MA1041 Unit 2: Rational Expressions & Equations 2.1.1-2.1 ...

1

MA1041

Unit 2: Rational Expressions & Equations

2.1.1-2.1.2 Restrictions & Simplifying Rational Expressions (2 classes)

Outcomes

1. Define and give an example of a rational expression.

2. Determine the restriction(s) on the variable(s) of a rational expression.

3. Simplify rational expressions by factoring.

nDef A rational expression is the quotient (divide) of two polynomials. It is a fraction that has a

polynomial in the numerator and a polynomial in the denominator and the denominator cannot equal 0

when values are substituted for the variable(s).

E.g.:

2

2 3 2 2

1 1 1 4 7, 2; , 2,3; , 1; , 3,0,1;

2 2 3 1 2 3 1

x x tx x r t

x x x r t t t x

Non e.g.: 3 4 4 1

, ,0 1

x

xw

If you look back at the examples of rational expressions, you should notice that most of rational

expressions with one or more variables in the denominator had restrictions on the variable(s). The

restrictions on the denominator occur because division by zero is undefined, so we are not permitted to

substitute certain values for the variable(s) in the denominator because they result in division by zero.

Finding the Restrictions on a Rational Expression

If the denominator (bottom) of the rational expression is factored, finding the restriction(s) on the

variable(s) becomes easier and can often be found by inspection (by looking).

E.g.: Find the restriction(s) on the variable(s) of 4 3x

x

.

Since we cannot divide by zero, we know that:

0x

E.g.: Find the restriction(s) on the variable(s) of 2

7

x

x

.


7 0

7 7 0 7

7

x

x

x

2

E.g.: Find the restriction(s) on the variable(s) of

8

1 3

t

t t

.


1 0 3 0and

1 1 0 1 3 3 0 3

1 3

t t

t t

t t

E.g.: Find the restriction(s) on the variable(s) of

2 2 8

2 9

r r

r r

.


2 09 0

and2 09 9 0 9

2 29

0

rr

rr

rr

E.g.: Find the restriction(s) on the variable(s) of 517 1

14

y

fg

.


14 0 0and14 0

14 14

0

f gf

f


2

4

6

q

q .

Note that 2 6q will always be 6 or greater no matter what value is substituted for q . This means there are

no restrictions on the variable q .

Complete #’s 1-14 Restrictions for Rational Expressions Worksheet

If the denominator (bottom) of the rational expression is NOT factored, we must use the factoring

methods used in Unit 1 before we can find the restrictions.

3


2

2 8

7

r r

r r

.

First we factor 2 7r r by taking out a common factor.

2 2

2

2 8 2 8

7 7

r r r r

r r r r


7 0and0

7 7 0 7

7

rr

r

r


4 8

7 8

t

t t

.

First we factor 2 7 8t t .

2 7 8

7; 8

#'s are 8 and 1

t t

s p

2

4 8 4 8

7 8 8 1

t t

t t t t


8 0 1 0and

8 8 0 8 1 1 0 1

8 1

t t

t t

t t


11 15

6 7 3

x

x x

.

First we factor 26 7 3x x .

26 7 3

7; 18

#'s are 2 and 9

x x

s p

2 2

11 15 11 15 11 15 11 15

6 7 3 2 3 1 3 3 1 3 1 2 36 2 9 3

x x x x

x x x x x x xx x x


4

3 1 0 2 3 0

3 1 1 0 1 2 3 3 0 3and

3 1 2 3

3 1 2 3

3 3 2 2

31

23

x x

x x

x x

x x

xx

Complete #’s 15-24 Restrictions for Rational Expressions Worksheet

Simplifying Rational Expressions

Simplifying rational expressions is very similar to simplifying numerical fractions. The idea is to factor

both the numerator and denominator and cancel like factors to get an equivalent simplified expression.

Don’t forget to pay attention to the restriction(s) on the variable(s). You have simplified a rational

expression when the numerator and the denominator have no common factors.

E.g.:

3

3 33 39

6 2 3

rr

r r r r

r

2 3 r 2

3 3; 0

2 2r

r r rr r

E.g.:

33 4

3 2

xx x

x x

4

3

x

x

4, 3, 2

22

xx

xx

Note that you determine the restrictions after you factor but before you cancel .

