Name:_______________ Date:_______________

5.0 Worksheet (2 pages) Simplify the following Rational Expressions….make sure answers are in FACTORED FORM.

1. 𝒙𝟐(𝒙+𝟐)

𝟔𝒙𝟓 2.

(𝒙+𝟐)(𝒙−𝟑)

𝒙+𝟐 3.

𝒉−𝟐

𝟐−𝒉

4. 𝒙+𝟐

𝒙+𝟐 5.

𝟒𝒙(𝟐𝒙−𝟏)

𝟓𝒙(𝟑𝒙−𝟏) 6.

𝟖(𝒌−𝟑)

𝟔𝒌

7. 𝟐(𝒙−𝟑)

𝟐𝒙+𝟕 8.

−𝟔𝒙𝟓𝒚𝟐(𝒙−𝟐)

𝟑𝒙𝟓𝒚(𝒙−𝟐) 9.

𝒎𝟐−𝟗

𝒎𝟐−𝟔𝒎+𝟗

10. 𝟔𝒙𝟐+𝟓𝒙+𝟏

𝟐𝒙𝟐−𝟓𝒙−𝟑 11.

𝒑𝟐−𝟑𝟔

𝒑−𝟔 12.

𝟑𝒂𝟐+𝟓𝒂−𝟐

𝒂𝟐−𝒂−𝟔

Name:_______________ Date:_______________

Multiplying and Dividing Rational Expressions…make sure answers are in FACTORED FORM.

13. 𝟏𝟔𝒙𝟑𝒚

𝟑𝒙𝟐∙𝟐𝟕𝒚

𝟒𝒙𝟐 14.

𝒎−𝟕

𝒎−𝟐∙𝒎𝟐−𝟒

𝒎−𝟕 15.

𝒂𝟐−𝟏𝟔

𝟐𝒂+𝟖∙

𝒂+𝟒

𝒂𝟐+𝟖𝒂+𝟏𝟔

16. 𝒎𝟐−𝟔𝒎+𝟗

𝟔∙

𝟗

(𝒎−𝟑)𝟐 17.

𝟗𝒑𝒏𝟑

𝟒𝒎𝟐∙𝟏𝟐𝒎𝟑

𝟔𝒑𝒏 18.

𝟔𝒎𝟐𝒏

𝟖∙

𝟏

𝟏𝟖𝒎𝒏

19. 𝒙𝟐−𝟑𝟔

𝒙𝟐−𝟒𝟗÷

𝒙+𝟔

𝒙−𝟕 20.

𝟑𝒚𝟑−𝟐𝟕𝒚

𝟐𝒚𝟐+𝟓𝒚−𝟑÷

𝟗𝒚𝟐

𝟒𝒚𝟐−𝟏 21.

𝒂𝟐−𝟓𝒂+𝟔

𝟒÷

𝒂−𝟑

𝟏𝟔

22. 𝒙𝟐+𝒙

𝟔𝒙+𝟏𝟐÷

𝒙𝟐−𝟏

𝒙𝟐−𝟒 23.

𝒙𝟐−𝟏𝟔

𝟐𝒙+𝟖÷

𝒙𝟐−𝟐𝒙−𝟖

𝒙𝟐−𝒙−𝟔 24.

𝒙𝟐 − 𝟐𝟓

𝟐𝒙 + 𝟏𝟎∙𝒙𝟐 + 𝟓𝒙 + 𝟔

𝒙𝟐 − 𝟑𝟔÷

𝒙 + 𝟑

𝟐𝒙 + 𝟏𝟐

Name:_______________ Date:_______________

5.1 Worksheet (2 pages) Find the SUM or DIFFERENCE.

a. 𝑥2+3𝑥

𝑥+2+

2𝑥+6

𝑥+2 b.

7𝑥+3

𝑥2−1−

6

𝑥−1

c. 7𝑥+3

𝑥2−1−

6𝑥+4

𝑥2−1

d. 6𝑦+13

𝑦2−4𝑦−21−

𝑦−2

𝑦2−4𝑦−21

e. 4𝑥

𝑥2+9𝑥+18+

5

𝑥+6

f. 3𝑎+6

𝑎2−16−

7

6𝑎+24

Name:_______________ Date:_______________

Simplify Each Complex Fraction.

g.

1

𝑦2−1

1+1

𝑦

h. 25

𝑎−𝑎

5+𝑎

i. 4+

12

2𝑥−3

5+15

2𝑥−3

j. 𝑥

3𝑥−2𝑥

9𝑥2−4

Name:_______________ Date:_______________

5.2 Worksheet (2 pages) Solve each Rational Equation. Make sure to include all restrictions on Domain and

Check solutions for any extraneous solutions.

1. 1

6𝑘2=

1

3𝑘2−

1

𝑘 2.

𝑏+6

4𝑏2+

3

2𝑏2=

𝑏+4

2𝑏2 3.

