15 proportions and the multiplier method for solving rational equations

Addition and Subtraction of Rational Expressions

Frank Ma © 2011

Addition and Subtraction of Rational ExpressionsOnly fractions with the same denominator may be added or subtracted directly.


Addition and Subtraction Rule(for rational expressions with the same denominator)

Only fractions with the same denominator may be added or subtracted directly.

A BD D

± = A±BD




A BD D

± = A±BD

Usually we put the result in the factored form, cancel the common factor and give the simplified answer.




A BD D

± = A±BD


Example A. Add and subtract and simplify the answer.

a. 5 78 8+ =




A BD D

± = A±BD



a. 5 78 8+ = 5 + 7

8




A BD D

± = A±BD



a. 5 78 8+ = 5 + 7

8 = 128




A BD D

± = A±BD



a. 5 78 8+ = 5 + 7

8 = 128

3

2




A BD D

± = A±BD



a. 5 78 8+ = 5 + 7

8 = 128 = 3

2

3

2




A BD D

± = A±BD



a. 5 78 8+ = 5 + 7

8 = 128 = 3

2

3

2

b. 3x2x – 3 – 6 – x

2x – 3




A BD D

± = A±BD



a. 5 78 8+ = 5 + 7

8 = 128 = 3

2

3

2

b. 3x2x – 3 – 6 – x

2x – 3 = 3x – (6 – x) 2x – 3




A BD D

± = A±BD



a. 5 78 8+ = 5 + 7

8 = 128 = 3

2

3

2

b. 3x2x – 3 – 6 – x

2x – 3 = 3x – (6 – x) 2x – 3 = 3x – 6 + x

2x – 3




A BD D

± = A±BD



a. 5 78 8+ = 5 + 7

8 = 128 = 3

2

3

2

b. 3x2x – 3 – 6 – x

2x – 3 = 3x – (6 – x) 2x – 3 = 3x – 6 + x

2x – 3

= 4x – 6 2x – 3




A BD D

± = A±BD



a. 5 78 8+ = 5 + 7

8 = 128 = 3

2

3

2

b. 3x2x – 3 – 6 – x

2x – 3 = 3x – (6 – x) 2x – 3 = 3x – 6 + x

2x – 3

= 4x – 6 2x – 3 = 2(2x – 3)

2x – 3




A BD D

± = A±BD



a. 5 78 8+ = 5 + 7

8 = 128 = 3

2

3

2

b. 3x2x – 3 – 6 – x

2x – 3 = 3x – (6 – x) 2x – 3 = 3x – 6 + x

2x – 3

= 4x – 6 2x – 3 = 2(2x – 3)

2x – 3 = 2

To add or subtract rational expressions with different denominators, they have to be converted to expressions with a common denominator.


To add or subtract rational expressions with different denominators, they have to be converted to expressions with a common denominator. The easiest common denominator to work with is their LCM.




Multiplier Method

Given the fraction , to convert it into denominator D,

the new numerator is

AB

AB

*

D.



Multiplier Method



Example B.

a. Convert to a fraction with denominator 12.

AB

AB

*

D.

54

In practice, we write this as AB = A

B * D D

new numerator



Multiplier Method



Example B.


AB

AB

*

D.

54


B * D D

54 =

new numerator



Multiplier Method



Example B.


AB

AB

*

D.

54

54 * 12


B * D D

54 = 12

new numerator

new numerator



Multiplier Method



Example B.


AB

AB

*

D.

54

54 * 12

3


B * D D

54 = 12

new numerator

new numerator



Multiplier Method



Example B.


AB

AB

*

D.

54

54 * 12

3 1512


B * D D

54 = 12 =

new numerator

b. Convert into an expression with denominator 12xy2.

Addition and Subtraction of Rational Expressions3x4y


3x4y



*12xy23x4y = 3x

4y 12xy2

new numerator



*12xy23x4y = 3x

4y 12xy23xy



*12xy23x4y = 3x

4y 12xy2 = 9x2y12xy2

3xy



*12xy2

x + 12x + 3

3x4y = 3x

4y 12xy2 = 9x2y12xy2

3xy


c. Convert into an expression denominator 4x2 – 9.