E.g.: 8 28 16

2

xx

x

2x

88, 2

1x

E.g.: 23 2 3 13 3

12 12

a aa a

a

12 1a

23 3

12

a

2

3

a

2

; 144

aa

E.g.: 1 11 11

1 1

xxx

x x

1x

11, 1

1x

E.g.:

1 51 5 1 55

5 5 5

tt tt

t t t

5t

1, 5t

5

E.g.: Simplify 236

6

x

x

2 636

6

xx

x

6

6

x

x

6 ; 6x x

E.g.: Simplify 2

2

10 8

24 30

x x

x x

2

2

2 5 4 210 8

24 30 6 4 5

x xx x

x x x x

x 5 4

6

x

3

x

5 4 5 4 5 4

3 4 5 3 1 5 44 5

x x x

x xx

1

3 5 4x

1 4; ; 0,

3 5x

E.g.: Simplify 2 3

2

14 21

35

a a

a

22 3

2

714 21

35

aa a

a

2

2 3

7

a

a

2 3; 0

55

aa

E.g.: Simplify 2 3

12

8 20

x

x x

2 3 2

44 3 4 312

8 20 4 2 5 4 2 5

x xx

x x x x x x x

3 x

4 x 3 2

; 0,2 5 52 5

xx xx x

E.g.: Find the error in the following “simplificaton” of 16 20

10

x

x

.

16 20 16 20

10

x x

x

2

10x16 2 18; 0x

The error occurs in canceling 20 and 10x x to get 2. The correct simplification is

216 20

10

x

x

8 10

2

x

8 10; 0

55

xx

xx

6

E.g.: Find the errors in the following “simplificaton” of 2

2 2

4 4

a

a

.

2 2

22 12 2

4 4 4 1

aa

a a

1a

4 2

1 1a a 2 1 2 2; 1a a a

There are 2 errors, one in the simplification

12 1 instead of

2 1a

a

and one in the restrictions.

The correct simplification is

2 2

22 12 2

4 4 4 1

aa

a a

1a

4 2

1 1a a 1 1

; 12 1 2 2

aa a

Complete Simplifying Rational Expressions Worksheet

7

2.1.3 Multiplying Rational Expressions (0.5 class)

Outcomes

1. Multiply Rational Expressions

Multiplying Rational Expressions

Multiplying rational expressions is similar to multiplying numerical fractions. However, you must watch

for the restriction(s) on the variable(s).

E.g.: Simplify 5 3

4 7

We multiply across the top and across the bottom to get

5 3 15

4 7 28

Sometimes, we can do some cancelling before we multiply.

E.g.: Simplify 15 7

14 3

15 7 15

14 3

5

142

7

1

31

5 1 5

2 1 2

Multiplying rational expressions is similar except that we have to determine restrictions on the variable.

E.g.: Simplify 27 9

3 14

x

x

2 2 2121 37 9 63

3 14 42 21 2

x xx x

x x x

3 x

21

x

2 x

3; 0

2

xx

E.g.: Simplify 4 5

156

x

4 5 15 4 5 1515

6 1 6

x x

5

4 5

1 6

x

2

5 4 5 20 25

2 2

x x

8

E.g.: Simplify 2

2

9 4

4 3

x x

x x x

We have to rewrite the expression so that everything is factored and cancel like factors. Don’t forget to

find the restrictions values before you cancel. You could multiply across the top and across the bottom

and then factor but this would greatly increase the difficulty of the factoring.

