ma1041 unit 2: rational expressions & equations 2.1.1-2.1
TRANSCRIPT
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
.
Since we cannot divide by zero, we know that:
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
.
Since we cannot divide by zero, we know that:
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
.
Since we cannot divide by zero, we know that:
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
.
Since we cannot divide by zero, we know that:
14 0 0and14 0
14 14
0
f gf
f
E.g.: Find the restriction(s) on the variable(s) of 3
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
E.g.: Find the restriction(s) on the variable(s) of 2
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
Since we cannot divide by zero, we know that:
7 0and0
7 7 0 7
7
rr
r
r
E.g.: Find the restriction(s) on the variable(s) of 2
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
Since we cannot divide by zero, we know that:
8 0 1 0and
8 8 0 8 1 1 0 1
8 1
t t
t t
t t
E.g.: Find the restriction(s) on the variable(s) of 2
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
Since we cannot divide by zero, we know that:
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
E.g.: Simplify 2 2 3
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 ?
E.g.: Simplify 4 3 4
1 1
x
x x
4 3 44
1 1
x
x x
3
1
x
x
1x
43; 1x x
10
E.g.: Simplify 4 12 4
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.
Step 2: Write both numerators over a single
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 .
Step 2: Write both numerators over a single
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
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: Subtract the numerators. 2
4
Step 4: Reduce, if necessary 2 1
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.
Step 2: Write both numerators over a single
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
E.g.: Simplify 5 1 3 4
8 8
x x
x x
Step 1: Find the lowest common denominator. Lowest common denominator is 8x .
Step 2: Write both numerators over a single
common denominator. 5 1 3 4
8
x x
x
Step 3: Distribute the negative sign in front of the
parentheses.
5 1 3 4
8
x x
x
Step 4: Combine like terms. 2 5
8
x
x
Step 5: Reduce, if necessary 2 5
8
x
x
is already reduced.
Step 6: Find the restriction(s). 8x
Step 7: Write the answer with any restrictions 2 5; 8
8
xx
x
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 .
Step 2: Write both numerators over a single
common denominator. 10 1 8 2
4 3
m m
m
Step 3: Distribute the negative sign in front of the
parentheses.
10 1 8 2
4 3
m m
m
Step 4: Combine like terms. 12 9
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 .
Step 2: Write both numerators over a single
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 1: Find the lowest common denominator. Lowest common denominator is 14.
Step 2: Write equivalent fractions with the
common denominator.
15
14
3 2 6
7 2 14
Step 3: Write both numerators over a single
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
Step 4: Write both numerators over a single
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
Step 7: Reduce, if necessary
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 .
Step 3: Write both expressions with the common
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
Step 4: Write both numerators over a single
common denominator.
4 3 1
1 1
p
p p
Step 5: Expand the numerator, if necessary.
4 3 3
1 1
p
p p
Step 6: Combine like terms.
3 7
1 1
p
p p
Step 7: Reduce, if necessary
3 7
1 1
p
p p
is already reduced.
Step 8: Find the restrictions. 1p
Step 9: Write the answer with any restrictions
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
Step 2: Find the lowest common denominator. Lowest common denominator is 22x .
Step 3: Write both expressions with the common
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
Step 4: Write both numerators over a single
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
E.g.: Simplify 3 3 2
22 2 1x x
Step 1: Factor both denominators, if possible. Already factored
Step 2: Find the lowest common denominator. Lowest common denominator is 2x .
Step 3: Write both expressions with the common
denominator.
3
2
2 22 2 2 4
1 2 2 2
x
xx x
x x x
Step 4: Write both numerators over a single
common denominator.
3 2 4
2
x
x
Step 5: Expand the numerator, if necessary. Not necessary
Step 6: Combine like terms. 2 7
2
x
x
Step 7: Reduce, if necessary 2 7
2
x
x
is already reduced.
Step 8: Find the restrictions. 2x
Step 9: Write the answer with any restrictions 2 7; 2
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 .
Step 3: Write both expressions with the common
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
Step 4: Write both numerators over a single
common denominator.
2 1 2
1 1
x x
x x x
Step 5: Expand the numerator, if necessary.
2 2 2
1 1
x x
x x x
Step 6: Combine like terms.
2
1 1x x x
Step 7: Reduce, if necessary
2
1 1x x x is already reduced.
Step 8: Find the restrictions. 0, 1x
Step 9: Write the answer with any restrictions
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
Step 2: Find the lowest common denominator. Lowest common denominator is
7 10x x .
Step 3: Write both expressions with the common
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
Step 4: Write both numerators over a single
common denominator.
4 10 3 7
7 10
x x x
x x
Step 5: Expand the numerator, if necessary.
24 40 3 21
7 10
x x x
x x
Step 6: Combine like terms.
24 37 21
7 10
x x
x x
Step 7: Reduce, if necessary
24 37 21
7 10
x x
x x
is already reduced.
Step 8: Find the restrictions. 7,2,5,10x
Step 9: Write the answer with any restrictions
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 1: Find the lowest common denominator. Lowest common denominator is 14.
Step 2: Write equivalent fractions with the
common denominator.
15
14
3 2 6
7 2 14
Step 3: Write both numerators over a single
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 .
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
Step 4: Write both numerators over a single
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.
2 24
3 2
n
n n
Step 7: Reduce, if necessary
2 24
3 2
n
n n
is already reduced.
Step 8: Find the restrictions. 2,3n
Step 9: Write the answer with any restrictions
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 .
Step 3: Write both expressions with the common
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
Step 4: Write both numerators over a single
common denominator.
4 3 1
1 1
p
p p
Step 5: Expand the numerator, if necessary.
4 3 3
1 1
p
p p
Step 6: Combine like terms.
1 3
1 1
p
p p
Step 7: Reduce, if necessary
1 3
1 1
p
p p
is already reduced.
Step 8: Find the restrictions. 1p
Step 9: Write the answer with any restrictions
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 .
Step 3: Write both expressions with the common
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
Step 4: Write both numerators over a single
common denominator.
2 1 2
1 1
x x
x x x
Step 5: Expand the numerator, if necessary.
2 2 2
1 1
x x
x x x
24
Step 6: Combine like terms.
4 2
1 1
x
x x x
Step 7: Reduce, if necessary
4 2
1 1
x
x x x
is already reduced.
Step 8: Find the restrictions. 0, 1x
Step 9: Write the answer with any restrictions
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
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
Step 2: Find the lowest common denominator. Lowest common denominator is 7 10x x .
Step 3: Write both expressions with the common
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
Step 4: Write both numerators over a single
common denominator.
4 10 3 7
7 10
x x x
x x
Step 5: Expand the numerator, if necessary.
24 40 3 21
7 10
x x x
x x
Step 6: Combine like terms.
24 43 21
7 10
x x
x x
Step 7: Reduce, if necessary
24 43 21
7 10
x x
x x
is already reduced.
Step 8: Find the restrictions. 7,2,5,10x
Step 9: Write the answer with any restrictions
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
Step 1: Factor both the numerators and the
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
Step 2: Find the lowest common denominator. Lowest common denominator is
3 2 1x x x .
Step 3: Write both expressions with the common
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
Step 4: Write both numerators over a single
common denominator.
1 1 2 2
3 2 1
x x x x
x x x
Step 5: Expand the numerator, if necessary.
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
Step 6: Combine like terms.
4 5
3 2 1
x
x x x
Step 7: Reduce, if necessary
4 5
3 2 1
x
x x x
is already reduced.
Step 8: Find the restrictions. 3, 1,2x
Step 9: Write the answer with any restrictions
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