Ch. 6 - Rational Expressions Notes with answers · Chapter 6 – Rational Expressions Created by...

Pre-Calculus 11 Chapter 6 – Rational Expressions

Created by Ms. Lee 1 of 19 Reference: McGraw-Hill Ryerson Pre-Calculus 11

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Ch. 6 – Rational Expressions Notes

6.1 – RATIONAL EXPRESSIONS 2

Ch. 6. 1 HW: p. 317 # 1 – 6 odd letters, 8, 9, 20, 25. 3

6.1 – EXTRA PRACTICE 4

6.2 – MULTIPLYING AND DIVIDING RATIONAL EXPRESSIONS 6

Ch. 6.2 HW: p. 326 # 1 – 4 odd letters, 5 – 7 all, 8, 15, 16, 18 8

6.3 – ADDING AND SUBTRACTING RATIONAL EXPRESSIONS 9

Ch. 6.3 HW: p. 336 # 1 – 11 (odd letters), 12, 15, 18 11

6.4 – RATIONAL EQUATIONS 12

Ch. 6.4 HW: p. 349 # 1 – 13, 14 17

CH. 6 ‐ REVIEW 17

Ch. 6 Review HW: p. 352 All #s, odd letters, p. 355 # 1 – 9 19



6.1–RationalExpressions Definition: Rational Expression is an algebraic fraction with a numerator and a denominator that are polynomials. Non-Permissible Values: Given a rational expression, non-permissible values are all values of a variable that make the denominator zero. Examples: Determine Non-permissible values given a rational expression. Rational Expressions Non-permissible Value(s)

x

x 12

5

2

x

32

3

x

x

)52(

3

xx

x

)3)(2(

1

xx

x

12

122

pp

p



Examples: Simplify Rational Expressions and determine non-permissible values Rational Expression Simplified Expression Non-permissible Values

102

632

xx

x

1

12

t

t

103

1022

2

yy

yy

9

262

m

m

Examples: Rational Expression in two variables

Simplify yx

yx

68

916 22

.

a) How can you express non-permissible values for x in terms of y?

b) How can you express non-permissible values for y in terms of x?

Ch. 6. 1 HW: p. 317 # 1 – 6 odd letters, 8, 9, 20, 25.



6.1–ExtraPracticeCreated by Mr. Ko Please use a separate sheet of paper for this worksheet. SHOW ALL WORK! 1) State weather each expression is a rational expression. If the expression is not rational, explain

why.

a) 4

x

b)

3

a c)

8

m

d)

x

x

e) 4

2

x

x

f) 2 6x x g)

3

x x h)

3

4

1x

i) 3 1

5

a

a b

j) 3

x y

x

2) Evaluate each rational expression for 2x and 3y .

a) 2

3 1

x

y

b) 3 2x y

xy

c)

3 1

3

x

y

d) 2

2

2 1

1

x x

x

3) For which value(s) of x is each expression not defined?

a) 2

1x

b) 2 1

5

x c)

2

2 7

x

x d)

2

3

1

8

x

x

e) 2

3 4

9

x

x

f)

6 5

2 4

x

x x

g) 3

3

x

h)

2

3

81

8

x

x

4) Reduce to lowest terms.

a) 36

4

x

y b)

25

15

x

x c)

26

8

a c

ab d)

2

5

9m n

m

5) Reduce to lowest terms.

a) 9

3 27

a b

a b

b) 5

10 2

x

x

c) 16 4

32 8

a

a

d) 7 14

5 10

x

x

6) Simplify each expression. Identify the nonpermissible value(s) of x .

a) 22 6

5

x x

x

b)

22 10

4 20

x x

x

c) 24 12

3

x x

x

d) 23 6

14 7

x x

x



7) Simplify.

a) c d

d c

b) 2

2

3 18

12 2

xy y

y xy

c) 2

2

10 15

6 4

xy x y

x x

d) 2 2

2 2

60 24

16 40

a b ab

ab a b

8) Simplify each expression, if possible. Identify any nonpermissible value(s) for each variable.

a) 2

2

5 14

6 8

a a

a a

b) 2

3

3 18

x

x x

c) 2

2

9

6 9

r

r r

d) 2

4

16

x

x

9) Simplify.

a) 2 2

2

9 16

6 8

a b

a ab

b) 2 2

2

6

2

a ab b

a ab

c) 2

2

8 32

2 12 16

t

t t

d) 2 2

2 2

5 24

7 12

c cd d

c cd d

e) 2 2

2 2

10 24

36

a ab b

a b

f) 2 2

2

36 25

30

x x

x x

10) Simplify.

a) 2

2

32 2

4 44 112

a

a a

b) 3 2

3

2 28 102

18 2

x x x

x x

c) 2

2

3 33 90

6 6 120

a a

a a

d) 4 4

2 2 2 25 4

x y

x y x xy y

Answers 1) a) Not rational, b) Rational, c) Rational, d) Not rational, e) Rational, f) Rational, g) Not rational ,

h) Rational, i) Not rational, j) Rational

2) a) 0, b) –2, c) 3

2 , d)

1

3c

3) a) –1, b) Expression is defined for all values of x , c) 7

2, d) 2, e) 3 , f) 0, –4, g) 0

h) Expression is defined for all values of x

4) a) 9 x

y

, b) 3

x, c)

3

4

ac

b, d)

3

9n

m 5) a)

1

3, b)

1

2 , c)

1

2, d)

7

5

6) a) 2 6

5

x , all real values of x , nonpermissible value: 0x

b) 2

x, all real values of x , nonpermissible value: 5x

c) 4x , all real values of x , nonpermissible value: 3x

d) 3

7

x , all real values of x , nonpermissible value: 2x

7) a) –1, b) 3

2

, c)

5

2

y, d)

3

2

8) a) 7

4

a

a

, 4, 2a , b) 1

6x, 6, 3x , c)

3

3

r

r

, 3r , d) 1

4x, 4, 4x

9) a) 3 4

2

a b

a

, b)

3a b

a

, c)

4 8

4

t

t

, d) 8

4

c d

c d

, e) 4

6

a b

a b

, f) 6 5x x

10) a) 4

14 2

a

a

, b) 17

3

x

x

, c) 6

2 8

a

a

, d) 4

x y

x y



6.2–MultiplyingandDividingRationalExpressions Recap: Multiply. Write the answer in simplest form.