*12xy2

x + 12x + 3

x + 12x + 3

3x4y = 3x

4y 12xy2 = 9x2y12xy2

3xy

= x + 12x + 3 * (4x2 – 9) (4x2 – 9)

new numerator




*12xy2

x + 12x + 3

x + 12x + 3

3x4y = 3x

4y 12xy2 = 9x2y12xy2

3xy

= x + 12x + 3 * (4x2 – 9) (4x2 – 9)

= x + 12x + 3 * (2x + 3)(2x – 3) (4x2 – 9)




*12xy2

x + 12x + 3

x + 12x + 3

3x4y = 3x

4y 12xy2 = 9x2y12xy2

3xy

= x + 12x + 3 * (4x2 – 9) (4x2 – 9)

= x + 12x + 3 * (2x + 3)(2x – 3) (4x2 – 9)

= (x + 1)(2x – 3) (4x2 – 9)




*12xy2

c. Convert into an expression denominator 4x2 – 9.x + 12x + 3

x + 12x + 3

3x4y = 3x

4y 12xy2 = 9x2y12xy2

3xy

= x + 12x + 3 * (4x2 – 9) (4x2 – 9)

= x + 12x + 3 * (2x + 3)(2x – 3) (4x2 – 9)

= (x + 1)(2x – 3) (4x2 – 9)

= 2x2 – x – 34x2 – 9


Addition and Subtraction of Rational ExpressionsWe give two methods of combining rational expressions below.

Addition and Subtraction of Rational ExpressionsWe give two methods of combining rational expressions below. The first one is the traditional method.

Addition and Subtraction of Rational ExpressionsWe give two methods of combining rational expressions below. The first one is the traditional method. The second one usesthe Multiplier Method directly.


Traditional Method (Optional) (Combining fractions with different denominators)

We give two methods of combining rational expressions below. The first one is the traditional method. The second one usesthe Multiplier Method directly.


Traditional Method (Optional) (Combining fractions with different denominators)I. Find the LCD of the expressions.



Traditional Method (Optional) (Combining fractions with different denominators)I. Find the LCD of the expressions.II. Convert each expression into the LCD.



Traditional Method (Optional) (Combining fractions with different denominators)I. Find the LCD of the expressions.II. Convert each expression into the LCD.III. Add or subtract the new numerators.



Traditional Method (Optional) (Combining fractions with different denominators)I. Find the LCD of the expressions.II. Convert each expression into the LCD.III. Add or subtract the new numerators.IV. Simplify the result.


Example C. Combine


Traditional Method (Optional) (Combining fractions with different denominators)I. Find the LCD of the expressions.II. Convert each expression into the LCD.III. Add or subtract the new numerators.IV. Simplify the result. 2

3xy – x2y2


Example C. Combine

The LCM of the denominators {3xy, 2y2} is 6xy2.



3xy – x2y2


Example C. Combine


Convert



3xy – x2y2

23xy = 6xy2

x2y2 = 3x2

6xy24y


Example C. Combine


Convert



3xy – x2y2

23xy = 6xy2

x2y2 = 3x2

6xy2

23xy – x

2y2 = 4y6xy2 – 3x2

6xy2Hence

4y


Example C. Combine


Convert



3xy – x2y2

23xy = 6xy2

x2y2 = 3x2

6xy2

23xy – x

2y2 = 4y6xy2 – 3x2

6xy2 =Hence 4y – 3x2

6xy2

4y


Example C. Combine


Convert



3xy – x2y2

23xy = 6xy2

x2y2 = 3x2

6xy2

23xy – x

2y2 = 4y6xy2 – 3x2

6xy2 =Hence 4y – 3x2

6xy2

This is simplified because the numerator is not factorable.