2

2

3 39 4

4 3

x xx x

x x x

4x x

4x

3x

3; 0,3,4

xx

x

E.g.: Simplify 2

2

2

4

x x

x x

2 2

2

2

4

x x x

x x

2

x

x

2

2

x

x

x; 0, 2

2

xx

x

E.g.: Simplify 3 3 2

2

12 4 8

3 6 5

x x x

x x

23 3 2 3

2

484 212 4 8 12

3 6 5 3 2 5

x xx x x x

x x x x

165x

4

2

x

x

155

x 2x

416; 2,0

5

xx


2 2 2

4 3 2 4

2 10 3 3 2

x x x x x

x x x x x x

22 2 3

2 2 2

3 1 2 1 24 3 2 4

2 10 3 3 2 2 5 2 3 2 1

3

x x x xx x x x x x

x x x x x x x x x x x x

x

1x

22 1 2

2 5 2

x x

x x x

3x

x

2 1x x

2 2

2

2 1 2 2 1 2 5; 3, 2, 1,0,

2 5 2 2 22 5 2

x x x xx

x x x x x

Complete Multiplying Rational Expressions Worksheet

9

2.1.3 Dividing Rational Expressions (0.5 class)

Outcomes

1. Divide Rational Expressions

Dividing Rational Expressions

Dividing rational expressions is similar to dividing numerical fractions. However, you must watch for

the restriction(s) on the variable(s).

E.g.: Simplify 5 3

4 7

First we multiply the first fraction by the reciprocal of the second fraction and then we multiply across

the top and across the bottom to get

5 3 5 7 5 7 35

4 7 4 3 4 3 12

Like multiplying numerical fractions, sometimes we can use cancellation after we multiply by the

reciprocal.

E.g.: Simplify 16 32

25 15

16 32 16 15 16

25 15 25 32

1

255

15

3

322

1 3 3

5 2 10

Dividing rational expressions is similar except that we have to determine the restrictions on the

variable(s) for the expression. This is a little bit trickier than with multiplying rational expressions.

E.g.: Simplify 2

2

4 2

3

t t

s s

2 2 2 2 2

2

4 2 4 4 2, 0, 0

3 3 2 6 3

t t t s t s sts t

s s s t st

Do you see why 0t ?


1 1

x

x x

4 3 44

1 1

x

x x

3

1

x

x

1x

43; 1x x

10


1 1

x

x x

44 12 4

1 1

x

x x

3

1

x

x

1x

43, 1x x

E.g.: Simplify 2

2 2

1

1

y y y

y y y

2

2 2

111 1

1 1 1 1

yy yy y y y

y y y y y y y

1y y

1y

1

1

y

y y

2

1, 1,0

yy

y

E.g.: Simplify 2

2 2

25 5

a a

a a

2

2 2

25 5

2 2

5 5 5

2 5

5 5 2

2

a a

a a

a a

a a a

a a

a a a

a

5 a

5

5

a

a

2a

1; 2, 5

5a

a

E.g.: Simplify 2 2

2 2

2 15 12

45 5 36

x x x x

x x x x

2 2

2 2

2 15 12

4 45 5 36

5 3 4 3

5 9 4 9

5 3 4 9

5 9 4 3

5

x x x x

x x x x

x x x x

x x x x

x x x x

x x x x

x

3x

5x 9x

4x

9x

4x 3x

1; 5, 4,3,9x

11

E.g.: Simplify 2 2

2 2

3 40 2 48

2 35 3 18

x x x x

x x x x

2 2

2 2

3 40 2 48

2 35 3 18

8 5 8 6

7 5 6 3

8 5 6 3

7 5 8 6

8

x x x x

x x x x

x x x x

x x x x

x x x x

x x x x

x

5x

7 5x x

6 3

8

x x

x

6

6 3; 8, 7 6,3,5,6

7 6

x

x xx

x x

Note that division and multiplication of rational expressions can be combined as in the example below.

E.g.: Simplify 3 12 12 2 6

4 3 4 4

x x

x x x

3 12 12 2 6

4 3 4 4

3 4 2 312

4 3 4 4

3 4 2 33 4

4 12 4

3

x x

x x x

x x

x x x

x xx

x x

4x

4x

3 4

12

x

2

2

3

4

3 4 3 4; 4,

2 4 3

x

x

x xx

x

Complete Dividing Rational Expressions Worksheet

12

2.1.4 Adding & Subtracting Rational Expressions with the Same Denominator (1 class)

Outcomes

1. Add & Subtract Rational Expressions with the Same Denominator

Adding Rational Expressions with Like Denominators

Adding rational expressions is similar to adding numerical fractions. As with numerical fractions,

rational expressions can only be added when they have identical denominators.

E.g.: Simplify 5 3

4 4

Step 1: Find the lowest common denominator. Lowest common denominator is 4.

Step 2: Write both numerators over a single

common denominator.