1) 15

16

18

5

2)

21

18

6

14

3)

x

y

xy

x

83

4 22

4)

yx

xy

y

x2

3

2

3 15

5

2

Divide. Write the answer in simplest form.

1) 15

12

25

16

2) 10

15

100

3

3) y

yx

x

yx

2

3

4

6 222

4) yx

x

xy

yx2

5

20

15

10

3

Similarly: Multiply. Write the answer in simplest form. Identify all non-permissible values.

1) 2

2

2

d

rh

r

d

2)

93

62

3

3

x

x

x

x



3)

3

9 2

3

2

y

rr

rr

y

4) 45

2

65

122

xx

x

xx

x

Divide. Write the answer in simplest form. State the non-permissible values.

1) 28

42

xx

2) 1243

3

x

x

x

x

3) 20

152

4

72

2

xx

xx

x

x



4) 1

43

22

232

1

328 2

2

2

y

y

y

yy

y

yy

Apply: A sphere is contained in a rectangular box such that the sides of the sphere are touching the sides of the box. What fraction of the volume of the box does the sphere occupy? Model the situation using a quotient of rational expression (in simplest form). Then write this value as a percent.

Note: 3lengthVcube 3

4 3rVsphere

Ch. 6.2 HW: p. 326 # 1 – 4 odd letters, 5 – 7 all, 8, 15, 16, 18



6.3–AddingandSubtractingRationalExpressions Recap: Reduce first if possible, get a common denominator and add/subtract. Add. Write the answer in simplest/reduced form.

1) 23

2

2) 3

1

4

3

3) 5

4

3

2

Subtract. Write the answer in simplest/reduced form.

1) 15

3

5

2

2) 3

2

2

5

3) 16

3

4

3

4) 37

3

Similarly: Add/Subtract. Write the answer in simplest form. Identify all non-permissible values.

1) 2

31

x

2) 23

2

2

1

aa

3) 13

2

32

xyxy

y



Add/Subtract. Write the answer in simplest form. Identify all non-permissible values.

1) n

m

n

m 1

2) 34

28

34

110

m

m

m

3) )1)(3(

564

)1)(3(

63

)1)(3(

5 22

xx

xx

xx

x

xx

xx

Add/Subtract. Write the answer in simplest form. Identify all non-permissible values.

5) 1

3

1

42

pp

6) 34

2

6

122

xx

x

xx

x



7)

yy

y4

42

8)

4

1

16

4

1

4

1

2

xx

xxx

Ch. 6.3 HW: p. 336 # 1 – 11 (odd letters), 12, 15, 18



6.4–RationalEquations Definition: Rational Equation: An equation containing at least one rational expression. Example: Solve each rational equation. State any non-permissible values.

1) 37

4

x

x

2) 2

1

126

10

4

22

xxx



3) 189

18

6

4

3

92

yyyy

4) 6

25

3

5

2

32

xxxx

x



5) 4

244

2

1

2

142

2

k

kk

k

k

k

k

6) Two friends share a paper route. Sheena can deliver the papers in 40 min. Jeff can cover the same route in 50 min. How long, to the nearest minute, does the paper route take if they work together? Fraction of papers delivered by Sheena in Fraction of papers delivered by Jeff in 1 minute:

1 minute:

2 minutes:

2 minute:

3 minutes:

3 minutes:

… … x minutes:

x minutes:



7) Stella takes 4 h to paint a room. It takes Jose 3 h to paint the same area. How long will the paint job take if they work together?

8) Kyra mows a lawn in 40 min. When Mark and Kyra work together, they can mow the lawn in 24 min. How long would it take Mark to mow the lawn on his own?



9) Given two consecutive numbers, 5 is added to the first number (smaller #) and two is subtracted from the second number (bigger #). The quotient of the new numbers (first divided by second) is

4

7. Determine the numbers algebraically.

10)



A boat travels 4 km upstream in the same time that it takes the boat to travel 12 km downstream. The average speed of the current is 3km/h. What is the average speed of the boat in still water? Let speed of the boat in still water = s What would be the speed of the boat going uphill if the current, with the speed of 3 km/h, is against the boat’s direction? What would be the speed of the boat going downhill if the current, with the speed of 3km/h is in the same direction as the boat?

Ch. 6.4 HW: p. 349 # 1 – 13, 14

Ch.6-Review 1. Simplify the rational expression:

32

232

3

32

3

2

6

3

yx

yx

yx

yx



2. Simplify. Identify any non-permissible values.

2

1

3

3

4

2322

2

xxx

x

x

xx

3. Simplify. Identify any non-permissible values.

32

1

35294

222

xxx

x

x

x



4. Solve. Identify all non-permissible values.

2

3

6

52

2

x

x

xx

5. True or False

a) ________ xxx

xxx

)2)(1(

)1)(2(

b) ________ xxx

xxx

)2)(1(

)1)(2(, 2,1x

Ch. 6 Review HW: p. 352 All #s, odd letters, p. 355 # 1 – 9

Ch. 6 - Rational Expressions Notes with answers · Chapter 6 – Rational Expressions Created by...

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