4y


Example D. Combine

Addition and Subtraction of Rational Expressionsx

4x – 2 – x – 1 2x2 + x – 1

Example D. Combine


4x – 2 – x – 1 2x2 + x – 1

Factor each denominator to find the LCD.4x – 2 = 2x2 + x – 2 =

Example D. Combine


4x – 2 – x – 1 2x2 + x – 1

Factor each denominator to find the LCD.4x – 2 = 2(2x – 1), 2x2 + x – 2 =

Example D. Combine


4x – 2 – x – 1 2x2 + x – 1

Factor each denominator to find the LCD.4x – 2 = 2(2x – 1), 2x2 + x – 2 = (2x – 1)(x + 1)

Example D. Combine


4x – 2 – x – 1 2x2 + x – 1

Factor each denominator to find the LCD.4x – 2 = 2(2x – 1), 2x2 + x – 2 = (2x – 1)(x + 1) Hence the LCD = 2(2x – 1)(x + 1)

Example D. Combine


4x – 2 – x – 1 2x2 + x – 1

Factor each denominator to find the LCD.4x – 2 = 2(2x – 1), 2x2 + x – 2 = (2x – 1)(x + 1) Hence the LCD = 2(2x – 1)(x + 1)Next, convert each fraction into the LCD

Example D. Combine


4x – 2 – x – 1 2x2 + x – 1


x4x – 2 = x

2(2x – 1)

Example D. Combine


4x – 2 – x – 1 2x2 + x – 1


x4x – 2 = x

2(2x – 1) * 2(2x – 1)(x + 1) LCD

Example D. Combine


4x – 2 – x – 1 2x2 + x – 1


x4x – 2 = x

2(2x – 1) * 2(2x – 1)(x + 1) LCD

= x(x + 1) LCD

Example D. Combine


4x – 2 – x – 1 2x2 + x – 1


x4x – 2 = x

2(2x – 1) * 2(2x – 1)(x + 1) LCD

= x(x + 1) = x2 + xLCD LCD

Example D. Combine


4x – 2 – x – 1 2x2 + x – 1


x4x – 2 = x

2(2x – 1) * 2(2x – 1)(x + 1) LCD

= x(x + 1) = x2 + xLCD LCD

x – 1 2x2 + x – 1 =

x – 1 (2x – 1)(x + 1)

Example D. Combine


4x – 2 – x – 1 2x2 + x – 1


x4x – 2 = x

2(2x – 1) * 2(2x – 1)(x + 1) LCD

= x(x + 1) = x2 + xLCD LCD

x – 1 2x2 + x – 1 =

x – 1 (2x – 1)(x + 1) * 2(2x – 1)(x + 1) LCD

Example D. Combine


4x – 2 – x – 1 2x2 + x – 1


x4x – 2 = x

2(2x – 1) * 2(2x – 1)(x + 1) LCD

= x(x + 1) = x2 + xLCD LCD

x – 1 2x2 + x – 1 =

x – 1 (2x – 1)(x + 1) * 2(2x – 1)(x + 1) LCD

= 2(x – 1) LCD

Example D. Combine


4x – 2 – x – 1 2x2 + x – 1


x4x – 2 = x

2(2x – 1) * 2(2x – 1)(x + 1) LCD

= x(x + 1) = x2 + xLCD LCD

x – 1 2x2 + x – 1 =

x – 1 (2x – 1)(x + 1) * 2(2x – 1)(x + 1) LCD

= 2(x – 1) = 2x – 2 LCD LCD

Example D. Combine


4x – 2 – x – 1 2x2 + x – 1


x4x – 2 = x

2(2x – 1) * 2(2x – 1)(x + 1) LCD

= x(x + 1) = x2 + xLCD LCD

x – 1 2x2 + x – 1 =

x – 1 (2x – 1)(x + 1) * 2(2x – 1)(x + 1) LCD

= 2(x – 1) = 2x – 2 LCD LCD

Hence x4x – 2 – x – 1

2x2 + x – 1 =x2 + xLCD – 2x – 2

LCD


4x – 2 – x – 1 2x2 + x – 1

= x2 + xLCD – 2x – 2

LCD


= x2 + x – (2x – 2) LCD

x4x – 2 – x – 1

2x2 + x – 1

= x2 + xLCD – 2x – 2

LCD


= x2 + x – (2x – 2) LCD

x4x – 2 – x – 1

2x2 + x – 1

= x2 + xLCD – 2x – 2

LCD

= x2 + x – 2x + 2LCD


= x2 + x – (2x – 2) LCD

x4x – 2 – x – 1

2x2 + x – 1

= x2 + xLCD – 2x – 2

LCD

= x2 + x – 2x + 2LCD

= x2 – x + 22(2x – 1)(x + 1)