5 3

4

Step 3: Add the numerators. 8

4 Step 4: Reduce, if necessary 8

24

However, adding rational expressions is different than adding numerical fractions in that we have to

determine the restriction(s) on the variable(s).

E.g.: Simplify 1m m

n n

Step 1: Find the lowest common denominator. Lowest common denominator is n.


common denominator.

1m m

n

Step 3: Combine like terms. 2 1m

n

Step 4: Reduce, if necessary 2 1m

n

is already reduced.

Step 5: Find the restriction(s). 0n

Step 6: Write the answer with any restrictions 2 1; 0

mn

n

13

E.g.: Simplify 10 1 8 2

4 3 4 3

m m

m m

Step 1: Find the lowest common denominator. Lowest common denominator is 4 3m .


common denominator.

10 1 8 2

4 3

m m

m

Step 3: Combine like terms. 8 7

4 3

m

m

Step 4: Reduce, if necessary 8 7

4 3

m

m

is already reduced.

Step 5: Find the restriction(s). 3

4m

Step 6: Write the answer with any restrictions 8 7 3;

4 3 4

mm

m

Complete #’s 1-15 Adding & Subtracting Rational Expressions with Like Denominators

Worksheet

Subtracting Rational Expressions with Like Denominators

Subtracting rational expressions is similar to subtracting numerical fractions. As with numerical

fractions, rational expressions can only be subtracted when they have identical denominators.

E.g.: Simplify 5 3

4 4



common denominator.

5 3

4

Step 3: Subtract the numerators. 2

4


4 2

However, subtracting rational expressions is different than subtracting numerical fractions in that we

have to determine the restriction(s) on the variable(s).

14

E.g.: Simplify 1m m

n n

Step 1: Find the lowest common denominator. Lowest common denominator is n.


common denominator. 1m m

n

Step 3: Distribute the negative sign in front of the

parentheses.

1m m

n

Step 4: Combine like terms. 1

n

Step 5: Reduce, if necessary 1

n

is already reduced.

Step 6: Find the restriction(s). 0n

Step 7: Write the answer with any restrictions 1; 0n

n


8 8

x x

x x

Step 1: Find the lowest common denominator. Lowest common denominator is 8x .


common denominator. 5 1 3 4

8

x x

x


parentheses.

5 1 3 4

8

x x

x


8

x

x


8

x

x

is already reduced.

Step 6: Find the restriction(s). 8x


8

xx

x


4 3 4 3

m m

m m

Step 1: Find the lowest common denominator. Lowest common denominator is 4 3m .


common denominator. 10 1 8 2

4 3

m m

m


parentheses.

10 1 8 2

4 3

m m

m


4 3

m

m

15

Step 5: Reduce, if necessary 3 4 33 4 312 9

4 3 4 3

mmm

m m

4 3m3

Step 6: Find the restriction(s). 3

4m

Step 7: Write the answer with any restrictions 33;

4m

E.g.: Simplify 2 2

3 6

3 10 3 10

x

x x x x

Step 1: Find the lowest common denominator. Lowest common denominator is 2 3 10x x .


common denominator. 2

3 6

3 10

x

x x

Step 3: Reduce, if necessary

2

3 23 23 6

3 10 5 2

xxx

x x x x

5 2x x

3

5x

Step 4: Find the restriction(s). 5,2x

Step 5: Write the answer with any restrictions 3, 5,2

5x

x

Complete #’s 16-40 Adding & Subtracting Rational Expressions with Like Denominators

Worksheet

16

2.1.5 Adding & Subtracting Rational Expressions with the Different Denominators (2 classes)

Outcomes

1. Add & subtract rational expressions with the different denominators.

Adding Rational Expressions with Different Denominators

Adding rational expressions with different denominators is similar to adding numerical fractions with

different denominators. We have to find a common denominator for each rational expression and then

add as we would for rational expressions with like denominators.

E.g.: Simplify 15 3

14 7


Step 2: Write equivalent fractions with the

common denominator.

15

14

3 2 6

7 2 14


common denominator.

15 6

14

Step 4: Add the numerators. 21

14 Step 5: Reduce, if necessary 21 7 3

14 7 2

E.g.: Simplify 6 4

3 2n n

Step 1: Factor both denominators. Both denominators cannot be factored.

