Algebra 3-4 Unit 4 Rational 4... · Rational expression: a fraction that has polynomials in the numerator and/or denominator. To multiply two rational expressions: factor, cancel

$Page 1: Algebra 3-4 Unit 4 Rational 4... · Rational expression: a fraction that has polynomials in the numerator and/or denominator. To multiply two rational expressions: factor, cancel$
Name __________________________________________________________________ Period ____________

Algebra 3-4 Unit 4

Rational

4.1-2 I can multiply and divide rational expressions.

4.3-5 I can add and subtract rational expressions.

4.6-8 I can solve rational equations.

4.9-12 I can graph rational functions.

My goal for this unit: _____________________________________________________ ______________________________________________________________________ What I need to do to reach my goal: ________________________________________ ______________________________________________________________________ ______________________________________________________________________

Name _________________________________________________________________ Period _____________

Algebra 3-4 Unit 4.1

Multiplying Rational Expressions

Rational expression: a fraction that has polynomials in the numerator and/or denominator.

To multiply two rational expressions: factor, cancel common factors, and simplify.

Remember, you cannot divide by 0 (ask siri why not) so values that makes the denominator 0 must be excluded.

Example Find the product and any excluded values. 2

2 2

2 8 1

1 6

x x x

x x x

− − −

− − −i

Step 1 Factor and multiply. ( 4)( 2)( 1)

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

( 4)( 2)

x x x

x x x x

x x

− + −=

+ − − +

− +=

( 1)x −

( 1)( 1)x x+ − ( 3)( 2)x x− +

4

( 1)( 3)

x

x x

−=

+ −

Step 2 Cancel common factors.

Step 3 Write simplified product.

Step 4 Note excluded values. 2x ≠ − , 1x ≠ − , 1x ≠ , 3x ≠

Directions: Multiply and state all excluded values.

1. 3

6 6

10 3

x x

xi

2. 427 12

19 11

x x

xi

3. 214 41

15 49

x x

xi

4. 2

6 5

65xi

5. 3 21 7 49

9 7

x x

x x

+

+ +i

6. 26 54 7

9 6

x x x

x x

−

−i

7. 18 36 2

4 8 9 18

x

x x

−

− +i

8. ( )2

2

1056 11 15

15 11 56x x

x x+ −

− −i

9. 2

3 2

169 104 48 11

44 169 104 48

x x x

x x x

+ − −

+ −i

10. 2

2 2

3 5 5 62 39

15 34 15 5 50

x x x

x x x x

+ − −

+ + −i

11. ( ) ( ) ( )

( ) ( )

5 1 6 2

3 1 2

x x x

x x

+ − +

− +i

12. 2

2 3 2

5 3 18

3 18 2 16

x x

x x x x

+ −

+ − −i

13. 2 2

2

5 14 20

216

x x x x

xx

− − − −

+−i

14. 2

2 2

14 45 3

6 3 2 50

x x x

x x x

− +

− −i

15. 2 2

2

6 6 8

220

x x x x

xx x

− − − +

++ −i

16. 2

2 2 4

3 36

x x

xx x

− +

−+i

Name _________________________________________________________________ Period _____________


Multiplying and Dividing Rational Expressions

To divide two rational expressions: take the reciprocal of the divisor then multiply like yesterday.

Remember, you cannot divide by 0 (ask siri why not) so values that makes the denominator 0 must be excluded.

Example Find the quotient and any excluded values. 2 212 9 18

5 2 10

x x x x

x x

− − + +÷

+ +

Step 1 Rewrite as multiplication by

reciprocal of divisor.

2

2

12 2 10

5 9 18

( 4)( 3) 2( 5)

5 ( 6)( 3)

2( 4)( 3)( 5)

( 5)( 6)( 3)

2( 4)( 3)

x x x

x x x

x x x

x x x

x x x

x x x

x x

− − +=

+ + +

− + +=

+ + +

− + +=

+ + +

− +=

i

i

( 5)x +

( 5)x + ( 6) ( 3)x x+ +

2( 4)

( 6)

x

x

−=

+

Step 2 Factor

Step 3 Multiply


Step 5 Write simplified product.

Step 6 Note excluded values. 6x ≠ − , 5x ≠ − , 3x ≠ −

Directions: Multiply or divide and state all excluded values.

