chapter 11 sections 11.1, 11.3-11.5 rational expressions

Chapter 11

Sections 11.1, 11.3-11.5

Rational Expressions

§ 11.1

Solving Proportions

Martin-Gay, Developmental Mathematics 3

Solve each proportion:

ANSWER

ANSWER

€

t

32=

7

16

1.)

€

18

24=b

122.)

€

16t = 22416 16

€

t =14

€

24b = 216

24 24

€

b = 9



ANSWER

ANSWER

€

21

n=

28

32

1.)

€

4

12=

11

p2.)

€

672 = 28n28 28

€

n = 24

€

4 p =132

4 4

€

p = 33



ANSWER

€

7

10=

9 + x

x

1.)

€

x 2 −16

x + 4=x − 4

3

€

7x =10(9 + x)

-10x - 10x

€

x = −30

€

7x = 90 +10x

€

−3x = 90-3 -3

€

3(x 2 −16) = (x + 4)(x − 4)

€

3x 2 − 48 = x 2 −16- x2 - x2

€

2x 2 − 48 = −16+ 48 + 48

€

2x 2 = 322 2

€

x 2 =16

€

x = 4ANSWER



€

5

x + 6=x − 6

x€

5x = (x + 6)(x − 6)

€

5x = x 2 − 36- 5x - 5x

€

0 = x 2 − 5x − 36

ANSWERS

€

0 = (x − 9)(x + 4)

€

x − 9 = 0

+ 9 + 9

x = 9

€

x + 4 = 0

− 4 − 4

x = −4


Solve the proportion for x.

3

5

2

1

x

x

2513 xx

10533 xx

72 x

27x

Solving Proportions

Example


What is the value of “x”

4x

83x –

A – 6 B – 3 C 3 D 6

SOLUTION

4x =

8x – 3

Write original proportion.

Cross products property4(x – 3) = x 8

4x – 12 = 8x Simplify.

Subtract 4x from each side. –12 = 4x

Divide each side by 4. –3 = x


Example Solve the equation below:

€

3(x − 7) = 2(5x)

€

3x − 21 =10x- 3x - 3x

€

−21 = 7x7 7

€

x = −3ANSWER


Solve each problem.

€

−4x +12 = 5x +15+4x +4x

€

12 = 9x +15 - 15 - 15

€

−3 = 9x

9 9

€

x = −1

3

€

11x + 8 = 2x 2 + 5x- 11x - 11x

€

8 = 2x 2 − 6x- 8 - 8

€

0 = 2x 2 − 6x − 8

€

0 = 2x 2 − 8x + 2x − 8

€

0 = 2x(x − 4) + 2(x − 4)

€

0 = (2x + 2)(x − 4)

€

x = −1, 4

§ 11.3

Simplifying Rational Expressions


Simplifying a Rational Expression

1) Completely factor the numerator and denominator.

2) Apply the Fundamental Principle of Rational Expressions to eliminate common factors in the numerator and denominator.

Warning!

YOU CAN ONLY ELIMINATE THINGS THAT ARE BEING MULTIPLIED!!!



Simplify the following expression.

€

56x 3

4x 2 =

€

7⋅ 2⋅ 2⋅ 2⋅ x⋅ x⋅ x2⋅ 2⋅ x⋅ x

=

€

14x


Example



€

4x

20=

€

2⋅ 2⋅ x2⋅ 2⋅ 5

=

€

x

5


Example



€

3x

9x 2 + 3=

€

3⋅ x3(3x 2 +1)

=

€

x

3x 2 +1


Example

GCF Bottom



xx

x

5

3572

)5(

)5(7

xx

x

x

7


Example

GCF Top and Bottom



20

432

2

xx

xx

)4)(5(

)1)(4(

xx

xx

5

1

x

x


Example



7

7

y

y 7

)7(1

y

y1


Example



€

6x 2

12x 4 +18x 2 =

€

2⋅ 3⋅ x⋅ x6x 2(2x 2 + 3)

=

€

1

2x 2 + 3


Example

GCF Bottom

€

2⋅ 3⋅ x⋅ x2⋅ 3⋅ x⋅ x(2x 2 + 3)

=



€

x 2 + 4x + 4

x 2 + 9x +14=

€

(x + 2)(x + 2)

(x + 7)(x + 2)=

€

x + 2

x + 7


Example

Factor Top and Bottom

Use Sum and Product Method



€

5z3 + z2 − z

3z=

€

z(5z2 + z −1)