Step 2: Find the lowest common denominator. Lowest common denominator is 3 2n n .

Step 3: Write both expressions with the common

denominator.

2 6 26

3 2 3 2

3 4 34

2 3 3 2

n n

n n n n

n n

n n n n


common denominator.

6 2 4 3

3 2

n n

n n

Step 5: Expand the numerator, if necessary.

6 12 4 12

3 2

n n

n n

Step 6: Combine like terms.

10

3 2

n

n n

17


10

3 2

n

n n is already reduced.

Step 8: Find the restrictions. 2,3n

Step 9: Write the answer with any restrictions

10

; 2,33 2

nn

n n

E.g.: Simplify 2

4 3

1 1p p

Step 1: Factor both denominators, if possible. 2 22 1 1 1 1

1 1

p p p p

p p

Step 2: Find the lowest common denominator. Lowest common denominator is 1 1p p .


denominator.

2

4 4

1 1 1

3 13 3 1

1 1 1 1 1

p p p

pp

p p p p p


common denominator.

4 3 1

1 1

p

p p


4 3 3

1 1

p

p p


3 7

1 1

p

p p


3 7

1 1

p

p p

is already reduced.

Step 8: Find the restrictions. 1p


3 7

; 11 1

pp

p p

18

E.g.: Simplify 2

2 3 1

2x x x

Step 1: Factor all denominators, if possible. All denominators are factored



denominator. 2

2 2

2

2 2 4

2 2

3 2 6

2 2

1

2 2

x x

x x x

x x

x x

x x x


common denominator. 2

4 6

2

x x

x

Step 6: Combine like terms. 2

5 6

2

x

x

Step 7: Reduce, if necessary 2

5 6

2

x

x

is already reduced.

Step 8: Find the restrictions. 0x

Step 9: Write the answer with any restrictions 2

5 6; 0

2

xx

x


22 2 1x x

Step 1: Factor both denominators, if possible. Already factored



denominator.

3

2

2 22 2 2 4

1 2 2 2

x

xx x

x x x


common denominator.

3 2 4

2

x

x

Step 5: Expand the numerator, if necessary. Not necessary


2

x

x


2

x

x

is already reduced.

Step 8: Find the restrictions. 2x


2

xx

x

19

E.g.: Simplify 2 2

3 2 32

5 20 25 1 5 20 25r r r r

2 2

2 2 2

2 3 2 3 2 3

1 5 20 25 1 1 5 5 15 4 5

5 5 1

5 5 1 10 5 1 10 5 1 32 3 3

1 5 5 1 5 5 1 5 5 1 5 5 1 5 5 1

10 4 5 3 10 40 50 3 10 40 47; 5,1

5 5 1 5 5 1 5 5 1

r r r rr r

LCD r r

r r r r r r

r r r r r r r r r r

r r r r r rr

r r r r r r

E.g.: Simplify 2 2

4 4 33

16 16 1

n n nn

n n

2

2

3 3

4 3 4 3

16 1 4 4 1

4 4

4 3 164 4 4 3 4 44 3

4 4 1 4 4 4 4 4 4

4 3 48 3 44; 4

4 4 4 4

n n n n

n n n

LCD n n

n n nn n n n n nn n

n n n n n n n n

n n n n nn

n n n n

20

E.g.: Simplify 2 22 3 5 4

x x

x x x x

2 2

2 2

2 2

2 3 5 4 3 1 1 4

3 1 4

4 3

3 1 4 1 4 3

4 3

3 1 4 1 4 3

4 34 3

3 1 4 3 1 4

4 3; 3,1, 4

3 1 4 3 1 4

x x x x

x x x x x x x x

LCD x x x

x x x x

x x x x x x

x x x x

x x x x x x

x x x xx x x x

x x x x x x

x x x x xx

x x x x x x

E.g.: Simplify 2 2

2 2

1x x x

Step 1: Factor both denominators.

2

2 22

1

1 1 1 1

x x x x

x x x x

Step 2: Find the lowest common denominator. Lowest common denominator is

1 1x x x .


denominator.

2

2

1 2 12 2

1 1 1 1

2 2 2

1 1 1 1 1

x x

x x x x x x x x

x x

x x x x x x x


common denominator.