1. 4 4

5 6

x x

x÷

2. 4 8

3 2

x xi

3. ( )

( ) ( )

6 2 2

1 10 10

x x

x x x

− −÷

− − −

4. ( )214 42

2 610

x xx

++ ÷

5. 227 9 3 8 3

10 10

x x x+ − −÷

6. 3 2

24 56 15 35

510 90

x x

x x

+ +÷

−

7. 3 2

2 20 2

14 3512 30

x

xx x

+÷

−−

8. 2

2 2

5 8 20 40

2 14 20 15 24

x x x

x x x x

− +

+ + −i

9. 215 12 7 14

40 32 3

x x x

x x

+ −

+i

10. ( )

2

5 13 5 30

13 14

x x

x x

+ −÷

+

11. 284 11 132

7 10 77 110

x x

x x

+÷

+ +

12. 2

2 2

2 10 12 1

22 42 20 11 100 100

x x

x x x x

+ −÷

− + + −

13. 77 11 21 3

3 42 3

x x

x

+ +÷

−

14. 2

2

4 2 2

22

x x

xx x

− +

+− −i

15. 2

2 2

6 16

3 3 2

x x x

x x x x

+ −

− − +i

16. 2

2 2

3 12 5

3 15 10 25

x x x

x x x x

− +÷

+ + +

17. A rectangular sandbox has a length of 3 24

2

x

x

+

+ feet and a width of

9

2 16

x

x

+

+ feet. Write a rational

expression that represents the area of the sandbox. Simplify the expression and state any excluded values.

Name _________________________________________________________________ Period _____________

Algebra 3-4 Unit 4.3 (alt)

Add and Subtract Rational Expressions Like Denominators

In order to add or subtract fractions, there must be a common denominator.

Add or subtract the numerators

Keep the denominator the same

Simplify if possible

Example 1: 7

5

7

3

7

2=+

Example 2: 3333 6

3

12

26

12

2

12

5

y

yx

y

yx

y

yx

y

x +=

+=

++

Example 3: 3

2

)3)(53(

)53(2

)3)(53(

106

15143

4

15143

6622 −

=−−

−=

−−

−=

+−−

+−

−

xxx

x

xx

x

xxxx

x

Add or subtract and simplify where possible.

1. 3232 30

4

30

4

yx

yx

yx

yx −+

− 2.

255

2

255

3

−

++

− x

x

x

3. 106

6

106

2

+

−+

+ x

x

x 4.

128

1

128

3222

+−

−−

+−

−

xx

x

xx

x

5. 1620

4

1620

64

+

−+

+

−

x

x

x

x 6.

67

4

67

3222

+−

++

+−

−

xx

x

xx

x

7. 164

2

164

6

+

−−

+ x

x

x 8.

3103

5

3103

122

++

−+

++

−

xx

x

xx

x

9. xx

x

xx

x

3012

45

3012

122

+

++

+

+ 10.

20295

1

20295

222

+−

++

+− xx

x

xx

11. 96

4

96

522

+−+

+− xxxx 12.

1253

2

1253

422

−−

++

−−

−

xx

x

xx

x

13. 67

3

67

15322

++

−−

++

+

xx

x

xx

x 14.

xx

x

xx

x

204

5

204

522

+

−−

+

+

15. 60186

2

60186

522

−−

−−

−−

−

xx

x

xx

x 16.

6193

1

6193

2322

+−+

+−

−

xxxx

x

17. 2012

2

2012

6

−

−+

−

−

x

x

x

x 18.

2323 366

3

366

1

xx

x

xx

x

−

+−

−

+

Name _________________________________________________________________ Period _____________


Add and Subtract Rational Expressions

The addition or subtraction of rational expressions can be compared to the addition or subtraction of fractions.

As with fractions, in order to add or subtract rational expressions, the denominators must be the same. The least

common multiple of the polynomials is the least common denominator (LCD).

Add Fractions Add Rational Expressions

3 1

4 6+

2

5 1

2

x

x x

++

Find the LCD. 12 2x2

Rename each term using

the LCD. 9 2

12 12+ 2 2

5 2 2

2 2

x x

x x

++

Add the numerators. 11

12 2

7 2

2

x

x

+

Directions: Find the least common multiple for each pair.