= x2 + x – (2x – 2) LCD

x4x – 2 – x – 1

2x2 + x – 1

= x2 + xLCD – 2x – 2

LCD

= x2 + x – 2x + 2LCD

= x2 – x + 22(2x – 1)(x + 1)


Multiplier Method


= x2 + x – (2x – 2) LCD

x4x – 2 – x – 1

2x2 + x – 1

= x2 + xLCD – 2x – 2

LCD

= x2 + x – 2x + 2LCD

= x2 – x + 22(2x – 1)(x + 1)


Multiplier Method The Multiplier Method is the same method used for fractional numbers.


= x2 + x – (2x – 2) LCD

x4x – 2 – x – 1

2x2 + x – 1

= x2 + xLCD – 2x – 2

LCD

= x2 + x – 2x + 2LCD

= x2 – x + 22(2x – 1)(x + 1)


Multiplier Method The Multiplier Method is the same method used for fractional numbers. We used the fact that x = (x * LCD) / LCD = x * 1


= x2 + x – (2x – 2) LCD

x4x – 2 – x – 1

2x2 + x – 1

= x2 + xLCD – 2x – 2

LCD

= x2 + x – 2x + 2LCD

= x2 – x + 22(2x – 1)(x + 1)


Multiplier Method The Multiplier Method is the same method used for fractional numbers. We used the fact that x = (x * LCD) / LCD = x * 1 then compute (x * LCD) using the Distributive Law.


= x2 + x – (2x – 2) LCD

x4x – 2 – x – 1

2x2 + x – 1

= x2 + xLCD – 2x – 2

LCD

= x2 + x – 2x + 2LCD

= x2 – x + 22(2x – 1)(x + 1)


Multiplier Method The Multiplier Method is the same method used for fractional numbers. We used the fact that x = (x * LCD) / LCD = x * 1 then compute (x * LCD) using the Distributive Law. Following is an example using this method.

Example E. Calculate

Addition and Subtraction of Rational Expressions712+ 5

8 – 49



8 – 49

The LCD is 72.



8 – 49

The LCD is 72. Multiply the problem by the LCD then put the result over the new denominator, the LCD.



8 – 49


712+ 5

8 – 49( )



8 – 49


712+ 5

8 – 49( )* 72 72



8 – 49


712+ 5

8 – 49( )* 72 72 Distribute the

multiplication


6


8 – 49


712+ 5


multiplication


6 9


8 – 49


712+ 5


multiplication


6 9 8


8 – 49


712+ 5


multiplication


6 9 8


8 – 49


712+ 5


multiplication= ( 42 + 45 – 32 )

72


6 9 8


8 – 49


712+ 5



72

5572=


6 9 8


8 – 49


712+ 5



72

5572=

Example F. Combine

34xy2

– 5x6y


6 9 8


8 – 49


712+ 5



72

5572=

Example F. Combine

34xy2

– 5x6y

The LCD is 12 xy2.


6 9 8


8 – 49


712+ 5



72

5572=

Example F. Combine

34xy2

– 5x6y

The LCD is 12 xy2. Multiply then divide the problem by the LCD.


6 9 8


8 – 49


712+ 5



72

5572=

Example F. Combine

34xy2

– 5x6y

The LCD is 12 xy2. Multiply then divide the problem by the LCD. 3

4xy2 – 5x

6y ( ) * 12xy2 / (12xy2)


6 9 8


8 – 49


712+ 5



72

5572=

Example F. Combine

34xy2

– 5x6y


4xy2 – 5x

6y ( ) * 12xy2 / (12xy2) Distribute


6 9 8


8 – 49


712+ 5



72

5572=

Example F. Combine

34xy2

– 5x6y


4xy2 – 5x

6y ( ) * 12xy2 / (12xy2) Distribute3


6 9 8


8 – 49


712+ 5



72

5572=

Example F. Combine

34xy2

– 5x6y


4xy2 – 5x

6y ( ) * 12xy2 / (12xy2) Distribute3 2xy


6 9 8


8 – 49


712+ 5



72

5572=

Example F. Combine

34xy2

– 5x6y


4xy2 – 5x

6y ( ) * 12xy2 / (12xy2) Distribute3 2xy

9 – 10x2y12xy2=


Example G. Combine

5x– 2

– 3x + 4

The LCD is (x – 2)(x + 4).