2 1 2

1 1

x x

x x x


2 2 2

1 1

x x

x x x


2

1 1x x x


2

1 1x x x is already reduced.

Step 8: Find the restrictions. 0, 1x


2

; 0, 11 1

xx x x

21

E.g.: Simplify 2

2 2

4 20 3 6

2 35 12 20

x x x

x x x x

Step 1: Factor both the numerators and the

denominators and cancel any like factors. 2

2

4 54 20

2 35

x xx x

x x

7 5x x

2

4

7

3 23 6

12 20

x

x

xx

x x

10 2x x

3

10x


7 10x x .


denominator.

10 4 104 4

7 7 10 7 10

7 3 73 3

10 10 7 7 10

x x xx x

x x x x x

x x

x x x x x


common denominator.

4 10 3 7

7 10

x x x

x x


24 40 3 21

7 10

x x x

x x


24 37 21

7 10

x x

x x


24 37 21

7 10

x x

x x

is already reduced.

Step 8: Find the restrictions. 7,2,5,10x


24 37 21; 7,2,5,10

7 10

x xx

x x

Complete Adding Rational Expressions with Unlike Denominators Worksheet

Subtracting Rational Expressions with Different Denominators

Subtracting rational expressions with different denominators is similar to subtracting numerical fractions

with different denominators. We have to find a common denominator for each rational expression and

then subtract as we would for rational expressions with like denominators.

22

E.g.: Simplify 15 3

14 7


Step 2: Write equivalent fractions with the

common denominator.

15

14

3 2 6

7 2 14


common denominator.

15 6

14

Step 4: Subtract the numerators. 9

14 Step 5: Reduce, if necessary 9

14is already reduced.

E.g.: Simplify

6 4

3 2n n

Step 1: Factor both denominators. The denominators cannot be factored.

Step 2: Find the lowest common denominator. Lowest common denominator is 3 2n n .


denominator.

2 6 26

3 2 3 2

3 4 34

2 3 3 2

n n

n n n n

n n

n n n n


common denominator.

6 2 4 3

3 2

n n

n n


6 12 4 12

3 2

n n

n n


2 24

3 2

n

n n


2 24

3 2

n

n n

is already reduced.

Step 8: Find the restrictions. 2,3n


2 24

; 2,33 2

nn

n n

23

E.g.: Simplify 2

4 3

1 1p p

Step 1: Factor both denominators. 2 22 1 1 1 1

1 1

p p p p

p p

Step 2: Find the lowest common denominator. Lowest common denominator is 1 1p p .


denominator.

2

4 4

1 1 1

3 13 3 1

1 1 1 1 1

p p p

pp

p p p p p


common denominator.

4 3 1

1 1

p

p p


4 3 3

1 1

p

p p


1 3

1 1

p

p p


1 3

1 1

p

p p

is already reduced.

Step 8: Find the restrictions. 1p


1 3

; 11 1

pp

p p

E.g.: Simplify 2 2

2 2

1x x x

Step 1: Factor both denominators.

2

2 22

1

1 1 1 1

x x x x

x x x x

Step 2: Find the lowest common denominator. Lowest common denominator is 1 1x x x .


denominator.

2

2

1 2 12 2

1 1 1 1

2 2 2

1 1 1 1 1

x x

x x x x x x x x

x x

x x x x x x x


common denominator.

2 1 2

1 1

x x

x x x


2 2 2

1 1

x x

x x x

24


4 2

1 1

x

x x x


4 2

1 1

x

x x x

is already reduced.

Step 8: Find the restrictions. 0, 1x


4 2

; 0, 11 1

xx

x x x

E.g.: Simplify 2

2 2

4 20 3 6

2 35 12 20

x x x

x x x x


denominators and cancel any like factors. 2

2

4 54 20

2 35

x xx x

x x

7 5x x

2

4

7

3 23 6

12 20

x

x

xx

x x

10 2x x

3

10x

Step 2: Find the lowest common denominator. Lowest common denominator is 7 10x x .


denominator.

10 4 104 4

7 7 10 7 10

7 3 73 3

10 10 7 7 10

x x xx x

x x x x x

x x

x x x x x


common denominator.

