32
Add and Subtract Rational Expressions Math-3 Lesson 3-2
Add and Subtract Rational Expressions
Math-3 Lesson 3-2
Citizenship Responsibility
He who is not ready today will be even less so
tomorrow.
Ideas don't work unless we do.
If everyone sweeps in front of his own front door,
all the world would be clean.
What is popular is not always right. What is right is
not always popular.
22 yx
What are “like terms” ?
x4
2x
Like variables: yx 32
Like powers: 3x
same ‘base’ and same exponent.
Multiples of the same variable
Like radicals: 32 23
same ‘radicand’ and same index number.
Like fractions: ,
3
2,
3
4
4
3
same denominator.
Adding Fractions
We can add “like fractions”
3
412
3
7
Combine the numerator over a common denominator.
3
2
3
1
3
4
Identity Property of Multiplication
The numeral “one” multiplied by any number does
not change the “value” of the number (the product
will have an “equivalent” value).
Used for:
5
5*
3
2
3
3*
5
3
3
2
5
3
15
9
15
10
Obtaining Common Denominators:
Rationalizing Denominators:
3
2
3
3*
3
32
5*6
1
Least Common Multiple: The smallest number that
two other numbers divide evenly.
Used for:
7*6
1
30
1
42
1
Obtaining Least Common Denominators:
Both denominators have a common factor of 6.
Left denominator has an uncommon factor of 5
Multiply by “one” in the form of 42
1
5
5
7*6
1
5
5*
right denominator has an uncommon factor of 7
Multiply by “one” in the form of 30
1
7
7
5*6
1
7
7*
5*315
Find the LCM
20
3
15
2
5*420
60
5*4*3
LCM
LCM
48
1
60
1
5*6*260
4*2*68*648
240
5*4*6*2
LCM
LCM
Vocabulary
A rational number can be written as a ratio of integers.
1
3,
5
2
A rational expression can be written as a ratio of expressions.
Since we just covered polynomials we will be looking at
ratios of polynomial expressions.
)1( x
x
2)1(
3.6
x
x
)1)(1(.5
xx
x
What type of rational expressions can you combine
together using addition or subtraction?
“like rational expressions”
)1(.1
x
x
)1(
2.4
2
x
x
Which of these expressions are “like expressions”?
)1(
3.2
x
x
)5(
4.3
x
12
2.7
2
2
xx
x
Which of these expressions are “like expressions”?
)1(
4.8
2 x
Adding/Subtraction Rational Expressions
The easy problem:
)5(
3
)5(
4
x
x
x )5(
34
x
x
Combine the numerator over a common denominator.
Can you combine
4 and 3x ? Why not?
7
2
7
3
7
32
7
5
The easy problem:
Combine the numerator over a common denominator.
xx
x
xx
x
**2
)2*2(
**2
2
Your turn: add/subtract
22 2
4
2
2
x
x
x
x
22
42
x
xx
22
22
x
x
Can you do it this way?
2*2
)1(2
x
x
2
)1(
x
x
22 2
4
2
2
x
x
x
x
What property are we using?
Inverse Property of Multiplication.
xx
21
Why not? You CANNOT use the Inverse Property of
Multiplication on addends.
Your turn: add/subtract
22 2
4
2
2
x
x
x
x
22
42
x
xx
22
22
x
x
2*2
)1(2
x
x
2
)1(*
2
2
x
x
I will not allow you to simplify using the
Inverse Property of Multiplication until you
have factored it into two fractions.
2
)1(
x
x
Only then will you be able to see how the
Inverse Property of Multiplication changes the
rational expression into multiplication by one.
Your turn: add/subtract
2
)4(
2
)72(22
x
x
x
x
2
)4()72(2
x
xx
Which one is correct?
2
4722
x
xx
Subtract every term in the right side numerator! (Half of you
will make this mistake on the HW and on the Test).
2
4722
x
xx
2
32
x
x Can you factor this into two
fractions multiplied together?
No common denominator.
x
x
x
x
3
)2(
2
)12(
Multiply the left side fraction by one in the form of 3/3
Multiply the right side fraction by one in the form of 2/2
x
x
2
)12( 3
3 * +
x
x
3
)2( 2
2 *
x
x
x
x
6
)2(2
6
)12(3
x
xx
6
)2(2)12(3
x
xx
6
4236
x
x
6
78
Can you factor this into two fractions multiplied together?