1. 2x6 and 6x

2. 10xy and 5x4y3

3. (x − 8)(x + 1) and (x + 1)

4. x2 − 5x + 6 and x − 3

5. x2 − 2x − 8 and x2 − 16

6. x2 + 2x − 15 and x2 − 7x + 12

Directions: Write the given expression as an equivalent rational expression that has the given denominator.

7. Expression: 1

2

x

x

+

+

Denominator: 2x2 + 4x

8. Expression: 5

2 3

x

x −

Denominator: 4x2 − 9

9. Expression: 24

4 8

x

x

−

−

Denominator: (2 − x)(x + 3)

10. Expression: 2

2

16

12

x

x x

−

− −

Denominator: (x + 3)(x + 4)(x − 4)

11. Expression: 183

1522

2

−+

−+

xx

xx

Denominator: (x + 6)(x − 3)(x + 3)

12. Expression: 2

122

2

−+

−+

xx

xx

Denominator: (x + 2)(x − 1)(x + 1)

To add or subtract rational expressions, they must have common denominators.

Example Add 2 1

3 1

x x

x x

+ −+

− +.

Step 1 Multiply each rational expression by a

common factor to get equivalent fractions

with a common denominator.

2 1

3 1

x x

x x

+ −+

− +

22 1 ( 2)( 1) 3 2

3 1 ( 3)( 1) ( 3)( 1)

x x x x x x

x x x x x x

+ + + + + += =

− + − + − +i

21 3 ( 1)( 3) 4 3

1 3 ( 1)( 3) ( 3)( 1)

x x x x x x

x x x x x x

− − − − − += =

+ − + − − +i

Step 2 Add the fractions. 2 2 23 2 4 3 2 5

( 3)( 1) ( 1)( 3) ( 1)( 3)

x x x x x x

x x x x x x

+ + − + − ++ =

− + + − + −

Step 3 Give excluded values that make the

denominator 0. 1x ≠ − , 3x ≠

Directions: Add or subtract. Identify any x-values for which the expression is undefined.

13.

2 3 4 5

4 4

x x

x x

− −+

+ +

14.

12 3 2

2 5 2 5

x x

x x

+ −−

− −

15. 2

1 1

x x

x x+

+ +

16. 4 1 2 7

4 4

x x

x x

+ ++

− −

17. 5 1 3

3 2 6

x x

x x

−+

+ +

18. 6 2

2

x

x x−

+

Common Denominators

Multiply

by

1x +

Multiply by

3

3

x

x

−

−

Name _________________________________________________________________ Period _____________


Add and Subtract Rational Expressions (Day 2)


1. 2

7 2

43

x

xx−

+

2.

1

4 3 1

x x

x x

++

− +

3.

3 3

4 2 4 2

x x

x x

++

− +

4.

3 1

2 5

x

x x

++

+ −

5.

4 7

( 2)

x x

x x x

++

−

6.

8 2

3 1

x x

x x

+ −−

− −

7.

3 4

6 ( 5)( 6)

x

x x x−

+ − +

8. 2

3 1

5 7 10x x x−

− − +

9. 2

4 2

412

x x

xx x

++

−− −

10.

2

2

3 1 2

63 18

x x

xx x

− +−

−− −

11. 2

5 10

3 9

x x

x x

− −+

+ −

12. 2

2

32 15

x x

xx x

++

+− −

13. 2 2

4 1

2 6

x x

x x x x

+ −−

− − + −

14. 2

6 2

97 18

x x

xx x

+−

−− −

15. 2 2

3 5

6 8 15

x

x x x x−

− − − +

16. 2

3

43 4

x

xx x+

−− −

Name _________________________________________________________________ Period _____________


Add and Subtract Rational Expressions (Day 3)


1. 2

2

11

x

xx−

+−

2.

3.

4.

5.

6.

7.

8.

9.

10.

11. The electric potential generated by a certain arrangement of electric charges is given by4 1

e e

x x+

− +,

where e is the fundamental unit of electric charge and x measures the location where the potential is being

measured. Express the electric potential as a rational expression.

12. A ferry shuttles from Seattle to Vancouver Island and back. Because of head winds, the return trip is slower than the trip to the island. The average speed of the ferry, in miles per hour, is given by the

expression: 2

.

50 60

d

d d+

What is the average speed of the ferry?

Name _________________________________________________________________ Period _____________


Solve Rational Equations (Day 1)

Multiply through by the LCD.