4 10 3 7

7 10

x x x

x x


24 40 3 21

7 10

x x x

x x


24 43 21

7 10

x x

x x


24 43 21

7 10

x x

x x

is already reduced.

Step 8: Find the restrictions. 7,2,5,10x


24 43 21; 7,2,5,10

7 10

x xx

x x

25

E.g.: Simplify 2 2

1 2

6 4 3

x x

x x x x


denominators and cancel any like factors.

2

2

1 1

6 3 2

2 2

3 1 4 3

x x

x x x x

x x

x x x x


3 2 1x x x .


denominator.

1 11 1

3 2 1 3 2 1

2 22 2

3 1 2 3 1 2

x xx x

x x x x x x

x xx x

x x x x x x


common denominator.

1 1 2 2

3 2 1

x x x x

x x x


2 2 2 21 4 4 1 4 4

3 2 1 3 2 1

x x x x x x

x x x x x x


4 5

3 2 1

x

x x x


4 5

3 2 1

x

x x x

is already reduced.

Step 8: Find the restrictions. 3, 1,2x


4 5

; 3, 1,23 2 1

xx

x x x

26

E.g.: Find the error in the “simplification” below and write the correct simplification.

3 2 2 4 3 2 2 4 6

; 22 2 2 2 2 2 2 2

x x x x xx

x x x x x x x x

The error occurs in the subtraction of 2 4x . The correct simplification is

3 2 2 4

2 2 2 2

3 2 2 4

2 2

3 2 2 4

2 2

2

x x

x x x x

x x

x x

x x

x x

x

2x 2

1; 2

2

x

xx

Complete Subtracting Rational Expressions with Unlike Denominators Worksheet

27

2.2 Complex Rational Expressions

Outcomes

1. Define and give an example of a complex rational expression.

2. Simplify complex rational expressions.

nDef A complex rational expression is a rational expression that contains fractions in the numerator

and fractions in the denominator.

E.g.:

2

2

2

2

4 11

4 ,16

120

x

x x x

x x

xx x

Complex rational expressions can be simplified by finding a common denominator for both fractions

and then multiplying by the reciprocal of the denominator (Method 1) or by multiplying the numerator

and the denominator by the LCD of all the rational expressions (Method 2).

E.g.: Simplify

42

4

y

yy

Method 1:

Finding the common denominator for the numerator and also for the denominator gives

2 2

4 2 4 2 42

2 4

4 4 4

y y

yy y y y

y y yyy y y y

y

2 2

2 22 4

4 4

yy

y y

2 2y y

2, 0, 2

2y

y

Method 2:

2 2

4 2 42

1

4 4

1

LCD of all denominators

2 4 2 42 21 1 2 41

4 44

11 1

y y

yy

y y

y

y y yyy yy

y y yy y

yy

2 2y y

2; 0, 2

2y

y

28

E.g.: Simplify

2

3 3

2 33 1

2 3

m m

m m

Method 1:

2 22

2 22 2 2

2

2

3 2 3 3 6 9 33 2 3 33 32 3 2 32 3 2 32 3

3 1 3 2 3 6 93 2 3 1

2 3 2 32 3 2 3 2 3

3 9

3 32 3

6 9

2 3

m m m mm m

m m m mm m m mm m

m m m mm m

m m m mm m m m m m

m

mm m

m m

m m

m 2 3m

2m

2 3

m

m

3 3m m

3 3, 3, ,0

3 2

mm

m

Method 2:

2

2

2 22 22

2 2 22 2

2 22

2

3 3

2 33 1

2 3

LCD of all denominators 2 3

2 3 2 33 3 3 2 3 3 2 32 33 31 2 3 12 3 2 31

3 1 2 3 3 2 3 2 32 3 2 33 12 3 1 2 31 2 3 1

3

m m

m m

m m

m m m m m m m mm mm mm m m m

m m m m m mm m m m

m m m mm m

m

2 3

m

m

m

23 2 3m m

2 3m

23 m 2

2 3m

m

2 2 3m m

2 3m

2 2 2 2

2 2 2 2

3 2 3 3 3 36 9 3 3 9

3 2 3 6 9 6 9 6 9

3 3

m m m m mm m m m m

m m m m m m m m

m m

3 3m m

3 3, 3, ,0

3 2

mm

m

29

E.g.: Simplify

3 2

4 5

9 2

5

s

s

s

s

Method 1:

