 when x 534

5. If y varies inversely with x, and y 5 9 when x 5 13, fi nd the constant of variation k.

6. If y varies inversely with x, solve for y if the constant of variation k 5 6 and

x 5 13.

Graph each inverse variation.

7. y 5 5x 8. xy 5 22 9. xy 5 9

10. Rate equals distancetime . If Sharon is bicycling at a constant rate of 18 mi/h, how many minutes does it take to go 45 mi? to go 90 mi? Is this a direct variation or an inverse variation? How do you know?

11. In a given equation, M varies inversely with N. If M is 25 when N 5 10, fi nd M when N is 25.

xy 5 16

xy 5 212.5

117

18

xy 5 38

xy 5 230

2.5 hr; 5 hr; direct variation; the rate is constant

10

xO

y

2

4

4

2

24 42x

O

y

2

4

4

2

24 42x

O

y

2

4

4

2

24 42

g

11-6 Practice (continued) Form GInverse Variation 17. Graph the equations xy 5 4 and xy 5 24. 18. Graph of xy 5 14? How is it like the How are the graphs of xy 5 4 and graph of xy 5 4? How is it diff erent? xy 5 24 alike? How are they diff erent?


19. 20. 21.

Tell whether each situation represents a direct variation or an inverse variation.

22. You buy strawberries for $2.99/pt. 23. Your earn $7.25/hour.

24. A 10-in. cake is shared equally by your study group.

Tell whether each table represents a direct variation or an inverse variation. Write an equation to model the data. Th en complete each table.

25. 26. 27.

x y

2

3

6

6

4

2

x y

2

3

6

6

9

18

x y

8

2

12

6

24

4

xy 5 4 xy 5 24

xy 5 24 looks the opposite of xy 5 4. The graph is in Quadrants 2 and 4 instead of in Quadrants 1 and 3 because k is negative.

The graphs of xy 5 4 and xy 5 14 are in the same quadrants and have the same shape; the graph of xy 5 14 lies closer to the origin.

inverse variation; y 5 12x

direct variation

direct variation; y 5 x12 direct variation; y 5 2 x9 inverse variation; y 5

25x

direct variation

inverse variation

direct variation; y 5 3x inverse variation; y 5 2 48x

xO

y

4

4

8

4

4

8

88x

O

y

4

4

8

4

4

8

88

xO

y

1

1

2

1

1

2

22

x y

24

36

2

1

3

12

x y

72

36

63

8

4

27

x y

50

5

25

5

112

g

11-6 Practice Form GInverse VariationSuppose y varies inversely with x. Write an equation for the inverse variation.

1. y 5 20 when x 5 5 2. y 5 16 when x 5 22 3. y 5 35 when x 5 15

4. y 5 1.2 when x 5 24 5. y 5 23 when x 545 6. y 5 20.5 when x 5 22.4

7. If y varies inversely with x, and y 5 12 when x 5 11, fi nd the constant of variation k.

8. If y varies inversely with x, solve for y if the constant of variation k 5 8 and x 5 23.

Graph each inverse variation.

9. y 5 12x 10. xy 5 26 11. xy 5 30 12. y 5210

x

13. Two fi fth graders play on the seesaw. One fi fth grader weighs 75 lb and the other weighs 90 lb. In order to balance the seesaw so both can ride, how far from the center pole does the 90 lb fi fth grader have to be if the 75 lb fi fth grader is 6 ft from the center? Weight and distance vary inversely.

14. In Exercise 13, what if the 75 lb fi fth grader and a friend of equal weight sit 4 ft from the center of the seesaw to balance the 90 lb fi fth grader? How far from the center should the 90 lb fi fth grader be?

15. Speed equals distancetime . If a car is traveling at a constant speed of 60 kmh , how

many minutes does it take to go 20 km and then 40 km? Is this a direct variation or an inverse variation? How do you know?