11

3

32 +=

− x

x

x

x

LCD is (2x − 3)(x + 11)

11

3)11)(32(

32)11)(32(

++−=

−+−

x

xxx

x

xxx

11

3)11)(32(

32)11)(32(

++−=

−+−

x

xxx

x

xxx

)3)(32())(11( xxxx −=+ (Yeah! No more denominators!)

x2 + 11x = 6x2 − 9x

5x2 − 20x = 0

5x(x − 4) = 0

5x = 0 x − 4 = 0

x = 0 x = 4

Directions: Solve for the variable using the method of cross multiplication. Check each answer.

1. 2

4 2

x x +=

2. 9

4 4

x

x=

3.

2 3 1

7 6

x x

x

+ +=

4. 4

5 9

x

x

−=

−

5. 2

3 7

x

x

−=

+

6. 2

23

x

x

+=

−

Substitute solutions in the original

problem to make sure they are

not extraneous solutions:

110

)0(3

3)0(2

0

+=

−

0 = 0

114

)4(3

3)4(2

4

+=

−

15

12

5

4=

5

4

5

4=

7. 2 4 2

5

x

x x

+=

8. 2 6

6 2

x x

x x

−=

− +

9. 25

3

+=

+

−

x

x

x

x

10. xx

x 1

22

−=

−

11. 2

102

−

+

x

x =

x

4

12. 2

4 1

24 xx=

−−

13. 4 2 8

4 4

x x

x x

+=

− −

14. 2

5 10

1 2

x x

x x x

−=

+ − −

15. 6

2 2 1

x x

x x

−=

+ −

16. 2

2

2 3 15

5 9 20

x x x

x x x

−=

− − +

Name _________________________________________________________________ Period _____________



Multiply through by the LCD.

Example Solve the rational equation algebraically. 2

1 2

2 4 6 8

x

x x x x+ =

− − − +

Step 1 Multiply each term by the LCD. 1 2

( 2)( 4) ( 2)( 4) ( 2)( 4)2 4 ( 2)( 4)

xx x x x x x

x x x x− − + − − = − −

− − − −


2

x

x −( 2)x −

1( 4)

4x

x− +

−( 2) ( 4)x x− −

2

( 2)( 4)x x=

− −( 2)( 4)x x− −

( 4) ( 2) 2x x x− + − =

Step 3 Simplify and solve the remaining equation.

2

2

4 2 2

3 4 0

( 4)( 1) 0

x x x

x x

x x

− + − =

− − =

− + =

4x = or 1x = −

Step 4 Check for extraneous solutions that are excluded values. 4x = is an excluded value.

1x = − is the solution.

Directions: Solve for the variable using the method of cross multiplication. Check each answer.

1. 9 2 2

1x

x x

++ =

2. 3 5

24 4x x

− =+ +

3. 6 2

43 3x x

− =− −

4. 3 1 4

12 x x

+ = +

Multiply by LCD

( 2)( 4)x x− −

1 2

2 4 ( 2)( 4)

x

x x x x+ =

− − − − Factor the

denominator2

( 2)( 4)x x− −

5. 1 2 4

3 3

x

x x x

−− =

6. 2 2

5 5 5 1

4 4

x

xx x x x

−− =

− −

7. 2 7 10 1

4x x

xx x

− ++ = +

8. 2

6 71

x x− + =

9. 2

2

8 121

3 2 13 21

x x x

x x x

− ++ =

+ + +

10. 2 27 10 13 40

5 30 6 5 30

x x x x x

x x x

+ + − ++ =

− − −

11. 1

2 4x

+ =

12. 3

2xx

+ =

13. 5 4

36 x

+ =

14. 12

4 3x

+ =

15. 1

3 2

x xx

x

++ =

+

16. 2 = 41

3−

+x

Name _________________________________________________________________ Period _____________



Directions: Solve each rational equation. https://my.hrw.com/content/hmof/math/hsm/common/video/video.html#videoId=ref:RW_HSM_ALG2_009_en

1. 4

4

82

)4(42 +

=−+

−

xxx

x

2. 128

4

26

12

+−=

−+

− xxx

x

x

3. 2 1 1

3 3

x

x x x

−+ =

− −

4. 3 5

3 5 5

x x x

x x x

+ ++ =

− − −

5. Fran can clean the garage in 3 hours, but it takes

Angie 4 hours to do the same job. How long would

it take them to clean the garage working together?