3 2

3 24 5 3 2 5

9 2 4 5 9 2

5

s

ss s s

s s s

s

4 5s

5s

9 2s

3 1; 5,2

36 12s

Method 2:

LCD of all denominators 4 5

3 2 43 2 3 2 4 5 3 2 4 5

4 5 4 5 1 4 5

9 2 9 2 4 5 9 2 4 5

5 5 1 5

s

ss s s s s

s s s

s s s s s

s s s

5s

4 5s

9 2 4 5s s

5s

3 2 3 1, 5,2

36 2 36 12

ss

s

Complete Simplifying Complex Expressions Worksheet

2.3 Rational Equations

Outcomes:

1. Define and give an example of a rational equation.

2. Solve rational equations algebraically.

3. Identify the restriction(s) on the variable(s) in a rational equation.

4. Identify extraneous roots when solving a rational equation.

nDef A rational equation is an equation that contains one or more rational expressions.

E.g.: 1

3, 8, 13 3 30 40

d d t t

x v v

We can often solve rational equations by multiplying each term of the equation by the LCD of all the

denominators in the equation. We have to determine the restrictions on the variables in the rational

equations to identify any extraneous roots.

nDef :An extraneous root is a solution to the equation that is not permissible in the original equation.

30

E.g.: Solve 6

4, 02

tt

t

The LCD of the denominators is 2t , so we will multiply each term by 2t to get rid of the fractions.

2

2

62 2 2 4

2

12 8

8 12 0

6 2 0

ZPP

6 0 or 2 0

6 or 2

tt t t

t

t t

t t

t t

t t

t t

Since the only restriction is 0t , we know that both 2 and 6 are solutions.

E.g.: Solve 3 1 1

, 2,02 5

xx x x

The LCD of the denominators is 5 2x x , so we will multiply each term by 5 2x x to get rid of the

fractions.

3 1 15 2 5 2 5 2

2 5

5 3 5 2 1 2

15 5 10 2

10 10 2

9 12

4

3

x x x x x xx x x

x x x

x x x

x x

x

x

Since the only restrictions are 2,0x , we know that 4

3 is a solution.

31

E.g.: Solve 10 4 5

, 0,22 2

xx x x x

The LCD of the denominators is 2x x , so we will multiply each term by 2x x to get rid of the

fractions.

10 4 52 2 2

2 2

10 4 2 5

10 4 8 5

2 4 5

2

x x x x x xx x x x

x x

x x

x x

x

But since the restrictions are 0,2x we know that 2x cannot be a solution so we write . There is

no solution to this rational equation as 2x is an extraneous root.

E.g.: Solve 2

9 4 18, 3,6

3 6 9 18y

y y y y

First we have to factor the denominators.

9 4 18

3 6 3 6y y y y

The LCD of the denominators is 3 6y y , so we will multiply each term by 3 6y y to get

rid of the fractions.

9 4 183 6 3 6 3 6

3 6 3 6

9 6 4 3 18

9 54 4 12 18

5 42 18

5 60

12

y y y y y yy y y y

y y

y y

y

y

y

Since the only restrictions are 3,6y , we know that 12 is a solution.

32

E.g.: Solve 2

3 5 25, 2,3

2 3 6

xx

x x x x

First we have to factor the denominators.

3 5 25

2 3 2 3

x

x x x x

The LCD of the denominators is 2 3x x , so we will multiply each term by 2 3x x to get

rid of the fractions.

2

2

2

3 5 252 3 2 3 2 3

2 3 2 3

3 3 5 2 25

3 9 5 10 25

3 14 15 0

3 9 5 15 0

3 3 5 3 0

3 3 5 0

ZPP

3 0 or 3 5 0

3

xx x x x x x

x x x x

x x x

x x x

x x

x x x

x x x

x x

x x

x

5

or 3

x

Since the restrictions are 2,3x , we know that 3x is an extraneous root and that 5

3 is the only

solution.

Complete Solving Rational Equations Worksheet

MA1041 Unit 2: Rational Expressions & Equations 2.1.1-2.1 ...

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