16. In a given equation, C varies inversely with D. If C is 50 when D 5 11.5, fi nd C when D is 20.

75 lb 90 lbx ft6 ft

xy 5 100 or y 5 100x

132

12

5 ft

28.75

20 min; 40 min; direct variation; when x doubles, y doubles

6 23 ft or 6 ft 8 in.

xy 5 232 or y 5 232x xy 5 9 or y 59x

xy 5 24.8 or y 5 24.8x xy 51815 or y 5

815x xy 5 1.2 or y 5

1.2x

xO

y

4

4

8

4

4

8

88x

O

y

4

4

8

4

4

8

88x

O

y

6

6

12

6

6

12

1212x

O

y

4

4

8

4

4

8

88


101

A N S W E R S

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11-6 Reteaching (continued)Inverse VariationExercises

Suppose y varies inversely with x. Write an equation for the inverse variation.

1. y 5 23 when x 5 15 2. y 5 4 when x 5 24

3. y 5 10 when x 5 100 4. y 5 1 when x 5 9

5. y 5 3 when x 5 24 6. y 5 40 when x 5 23


7. 8.

9. 10.

Each pair of points is on the graph of an inverse variation. Find the missing value.

11. (2, 3) and (6, y)

12. (23, 23) and (x, 6)

13. (0.2, 6) and (20.3, y)

x y

2

1

1

2

32

34

3234

x y

7

8

9

10

14

16

18

20

x y

12

13

14

15 1

1514151354

x y

4

3

2

1

30

40

60

120

y 5 245x

y 5 212x


direct variation; y 5 2x



y 5 2120x

y 5 1000x

y 5 96x

y 5 9x

1

1.5

24

g

11-6 ReteachingInverse VariationAn inverse variation is shown by the equation xy 5 k where k 2 0.

Problem

Suppose y varies inversely with x, and y 5 2 when x 5 26. What is an equation for the inverse variation?

Solve

Because y varies inversely with x, the equation has the form xy 5 k. Use the given values of x and y to fi nd the value of k.

xy 5 k Use the general form of an inverse variation.

26(2) 5 k Substitute 26 for x and 2 for y.

212 5 k Simplify.

Use the value of k to write an equation for the inverse variation.

xy 5 212 or y 5 212x

Recall that a direct variation is shown by the equation y 5 kx , or yx 5 k. To

determine whether data represent a direct variation or an inverse variation, investigate the ratios of their data pairs and the products of their data pairs. If the ratios are constant, the data represent a direct variation. If the products are constant, the data represent an inverse variation.

Problem

Do the data in the table represent a direct variation or an inverse variation? Write an equation to model the data.

Solve

Check each product xy.

2(8) 5 16 4(16) 5 64 5(20) 5 100

Th e products are not the same. So, the data do not vary inversely.

Check each ratio yx .

82 5 4

164 5 4

205 5 4

Th e ratios are the same. So, this is a direct variation, and k 5 4.

An equation for this data is xy 5 4, or y 5 4x .

x y

2

4

5

8

16

20

g

11-6 EnrichmentInverse Variation 1. Suppose p varies directly with l and inversely with m. Write the equation to

solve for k. If p 5 30 when l 5 2.5 and m 5 6, what is the value of p when l 5 4 and m 5 7?

2. Th e pressure P of a perfect gas varies inversely to the volume V. Th e pressure of a gas in a 4-L container under specifi c conditions is 100 atm. What is the pressure of the gas in a 45-L container under the same conditions? Use the

equation P 5 KV .

3. Th e volume of a cylinder equals the product of its base and height. In a cylinder fi lled with liquid, the height (h) of a liquid varies inversely with the area of the cylinder’s base (B), B 5 pr2. Find the radius of a cylinder with volume 30 in3 and height 15 in. What happens to the height if the area of the base is doubled and the volume stays the same? What kind of variation is this?

4. A group of students decides to drive to college instead of fl ying. Th ey leave from Miami, FL and head north toward college in Boston, MA. If the students have 1496 miles to drive to reach their destination and want to arrive in 24 hours, what rate of speed must they maintain in order to arrive on time? How much faster can they arrive there if they travel only the major highways at speeds of 65 mi/h? Is this a direct or inverse variation?

p 5 klm ; k 5 72 ; 4117

8.89 atm

r 5 0.798 in.; the height is halved; inverse variation

62.3 mi/h; 1 h; direct

g

11-6 Standardized Test PrepInverse VariationGridded Response

Solve each exercise and enter your answer on the grid provided.