1

𝑥=

6

5𝑥+ 1

4. 1

𝑣+

3𝑣+12

𝑣2−5𝑣=

7𝑣−56

𝑣2−5𝑣 5.

1

𝑚2−𝑚+

1

𝑚=

5

𝑚2−𝑚 6.

1

𝑛−8− 1 =

7

𝑛−8

7. 1

𝑟−2+

1

𝑟2−7𝑟+10=

6

𝑟−2 8. 1 =

𝑣+2

𝑣−4+

7𝑣−42

𝑣−4 9.

𝑟−4

5𝑟=

1

5𝑟+ 1

Name:_______________ Date:_______________

10. 1 +𝑥2−5𝑥−24

3𝑥=

𝑥−6

3𝑥

11. 1 =1

𝑥2+2𝑥+

𝑥−1

𝑥

12. 𝑛+5

𝑛+8= 1 +

6

𝑛+1 13.

𝑟+5

𝑟2−2𝑟− 1 =

1

𝑟2−2𝑟

14. 𝑎−2

𝑎+3− 1 =

3

𝑎+2 15.

5

𝑛3+2𝑛2=

4

𝑛+2+

1

𝑛2

Name:_______________ Date:_______________

5.3 Worksheet (2 pages) Given the following Rational Functions. Identify the asymptotes, domain, and range of each function.

a) 𝒇(𝒙) =𝟓

𝒙+𝟒 b) 𝒈(𝒙) =

𝟖

𝒙− 𝟑

c) 𝒇(𝒙) =𝟐

𝒙+𝟔− 𝟐 d) 𝒉(𝒙) =

−𝟑

𝒙−𝟏+ 𝟓

e) 𝒈(𝒙) =𝟓

𝒙 f) 𝒓(𝒙) =

−𝟏

𝒙−𝟐+ 𝟒

g) 𝒃(𝒙) =𝟑

𝟐𝒙−𝟒 h) 𝒎(𝒙) =

𝟐

𝟒𝒙+𝟏− 𝟕

Name:_______________ Date:_______________

Graph each function. State the Domain and Range.

i) 𝒇(𝒙) =𝟑

𝒙−𝟕− 𝟖

j) 𝒉(𝒙) =−𝟔

𝒙+𝟒− 𝟐

k) 𝒈(𝒙) =𝟑

𝒙− 𝟓

l) 𝒎(𝒙) =−𝟏

𝟏

𝟒𝒙−𝟐

+ 𝟐

Name:_______________ Date:_______________

5.4 Worksheet (1 page) For the following Functions, find (if they apply to the function):

a) the vertical asymptote(s)

b) the horiztonal/slant asymptotes

c) the holes

1. 𝑓(𝑥) =2𝑥

𝑥2−1 2. 𝑚(𝑥) =

𝑥2−5𝑥+6

𝑥2−4𝑥+3

3. 𝑔(𝑥) =8

𝑥2−𝑥−6 4. 𝑓(𝑥) =

𝑥2+11𝑥+18

2𝑥+1

5. ℎ(𝑥) =𝑥2−9

2𝑥2+1 6. 𝑏(𝑥) =

𝑥−4

𝑥2−3𝑥

Name:_______________ Date:_______________

5.5 Worksheet (2 pages) For the following Functions, find (if they apply to the function):

a) the 𝒙-intercepts, and 𝒚-intercepts

b) the vertical asymptote(s)

c) the horiztonal/slant asymptotes

d) the holes

e) any additional points needed

f) then, graph each function

7. 𝑓(𝑥) =2𝑥

𝑥2−1

8. 𝑚(𝑥) =𝑥2−5𝑥+6

𝑥2−4𝑥+3

Name:_______________ Date:_______________

9. 𝑔(𝑥) =8

𝑥2−𝑥−6

10. 𝑓(𝑥) =𝑥2+11𝑥+18

2𝑥+1

11. ℎ(𝑥) =𝑥2−9

2𝑥2+1

12. 𝑏(𝑥) =𝑥−4

𝑥2−3𝑥

Name:_______________ Date:_______________

5.6 Worksheet (2 pages) Solve Each Rational Inequality. Write answers in interval notation.

1. 3 −4

𝑥>

5

4𝑥 2.

5

3𝑎−

3

4𝑎>

5

6

3. 𝑥−2

𝑥+2+

1

𝑥−2>

𝑥−4

𝑥−2 4.

3

4−

1

𝑥−3>

𝑥

𝑥+4

Name:_______________ Date:_______________

5. 𝑥

5+

2

3<

3

𝑥−4 6.

𝑥

𝑥+2+

1

𝑥−1<

3

2

7. −7

𝑥+5≤ −

8

𝑥+6 8.

(𝑥+7)(𝑥−3)

(𝑥−5)2> 0

9. Upper Level…..Extra Credit

Solve 1+

9

𝑥+20

𝑥2

1−25

𝑥2

=𝑥+4

𝑥−5