Example G. Combine

5x– 2

– 3x + 4

The LCD is (x – 2)(x + 4). Hence

5x– 2

– 3x + 4 ( ) (x – 2)(x + 4) / (x – 2)(x + 4)


Example G. Combine

5x– 2

– 3x + 4


5x– 2

– 3x + 4 ( ) (x – 2)(x + 4) / (x – 2)(x + 4)

(x + 4)


Example G. Combine

5x– 2

– 3x + 4


5x– 2

– 3x + 4 ( ) (x – 2)(x + 4) / (x – 2)(x + 4)

(x + 4) (x – 2)


Example G. Combine

5x– 2

– 3x + 4


= [5(x + 4) – 3(x – 2)] / (x – 2)(x + 4)

5x– 2

– 3x + 4 ( ) (x – 2)(x + 4) / (x – 2)(x + 4)

(x + 4) (x – 2)


Example G. Combine

5x– 2

– 3x + 4


= [5(x + 4) – 3(x – 2)] / (x – 2)(x + 4)

5x– 2

– 3x + 4 ( ) (x – 2)(x + 4) / (x – 2)(x + 4)

(x + 4) (x – 2)

2x + 26(x – 2)(x + 4)=

2(x + 13)(x – 2)(x + 4)or


Example G. Combine

5x– 2

– 3x + 4


= [5(x + 4) – 3(x – 2)] / (x – 2)(x + 4)

5x– 2

– 3x + 4 ( ) (x – 2)(x + 4) / (x – 2)(x + 4)

(x + 4) (x – 2)

2x + 26(x – 2)(x + 4)=

Example H. Combine

xx2 – 2x

– x – 1 x2 – 4

2(x + 13)(x – 2)(x + 4)or


Example G. Combine

5x– 2

– 3x + 4


= [5(x + 4) – 3(x – 2)] / (x – 2)(x + 4)

5x– 2

– 3x + 4 ( ) (x – 2)(x + 4) / (x – 2)(x + 4)

(x + 4) (x – 2)

2x + 26(x – 2)(x + 4)=

Example H. Combine

xx2 – 2x

– x – 1 x2 – 4

Factor each denominator to find the LCD.

2(x + 13)(x – 2)(x + 4)or


Example G. Combine

5x– 2

– 3x + 4


= [5(x + 4) – 3(x – 2)] / (x – 2)(x + 4)

5x– 2

– 3x + 4 ( ) (x – 2)(x + 4) / (x – 2)(x + 4)

(x + 4) (x – 2)

2x + 26(x – 2)(x + 4)=

Example H. Combine

xx2 – 2x

– x – 1 x2 – 4

Factor each denominator to find the LCD. x2 – 2x = x(x – 2)

2(x + 13)(x – 2)(x + 4)or


Example G. Combine

5x– 2

– 3x + 4


= [5(x + 4) – 3(x – 2)] / (x – 2)(x + 4)

5x– 2

– 3x + 4 ( ) (x – 2)(x + 4) / (x – 2)(x + 4)

(x + 4) (x – 2)

2x + 26(x – 2)(x + 4)=

Example H. Combine

xx2 – 2x

– x – 1 x2 – 4

Factor each denominator to find the LCD. x2 – 2x = x(x – 2)x2 – 4 = (x – 2)(x + 2)

2(x + 13)(x – 2)(x + 4)or


Example G. Combine

5x– 2

– 3x + 4


= [5(x + 4) – 3(x – 2)] / (x – 2)(x + 4)

5x– 2

– 3x + 4 ( ) (x – 2)(x + 4) / (x – 2)(x + 4)

(x + 4) (x – 2)

2x + 26(x – 2)(x + 4)=

Example H. Combine

xx2 – 2x

– x – 1 x2 – 4

Factor each denominator to find the LCD. x2 – 2x = x(x – 2)x2 – 4 = (x – 2)(x + 2)

Hence the LCD = x(x – 2)(x + 2).

