8.5 – add and subtract rational expressions. when you add or subtract fractions, you must have a...

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5 – Add and Subtract Rational Expressio

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Page 1: 8.5 – Add and Subtract Rational Expressions. When you add or subtract fractions, you must have a common denominator. When you subtract, make sure to distribute

8.5 – Add and Subtract Rational Expressions

Page 2: 8.5 – Add and Subtract Rational Expressions. When you add or subtract fractions, you must have a common denominator. When you subtract, make sure to distribute

When you add or subtract fractions, you must have a common denominator.

a b a b

c c c

a b a b

c c c

When you subtract, make sure to distribute the negative!

Page 3: 8.5 – Add and Subtract Rational Expressions. When you add or subtract fractions, you must have a common denominator. When you subtract, make sure to distribute

1. Perform the indicated operation and simplify.

a. 7 5

12 12x x =

2

12x6

1

6x=

1

Page 4: 8.5 – Add and Subtract Rational Expressions. When you add or subtract fractions, you must have a common denominator. When you subtract, make sure to distribute

1. Perform the indicated operation and simplify.

b. 2

2 2

6 5

3 10 3 10

x x

x x x x

=

2

2

6 5

3 10

x x

x x

5 1

5 2

x x

x x

1

2

x

x

=

Page 5: 8.5 – Add and Subtract Rational Expressions. When you add or subtract fractions, you must have a common denominator. When you subtract, make sure to distribute

1. Perform the indicated operation and simplify.

c.

2

2 2

2 2 4

1 1

x x

x x

=

2

2

2 2 4

1

x x

x

2

2

2( 2)

1

x x

x

2( 1)( 2)

1 1

x x

x x

2 2

1

x

x

=

Page 6: 8.5 – Add and Subtract Rational Expressions. When you add or subtract fractions, you must have a common denominator. When you subtract, make sure to distribute

What if you have unlike denominators?

Find the least common denominator!

Page 7: 8.5 – Add and Subtract Rational Expressions. When you add or subtract fractions, you must have a common denominator. When you subtract, make sure to distribute

2. Find the least common multiple.

35x and 210 15x x5x(2x – 3)

5x3(2x – 3)

a.

Page 8: 8.5 – Add and Subtract Rational Expressions. When you add or subtract fractions, you must have a common denominator. When you subtract, make sure to distribute

2. Find the least common multiple.

24 16x and 26 24 24x x 4(x2 – 4) 6(x2 – 4x + 4)

6(x – 2)24(x – 2)(x + 2)

12(x – 2)2(x + 2)

b.

Page 9: 8.5 – Add and Subtract Rational Expressions. When you add or subtract fractions, you must have a common denominator. When you subtract, make sure to distribute

3. Perform the indicated operation.

3 1

4 6x 9 2

12 12

x

x x =

2x32x3 9 2

12

x

x

2 9

12

x

x

Page 10: 8.5 – Add and Subtract Rational Expressions. When you add or subtract fractions, you must have a common denominator. When you subtract, make sure to distribute

4. Perform the indicated operation.

4 2

4 6x x

x + 4

x + 4

x + 6

x + 6 =

4 6 2 4

4 6

x x

x x

4 24 2 8

4 6

x x

x x

6 32

4 6

x

x x

2 3 16

4 6

x

x x

Page 11: 8.5 – Add and Subtract Rational Expressions. When you add or subtract fractions, you must have a common denominator. When you subtract, make sure to distribute

5. Perform the indicated operation.

2

2 2 1

2 2 4 3

x x

x x x

=

2x – 3

2x – 3

2 6 4 2

2 1 3

x x x

x x

4

2 3

x

x

2 2 1

2( 1) 3 1

x x

x x x

2 3 2 2 1

2 1 3

x x x

x x

2 3 4

2 1 3

x x

x x

1 4

2 1 3

x x

x x

Page 12: 8.5 – Add and Subtract Rational Expressions. When you add or subtract fractions, you must have a common denominator. When you subtract, make sure to distribute

6. Perform the indicated operation.

2

3 1 3

5 325

x x

x xx

=

3 2 3

( 5)( 5) 5 3

x

x x x x

3 3 2 3 5 3 5 5

3 ( 5)( 5)

x x x x x x

x x x

(x+3)(x+3)(x+5) (x-5)(x+5)

2 2 26 9 2 8 15 3 25

3 ( 5)( 5)

x x x x x

x x x

2 2 26 9 2 16 30 3 75

3 ( 5)( 5)

x x x x x

x x x

Page 13: 8.5 – Add and Subtract Rational Expressions. When you add or subtract fractions, you must have a common denominator. When you subtract, make sure to distribute

2 2 26 9 2 16 30 3 75

3 ( 5)( 5)

x x x x x

x x x

6. Perform the indicated operation.

22 10 96

3 ( 5)( 5)

x x

x x x

22 5 48

3 ( 5)( 5)

x x

x x x

Page 14: 8.5 – Add and Subtract Rational Expressions. When you add or subtract fractions, you must have a common denominator. When you subtract, make sure to distribute

Complex Fraction:

Fraction in the numerator or denominator

How To solve:

Multiply the numerator and denominator by the least common multiple of every fraction in the numerator and denominator

Page 15: 8.5 – Add and Subtract Rational Expressions. When you add or subtract fractions, you must have a common denominator. When you subtract, make sure to distribute

7. Simplify.

6 37

5 10

x x

x

6060 10 20

12 42

x x

x

= =

10

2(6 21)

x

x

=

5

6 21

x

x

-5

Page 16: 8.5 – Add and Subtract Rational Expressions. When you add or subtract fractions, you must have a common denominator. When you subtract, make sure to distribute

8. Simplify.

35

2 13 5

x

x x

(x+5)(x-3)(x+5)(x-3)

=

3 3

2 5 3

x

x x

3 3

2 10 3

x

x x

3 3

3 7

x

x