6. Kent can paint a certain room in 6 hours, but

Kendra needs 4 hours to paint the same room.

How long would it take them working together?

7. An artist is designing a picture frame whose

length l and width w satisfy the Golden Ratio,

which is .w l

l l w=

+ If the length of the frame is

24 inches, what is the width of the frame?

8. John can pick a bushel of peaches in 30 minutes.

His little sister can pick a bushel of peaches in 45

minutes. How long will it take them to pick a

bushel of peaches working together?

9. Marco can build a lap top twice as fast as Cliff.

Working together, it takes them 5 hours. How

long would it have taken Marco working alone?

10. Kiyoshi can paint a certain fence in 3 hours by

himself. If Red helps, the job takes only 2 hours.

How long would it take Red to paint the fence by

himself?

11. Every week, Linda must stuff 1000 envelopes.

She can do the job by herself in 6 hours. If Laura

helps, they get the job done in 5 hrs and 30 mins.

How long would it take Laura to do the job by

herself?

12. Mr. McGregor has discovered that a large dog can

destroy his entire garden in 2 hours and that a small

boy can do the same job in 1 hour. How long

would it take the large dog and the small boy

working together to destroy Mr. McGregor’s

garden?

13. Edgar can blow the leaves off the sidewalks

around the capitol building in 2 hours using a

gasoline-powered blower. Ellen can do the same

job in 8 hours using a broom. How long would it

take them working together?

14. It takes a computer 8 days to print all of the

personalized letters for a national sweepstakes. A

new computer is purchased that can do the same

job in 5 days. How long would it take to do the job

with both computers working on it?

Name _________________________________________________________________ Period _____________


Graphing Rational Functions (Day 1)

Graph of 1

yx

=

Vertical Asymptote: 0x =

Horizontal Asymptote: y = 0

How to graph a rational function: 21

1+

−=

xy

Find intercepts if there are

any (for x-intercepts y = 0;

for y-intercepts x = 0)

x-intercept: y = 0

21

10 +

−=

x

1

12

−=−

x

−2(x − 1) = 1

−2x + 2 = 1

−2x = −1

x = 0.5

y-intercept: x = 0

210

1+

−=y

21

1+

−=y

y = −1 + 2

y = 1

Asymptotes (a line that approaches a given curve but does not meet it):

Horizontal and slant:

• If numerator power = denominator power; asymptote at y = b

a where a and b

are leading coefficients.

• If power in denominator is larger; asymptote at y = 0.

• If power in numerator is larger; no horizontal asymptote.

o Slant asymptote (numerator ÷ denominator; ignore remainder)

Vertical:

• Asymptote at x = a where a is any value that makes the denominator = 0

Holes:

• A factor in numerator and denominator. Cancels out but leaves a hole at this

point on the graph.

Denominator power is

larger; horizontal

asymptote at y = 0 then

translate up 2 so y = 2

Vertical asymptote when

denominator would be 0

therefore at x = 1

Domain: all real numbers except x-values that make

the denominator 0.

Domain: (-∞, 1) ∪ (1, ∞)

Range: given: ( ) = +−

af x k

x h; range is all y-values

except y ≠ k.

Range: (-∞, 2) ∪ (2, ∞)

f(x) = k shows vertical

h shows horizontal

Using the graph of f(x) = x

1 as a guide, graph each transformation. Fill in each blank.

1. 2

1)(

−=

xxg

Describe the transformation: ___________________

Asymptotes: ________________________________

Hole: ______________________________________

Domain: ___________________________________

Range: ____________________________________

Intercepts: __________________________________

2. 3

1)( +=

xxh


Asymptotes: ________________________________

Hole: ______________________________________

Domain: ___________________________________

Range: ____________________________________

Intercepts: __________________________________

3. ( ) = +−

15

3g x

x


Asymptotes: ________________________________

Hole: ______________________________________

Domain: ___________________________________

Range: ____________________________________

Intercepts: __________________________________

4. ( ) = −+

11

8g x

x


Asymptotes: ________________________________

Hole: ______________________________________

Domain: ___________________________________

Range: ____________________________________

Intercepts: __________________________________

Name _________________________________________________________________ Period _____________



Directions: Fill in each blank and use that information to sketch the graph.