2(x + 13)(x – 2)(x + 4)or

Multiply the problem by the LCD then put the result over the new denominator, the LCD.




xx(x – 2)

– (x – 1) (x – 2)(x + 2)

=

xx2 – 2x

– x – 1 x2 – 4

* x( x – 2)(x + 2)



xx(x – 2)

– (x – 1) (x – 2)(x + 2)

[ ] LCD=

xx2 – 2x

– x – 1 x2 – 4

* x( x – 2)(x + 2)(x + 2)



xx(x – 2)

– (x – 1) (x – 2)(x + 2)

[ ] LCD=

xx2 – 2x

– x – 1 x2 – 4

* x( x – 2)(x + 2)(x + 2) x



xx(x – 2)

– (x – 1) (x – 2)(x + 2)

[ ] LCD=

xx2 – 2x

– x – 1 x2 – 4

* x( x – 2)(x + 2)(x + 2) x



xx(x – 2)

– (x – 1) (x – 2)(x + 2)

[ ] LCD=

xx2 – 2x

– x – 1 x2 – 4

=

[x(x + 2) – x(x – 1)] LCD

* x( x – 2)(x + 2)(x + 2) x



xx(x – 2)

– (x – 1) (x – 2)(x + 2)

[ ] LCD=

xx2 – 2x

– x – 1 x2 – 4

=

[x(x + 2) – x(x – 1)] LCD

=

[x2 + 2x – x2 + x)] LCD

* x( x – 2)(x + 2)(x + 2) x



xx(x – 2)

– (x – 1) (x – 2)(x + 2)

[ ] LCD=

xx2 – 2x

– x – 1 x2 – 4

=

[x(x + 2) – x(x – 1)] LCD

=

[x2 + 2x – x2 + x)] LCD

=

3xx (x – 2)(x + 2)

* x( x – 2)(x + 2)(x + 2) x



xx(x – 2)

– (x – 1) (x – 2)(x + 2)

[ ] LCD=

xx2 – 2x

– x – 1 x2 – 4

=

[x(x + 2) – x(x – 1)] LCD

=

[x2 + 2x – x2 + x)] LCD

=

3xx (x – 2)(x + 2)=

3(x – 2)(x + 2)

Ex. A. Combine and simplify the answers.


xx – 2

– 2x – 2 1. 2x

x – 2 + 4 x – 22.

3xx + 3

+ 6x + 3 3. – 2x

x – 4 + 8x – 44.

x + 22x – 1

– 2x – 1 5. 2x + 5

x – 2 – 4 – 3x2 – x6.

x2 – 2x – 2

– xx – 2 7. 9x2

3x – 2 – 43x – 28.

Ex. B. Combine and simplify the answers.

312+ 5

6 – 239. 11

12+ 58 – 7

610. –56 + 3

8 – 3 11.

12. 65xy2

– x6y 13.

34xy2

– 5x6y 15. 7

12xy – 5x

8y3 16.

54xy

– 7x6y2 14.

34xy2

– 5y12x2 17.

–56 – 7

12+ 2

+ 1 – 7x9y2

4 – 3x

Ex. C. Combine and simplify the answers.


x2x – 4

– 23x – 6 18. 2x

3x + 9 – 42x + 619

. –32x + 1

+ 2x4x + 2 20. 2x – 3

x – 2 – 3x + 45 – 10x21.

3x + 16x – 4

– 2x + 32 – 3x22. –5x + 7

3x – 12 + 4x – 3–2x + 823.

xx – 2

– 2x – 3 24. 2x

3x + 1 + 4x – 625.

–32x + 1

+ 2x3x + 2 26. 2x – 3

x – 2 + 3x + 4x – 527.

3x + 1 + x + 3x2 – 428. x2 – 4x + 4

x – 4 – x + 5x2 – x – 229. x2 – 5x + 6

3x + 1 + 2x + 39 – x230. x2 – x – 6

3x – 4 – 2x + 5x2 + x – 631. x2 + 5x + 6

3x + 4 + 2x – 3–x2 – 2x + 332. x2 – x

5x – 4 – 3x – 51 – x233. x2 + 2x – 3

15 proportions and the multiplier method for solving rational equations

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