Your turn: simplify
6
1
3
xx
6
)1(
3*
2
2
xx
6
)1(
6
2
xx
6
)1(2
xx
6
12
xx
6
1
x
Your turn: simplify
x
x
x
x 32
2
1
x
x
x
x )32(*
2
2
2
)1(
x
x
x
x
2
)32(2
2
)1(
x
x
x
x
2
)64(
2
)1(
x
xx
2
)64()1(
x
x
2
55
Can you factor this into two fractions multiplied together?
Your turn: simplify
5
1
2
13
x
x
x
x
x
x
2
2*
5
1
2
)13(*
5
5
x
x
x
x
10
2
10
)13(5
x
xx
10
2)13(5
x
xx
10
2515
x
x
10
513
Can you factor this into two fractions multiplied together?
3
3
245
122
xxx
What is the factored version of the left denominator?
What is the least common denominator?
(step by step)
)3(
3
)3)(8(
12
xxx
)3)(8( xx
Your Turn: add the two rational expressions
)3(
3
)3)(8(
12
xxx
)8(
)8(*
)3(
3
)3)(8(
12
x
x
xxx
)3)(8(
)8(312
xx
x
)3)(8(
24312
xx
x
)3)(8(
363
xx
x
)3)(8(
)12(3
xx
xCan you factor this into two
fractions multiplied together?
Your Turn: simplify
3
2
32
)1(2
xxx
x
3
2
)3)(1(
)1(
xxx
x
3
2
3
1
xx
3
3
x
65
21
Simplify the complex fraction.
Fraction: numerator ÷ denominator
6
5
2
1
Vocabulary:
Division: Division by a number is the same thing as…
Multiplication by the reciprocal of the number.
65
21
6
5
2
1
5
6*
2
1
5
3
Vocabulary
Complex Fraction is a fraction in the numerator
and a fraction in the denominator.
54
32
Simplifying complex fractions:
5
4
3
2
4
5*
3
2
2*3
5*
2
2
6
5
How do you divide fractions?
Multiply by reciprocal.
A more complicated Complex Fraction
4
5
x
4
3
x
4
3
4
5
xx
3
)4(*
)4(
5
x
x
3
5
Put binomials into parentheses!
3
5*
)4(
)4(
x
x
Your turn:
3
3
2
xx
x
18. Simplify the complex fraction.
3
32
x
x
x
3
3*
2
x
x
x
3
3*
2
x
x
x
)2(3
)3(
x
xx
Sum/difference in Numerator and Denominator
63
x
x
32
1. Combine the numerator fractions into one fraction.
x
32
= 1
6
3
x
x
32
= 3
18x
2. Combine the denominator fractions into one fraction.
x
x 32
3
18x
= 32
*3
18
x
xx
)32(3
)18(
x
xx
Your turn:
4
33
21
x
xx
Simplify the complex fraction.
Another Example: Electrical Circuits
1R
2R
321 RRRRtotal
3R
Voltage
Source
Parallel Resistors 321
1111
RRRR total
1R2R 3R
Voltage
Source
321
21
321
31
321
321
RRR
RR
RRR
RR
RRR
RR
RT
321
2131321
RRR
RRRRRR
RT
213132
321
RRRRRR
RRRRT
Real World Application
qp
f11
1
Camera
Lens Film
p f
q
pq
pqf
1
pq
pqf
simplify
9
2
3
42
x
x
x
2
3
3
2
xx
harder example: xx
x
x 333
722
Find the LCM of the two denominators.
1. Factor both denominators:
xx**3 )1(**3 xx
2. What factors are missing on the left/right sides?
Left side needs: (x + 1) Right side needs: x
What is the LCM of the denominators?
)1(***3 xxx )1(3 2 xx
Another example: xx
x
x 333
722
)1(***3 xxxLCM is:
Left side needs: (x + 1)
xx
x
33 2
23
7
x
x
x *
(x + 1)
(x + 1) *
Right side needs: x
)1(3)1(3
)1(72
2
2
xx
x
xx
x
)1(3 2 xx
)1(3
)1(72
2
xx
xx
)1(3
772
2
xx
xx
Can we simplify this?