1. Suppose y varies inversely with x, and y 5 16 when x 5 4. What is the constant of variation k?

2. If 9 students sharpen 18 pencils in 2 minutes, how many students will it take to sharpen 18 pencils in 1 minute?

3. If y varies inversely with x, and y 5 216 when x 5 264, what is the constant of variation?

4. Th e pair of points (3, 8) and (x, 6) are on the graph of an inverse variation. What is the missing value?

5. Th e weight needed to balance a lever varies inversely with the distance from the fulcrum to the weight. A 120-lb weight is placed on a lever, 5 ft from the fulcrum. What amount of weight; in pounds; should be placed 8 ft from the fulcrum to balance the lever?

1. 2. 3. 4. 5.

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A N S W E R S

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Page 64age 6

11-7 Practice (continued) Form GGraphing Rational FunctionsDescribe the graph of each function.

16. y 5 5x 17. f (x) 5 2x

18. f (x) 5 u x 2 3 u 19. y 5 !x 1 5

20. g(x) 5 4x 1 12 21. y 5 2x 1 6

22. y 5 x2 2 16 23. g(x) 5 2x 1 7 2 3

24. Graphing Calculator Maria invites 14 friends over to eat a cake she baked that contains 3000 calories. Everyone also has a glass of milk worth 100 calories each, but two friends do not eat the cake. Write an equation for how many calories each person who has cake consumes. Solve the equation. Th en use your graphing calculator to graph it. Find the number of calories each person would consume if only 8 people showed up.

25. Open-Ended Write an example of a rational function with a vertical asymptote at x 5 1 and a horizontal asymptote at y 5 25.

26. Graph and describe the similarities and diff erences between each of these functions.

a. f (x) 5 1x b. f (x) 51

x 2 1 c. f (x) 51

x 1 1 1 5

f (x) 5 1x is the most basic inverse variation function with the x- and y-axes as asymptotes.

f (x) 5 1x 2 1 translates the graph of (x) 5 1x 1 unit to the right and moves the vertical asymptote to x 5 1.

f (x) 5 1x 2 1 1 5 translates the graph of (x) 5 1x right 1 unit and up 5 units. The horizontal asymptote is y 5 5 and the vertical asymptote is x 5 1.

y 5 3000x 2 2 1 100; 331 cal; 433 cal;

Answers may vary. Sample: y 5 1x 2 1 2 5

a line that is steeper than y 5 x as x increases, y increases exponentially

a V-shaped graph with a single point a minimum at x 5 3

a concave down curve starting at x 5 25 with y L 0

a line with a y-intercept at 0.5 a line with a negative slope and a y-intercept of 6

a parabola that opens up with a y-intercept of 216 and x-intercepts at 4 and 24

a hyperbola with a vertical asymptote at x 5 27 and horizontal asymptote at y 5 23

xO4

4

8

4

4

8

88

f(x)

xO4

4

8

4

4

8

88

f(x)

xO4

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f(x)

age 63

11-7 Practice Form GGraphing Rational FunctionsIdentify the excluded value of each rational function.

1. y 5 2x 2. f (x) 523

x 1 1 3. y 510

2x 1 2

4. f (x) 5 24x 1 6 5. y 59x 6. y 5

x4x 1 8

Identify the asymptotes of the graph of each function. Th en graph the function.

7. y 5 2.5x 8. f (x) 54

3 1 x

9. Find the domain for x in the equation xy 5 1.

10. Graph the equation xy 5 1. What happens to the values of y as x approaches 0 from the left and from the right?

11. Identify the horizontal asymptote for the graph of the equation

y 5 22.5x 1 17 1 3. How many units has the graph of y 522.5

x been vertically

translated?

12. Identify the horizontal and vertical asymptotes of y 5 1x 1 1 without graphing or making a table. Explain how you fi gured out what they were.

Describe how the graph of each function is a translation of the graph of y 5 21x .

1

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