1. ( ) = +−

12

3f x

x


Asymptotes: ________________________________

Hole: ______________________________________

Domain: ___________________________________

Range: ____________________________________

Intercepts: __________________________________

2. g(x) = 14.

( 2)x+

− −


Asymptotes: ________________________________

Hole: ______________________________________

Domain: ___________________________________

Range: ____________________________________

Intercepts: __________________________________

3. 5

13

yx

= −+


Asymptotes: ________________________________

Hole: ______________________________________

Domain: ___________________________________

Range: ____________________________________

Intercepts: __________________________________

4. ( ) = ++

17

5g x

x


Asymptotes: ________________________________

Hole: ______________________________________

Domain: ___________________________________

Range: ____________________________________

Intercepts: __________________________________

5. ( ) =+

2

4g x

x


Asymptotes: ________________________________

Hole: ______________________________________

Domain: ___________________________________

Range: ____________________________________

Intercepts: __________________________________

6. 2

64

yx

−= +

−


Asymptotes: ________________________________

Hole: ______________________________________

Domain: ___________________________________

Range: ____________________________________

Intercepts: __________________________________

7. ( ) −+

17

3f x

x====


Asymptotes: ________________________________

Hole: ______________________________________

Domain: ___________________________________

Range: ____________________________________

Intercepts: __________________________________

8. 2

35

yx

= −−


Asymptotes: ________________________________

Hole: ______________________________________

Domain: ___________________________________

Range: ____________________________________

Intercepts: __________________________________

Name _________________________________________________________________ Period _____________




1. ( )+ −

=+

2 4 5

1

x xf x

x


Asymptotes: ________________________________

Hole: ______________________________________

Domain: ___________________________________

Range: ____________________________________

Intercepts: __________________________________

2. 1

34)(

2

+

+−=

x

xxxf


Asymptotes: ________________________________

Hole: ______________________________________

Domain: ___________________________________

Range: ____________________________________

Intercepts: __________________________________

3. 1

7yx

−= +


Asymptotes: ________________________________

Hole: ______________________________________

Domain: ___________________________________

Range: ____________________________________

Intercepts: __________________________________

4.


Asymptotes: ________________________________

Hole: ______________________________________

Domain: ___________________________________

Range: ____________________________________

Intercepts: __________________________________

( ) = −−

15

6f x

x

5.


Asymptotes: ________________________________

Hole: ______________________________________

Domain: ___________________________________

Range: ____________________________________

Intercepts: __________________________________

6. 1

( ) 3.3 6

f xx

= −+


Asymptotes: ________________________________

Hole: ______________________________________

Domain: ___________________________________

Range: ____________________________________

Intercepts: __________________________________

7. ( ) = −−

4 1

9 4g x

x


Asymptotes: ________________________________

Hole: ______________________________________

Domain: ___________________________________

Range: ____________________________________

Intercepts: __________________________________

8.

( ) = −

+

112

2

3

g x

x


Asymptotes: ________________________________

Hole: ______________________________________

Domain: ___________________________________

Range: ____________________________________

Intercepts: __________________________________

( ) = ++

11

4f x

x

Name _________________________________________________________________ Period _____________




1. ( )2 2

.x

f xx

+=


Asymptotes: ________________________________

Hole: ______________________________________

Domain: ___________________________________

Range: ____________________________________

Intercepts: __________________________________

2. 2

2

12( ) .

3 4

x xf x

x x

+ −+ −+ −+ −====

+ −+ −+ −+ −

Asymptotes: ________________________________

Hole: ______________________________________

Domain: ___________________________________

Range: ____________________________________

Intercepts: __________________________________

3. Write the functions in the form ( )a

f x kx h

= += += += +−−−−

by using the graph.


f x kx h

= += += += +−−−−

by using the graph.


f x kx h

= += += += +−−−−

by using the graph.


f x kx h

= += += += +−−−−

by using the graph.

7. The number n of daily visitors to a new store can be modeled by the function ( )+

=250 1000x

nx

, where x

is the number of days the store has been open.

a. What is the horizontal asymptote of this function and what does it represent? _______________________

_____________________________________________________________________________________

b. To the nearest integer, how many visitors can be expected on day 30? ____________________________

8. The annual transportation costs, C, incurred by a company follow the formula 2500

,C ss

= + where C is

in thousands of dollars and s is the average speed the company’s trucks are driven, in miles per hour. Use

your graphing calculator to find the speed at which cost is at a minimum.

