Section 7.3
Adding and
Subtracting
Rational
Expressions
with the Same
Denominator
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Adding Rational Expressions
Now, we will practice adding and subtracting rational expressions. Remember that when adding or subtracting fractions, it is necessary to rewrite the fractions as fractions having the same denominator, which is called the common denominator for the fractions being combined. In this section, we will be combining fractions under addition or subtraction that already share the same denominator.

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Adding Rational Expressions
Adding Rational Expressions With
Common Denominators
If P and Q are rational expressions, then
R
R
P Q P Q
 
R R
R
To add rational expressions with the same denominator, add numerators and place the sum over the common denominator. If possible, simplify the result.

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Adding Rational Expressions
Add: 3x  1  5 x  6

EXAMPLE

7

3x 1 5 x  6

7
7
3x  1  5 x  6

7
8x  5

7

7

Add numerators. Place the sum over the common denominator.

Combine like terms. Factor and simplify if possible. This does not simplify.

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Adding Rational Expressions
Add: 3x  1  5 x  6

EXAMPLE

7

3x 1 5 x  6

7
7
3x  1  5 x  6

7
8x  5

7

7

Add numerators. Place the sum over the common denominator.

Combine like terms. Factor and simplify if possible. This does not simplify.

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Adding Rational Expressions
EXAMPLE

x 2 1 5x  7

x 1 x 1 x 2 1  5x  7

x 1 x 2  5x  6

x 1
( x  1)( x  6)

x 1
x  6

Add:

x 2 1 5x  7

x 1 x 1

Add numerators. Place the sum over the common denominator.

Combine like terms.

Factor and simplify.

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Adding Rational Expressions
EXAMPLE

x 2 1 5x  7

x 1 x 1 x 2 1  5x  7

x 1 x 2  5x  6

x 1
( x  1)( x  6)

x 1
x  6

Add:

x 2 1 5x  7

x 1 x 1

Add numerators. Place the sum over the common denominator.

Combine like terms.

Factor and simplify.

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Adding Rational Expressions
EXAMPLE

x2  2x x2  x
 2
.
Add: 2 x  3x x  3x
SOLUTION

x2  2x x2  x
 2
2
x  3x x  3x x2  2x  x2  x

x 2  3x
2x2  x
 2 x  3x x 2 x  1

x  x  3

This is the original expression.
Add numerators. Place this sum over the common denominator. Combine like terms.
Factor.

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Adding Rational Expressions
CONTINUED

x 2 x  1

x  x  3
2x  1

x 3

Factor and simplify by dividing out the common factor, x.
Simplify.

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Objective #1: Example

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Objective #1: Example

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Objective #1: Example

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Objective #1: Example

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Subtracting Rational Expressions
Subtracting Rational Expressions with
Common Denominators
If P and Q are rational expressions, then
R

R

P Q P Q
 
.
R R
R

To subtract rational expressions with the same denominator, subtract numerators and place the difference over the common denominator. If possible, factor and simplify the result.

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Subtracting Rational Expressions
EXAMPLE
2

x  1 5 x  13

x2 x2 x 2  1  (5 x  13)

x2 x 2  1  5 x  13

x2 x 2  5 x  14

x2
( x  7)( x  2)

x2
x  7

Subtract:

x 2  1 5 x  13

x2 x2 Subtract numerators. Place the difference over the common denominator. Remove