3z=

€

5z2 + z −1

3


Example

Factor Top by grouping

BUT NOTHING WILL CANCEL OUT EXCEPT THE “Z”



€

4a2 − 9

10a+15=

€

(2a − 3)(2a+ 3)

5(2a+ 3)=

€

2a − 3

5


Example

Factor Top by “difference of squares”

Factor Bottom by “GCF”



€

35n

35n2 =

€

7⋅ 5⋅ n7⋅ 5⋅ n⋅ n

=

€

1

n


Example

Factor Top by “difference of squares”

Factor Bottom by “GCF”



€

x − 8

x 2 + x − 72=

€

x − 8

(x + 9)(x − 8)=

€

1

x + 9


Example

Factor Top by “Sum and Product”



€

56v − 72

32v=

€

8(7v − 9)

8⋅ 4⋅ v=

€

7v − 9

4v


Example

Factor Top by “GCF”

Factor Bottom

Multiplying and Dividing Rational Expressions

§ 11.4


Multiplying Rational Expressions

QS

PR

S

R

Q

P

Just remember:

1.) FACTOR IF POSSIBLE

2.) “TOP times TOP” and “BOTTOM times BOTTOM”

3.) THEN SIMPLIFY


Multiply the following rational expressions.

12

5

10

63

2 x

x

x

4

1

32252

532

xxx

xxx

Example




mnm

m

nm

nm2

2)(

)()(

))((

nmmnm

mnmnm

nm

nm


Example



€

10r5

21s2 ⋅3s

5r3 =

€

2r2

7s

€

2⋅ 5⋅ r⋅ r⋅ r⋅ r⋅ r⋅ 3⋅ s3⋅ 7⋅ s⋅ s⋅ 5⋅ r⋅ r⋅ r

=

Example




€

4 p4q

9r⋅

9r3

10p2q2 ⋅15pq

2r=

€

3p3r

€

2⋅ 2⋅ p⋅ p⋅ p⋅ p⋅ q⋅ 3⋅ 3⋅ r⋅ r⋅ r⋅ 5⋅ 3⋅ p⋅ q3⋅ 3⋅ r⋅ 2⋅ 5⋅ p⋅ p⋅ q⋅ q⋅ 2⋅ r

=

Example



JUST REMEMBER:

Change it to multiplication of the reciprocal

QR

PS

R

S

Q

P

S

R

Q

P

Dividing Rational Expressions


Divide the following rational expression.

25

155

5

)3( 2 xx

155

25

5

)3( 2

x

x

)3(55

55)3)(3(

x

xx3x

Dividing Rational Expressions

Example



€

9a

10b÷

3a3

20b=

€

6

a2

€

3⋅ 3⋅ a⋅ 2⋅ 2⋅ 5⋅ b2⋅ 5⋅ b⋅ 3⋅ a⋅ a⋅ a

=

Example


€

9a

10b⋅

20b

3a3 =

Adding and Subtracting Rational Expressions with the Same

Denominators

§ 11.5


Rational Expressions

Remember how to add or subtract fractions?

R

QP

R

Q

R

P

R

QP

R

Q

R

P


Add the following rational expressions.

72

83

72

34

p

p

p

p72

57

p

p

72

8334

p

pp

Adding Rational Expressions

Example


Subtract the following rational expressions.

2

16

2

8

yy

y

2

168

y

y

2

)2(8

y

y8

Subtracting Rational Expressions

Example



103

6

103

322 yyyy

y

103

632 yy

y

)2)(5(

)2(3

yy

y

5

3

y


Example



€

11

4c+

5

4c=

€

16

4c=

€

4⋅ 4

4⋅ c=

€

4

c


Example



€

4x +12

16x+

8x + 4

16x=

€

12x +16

16x=

€

4(x + 4)

4⋅ 4⋅ x=

€

x + 4

4x


Example



€

a

a+1+

1

a+1=

€

a+1

a+1=

€

1


Example



€

2x

x + 3+

6

x + 3=

€

2x + 6

x + 3=

€

2(x + 3)

x + 3=

€

2


Example



€

y

y 2 − 4−

2

y 2 − 4=

€

y − 2

y 2 − 4=

€

y − 2

(y + 2)(y − 2)=

€

1

y + 2


Example

chapter 11 sections 11.1, 11.3-11.5 rational expressions

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