9. Li Ming rowed 2 miles upstream and then 3 miles downstream. His average speed rowing in still water is

2 miles per hour. The model that describes the situation is 2 3 10

( )2 2 (2 )(2 )

sf s

s s s s

− += + =

− + − + where s is

the average speed of the current and f(s) is the total amount of time that Li Ming rowed. Sketch a graph of

the function and use it to determine the average speed of the current if Li Ming rowed for 3 hours.

(3, -2)

Name _________________________________________________________________ Period _____________


Are You Ready for Unit 4 Assessment

I can add, subtract, multiply, and divide rational expressions.

1. Multiply and state the excluded values:

23

2

549

6

162

8018

xx

x

x

xx

+

+•

+

++

2. Divide and state the excluded values:2 2

2

7 18 2

2 82 32

x x x x

xx

− − + −÷

−−

3. Divide and state the excluded values:

168

54

16

42

2

2++

−+÷

−

+

xx

xx

x

x

4. Divide and state the excluded values:

86

20

6

1582

2

2

2

+−

−+÷

−−

+−

xx

xx

xx

xx

5. Multiply the rational expresssions and state the excluded values.

186

93

3

9 22

+

+•

−

−

x

xx

x

x

6. Divide and simplify:

20

183

6

1272

2

2

2

−−

−+÷

−−

+−

xx

xx

xx

xx

7. What is the least common denominator

(LCD) of the expressions 20

32

−− xx and

82

42

−+ xx?

8. What is the least common denominator

(LCD) of the expressions 352

22

−− xx

x and

152

12

−+

+

xx

x?

9.

10. Evaluate the problem and identify the mistake. Fully explain the error then simplify the problem correctly.

123

2

209

32 +

−++

+

xxx

x

Step 1: )4(3

2

)5)(4(

3

+−

++

+

xxx

x

Step 2: )5)(4(3

)5(2

)5)(4(3

)3(3

++

+−

++

+

xx

x

xx

x

Step 3: )5)(4(3

102

)5)(4(3

93

++

+−

++

+

xx

x

xx

x

Step 4: )5)(4(3

19

++

+

xx

x

11. Add: 209

36

209

322

++

++

++ xx

x

xx

12. Add: 158

4

158

2422

2

++

++

++

++

xx

x

xx

xx

13. Simplify: 103

3522

2

−−

−+

xx

xx

14. Simplify: 56

982

2

++

−−

xx

xx

I can solve rational equations.

15. Solve: 7

15

1

3

+=

+

+

xx

x

16. Solve: 13

22

7

1

−

−=

−

x

xx

17. Solve: 24

331

−

+=

+

x

x

x

x

18. Solve: 1

42

2

2

+

+=

−

+

x

x

x

x

19. Solve: x

x

xx

1

2

11

2

−+

+=

20. Solve: 1

61522 +

−−

+=

+

+

x

x

xxxx

x

21. Solve: xx 2

1

2

132

=+−

22. Solve: 1

3

1

2

+=

− xx

23. Solve: 4

1

2

9

22

5 −=−

− xx

24. Solve: 2 1 1

3 3

x

x x x

−+ =

− −

25. Jan and Stan both work at the library. It takes Jan 42 minutes to put a cart of books away. The same cart would take Stan 58 minutes. How long will it take if they work together?

26. Working together it took Sara and Cara took 4 hours to complete a task. If it would have taken Sara 7 hours to complete it by herself, how long would it have taken Cara by herself?

I can graph rational functions.

27. Graph the function f(x) = 32

4−

+x using

technology and fill in the blanks. Vertical asymptote: ______________________ Horizontal asymptote: ____________________


1+

−

−

x using



14

+

+

x

x using



7

−

+

x

x using



52

−

+

x

x using



52

−−

+

xx

x using



22

−+

+

xx

x using

technology and fill in the blanks. Domain: ______________________________ Range: _______________________________


52

−+

+

xx

x using

technology and fill in the blanks. Domain: ______________________________ Range: _______________________________































Documents

Algebra 3-4 Unit 4 Rational 4... · Rational expression: a fraction that has polynomials in the numerator and/or denominator. To multiply two rational expressions: factor, cancel