Download 6.2

Chapter 6
Rational
Expressions
and Equations
Copyright © 2015, 2011, 2007 Pearson Education, Inc.
Chapter 6-1
1
Chapter Sections
6.1 – The Domains of Rational Functions and
Multiplication and Division of Rational
Expressions
6.2 – Addition and Subtraction of Rational
Expressions
6.3 – Complex Fractions
6.4 – Solving Rational Equations
6.5 – Rational Equations: Applications and Problem
Solving
6.6 – Variation
Copyright © 2015, 2011, 2007 Pearson Education, Inc.
Chapter 6-2
2
§ 6.2
Addition and
Subtraction of Rational
Expressions
Copyright © 2015, 2011, 2007 Pearson Education, Inc.
Chapter 6-3
3
Add and Subtract Expressions with a Common
Denominator
To Add or Subtract Rational Expressions with a
Common Denominator
To add or subtract rational expressions, use the following
rules.
ADDITION
SUBTRACTION
a b ab
 
, c0
c c
c
a b a b
 
, c0
c c
c
To add or subtract rational expressions with a common
denominator,
1. Add or subtract the expressions using the rules given
above.
2. Simplify the expression if possible.
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Chapter 6-4
4
Example
3
x  4 3  ( x  4)
a)


x6 x6
x6
x 1

x6
Copyright © 2015, 2011, 2007 Pearson Education, Inc.
Chapter 6-5
5
Find the Least Common Denominator (LCD)
To Find the Least Common Denominator (LCD) of
Rational Expressions
1. Write each nonprime coefficient (other than 1) of monomials
that appear in denominators as a product of prime numbers.
2. Factor each denominator completely. Any factors that occur
more than once should be expressed as powers. For
example, (x + 5)(x + 5) should be expressed as (x + 5)2.
3. List all different factors (other than 1) that appear in any of
the denominators. When the same factor appears in more
than one denominator, write the factor with the highest
power that appears.
4. The least common denominator is the product of all the
factors found in step 3.
Copyright © 2015, 2011, 2007 Pearson Education, Inc.
Chapter 6-6
6
Find the Least Common Denominator (LCD)
Example Find the LCD of the expression.
Factor both denominators.
3
8x
3
8x
 2


2
2
2 x  4 x x  4 x  4 2 x( x  2) ( x  2)
The factors are 2, x, and x – 2. Multiply the factors
raised to the highest power that appears for each
factor.
LCD  2  x  ( x  2)  2 x( x  2)
2
Copyright © 2015, 2011, 2007 Pearson Education, Inc.
2
Chapter 6-7
7
Unlike Denominators
To Add or Subtract Rational Expressions with Different
Denominators
1. Determine the least common denominator (LCD).
2. Rewrite each fraction as an equivalent fraction with the
LCD. This is done by multiplying both the numerator and
denominator of each fraction by any factors needed to
obtain the LCD.
3. Leave the denominator in factored form, but multiply out
the numerator.
4. Add or subtract the numerators while maintaining the LCD.
5. When it is possible to reduce the fraction by factoring the
numerator, do so.
Copyright © 2015, 2011, 2007 Pearson Education, Inc.
Chapter 6-8
8
Unlike Denominators
2 9
Example Add 
x y
First determine the LCD. The LCD = xy. Now write
each fraction with the LCD. Do this by multiplying
both numerator and denominator of each fraction by
any factors needed to obtain the LCD.
2 9 y 2 9 x 2 y 9x
     

x y y x y x xy xy
continued
Copyright © 2015, 2011, 2007 Pearson Education, Inc.
Chapter 6-9
9
Unlike Denominators
Now, add the numerators while maintaining
the LCD.
2 y 9x 2 y  9x
9x  2 y
 
or
xy xy
xy
xy
2 9 9x  2 y
Therefore,  
x y
xy
Copyright © 2015, 2011, 2007 Pearson Education, Inc.
Chapter 6-10
10