Simplifying Complex Fractions complex fraction is a fraction whose ...

Simplifying Complex Fractions

A complex fraction is a fraction whose numerator or denominator contains one or more

fractions. Examples of complex fractions:

1

1

1 6

?1

2?

1? ? 2

2

y

3

x x

2

4 3

1

8?

1? ? 2

1?

3

x x

y

TO SIMPLIFY COMPLEX FRACTIONS

Step 1.

Multiply the numerator and the denominator of the complex fraction by the

LCD of the ALL the fractions in both the numerator and denominator.

Simplify the resulting fraction. Reduce the fraction, if possible.

Step 2.

Example 1

1

3

Simplify

2

8?

3

2?

1

(2 ? )

3 * 3 multiply top and bottom by the lcd

2

(8 ? ) 3

3

6 ?1

combine like terms and simplify

24 ? 2

5

26

1

Example 2

1

?1

y2

Simplify

1

1?

y

1

? 1) 2

y

y2

* 2 multiply top and bottom by the lcd

1

(1 ? ) y

y

(

1? y2

y2 ? y

(1 ? y )(1 ? y )

factor top and bottom and cancel common factors

y ( y ? 1)

1? y

y

6.6 ¨C SOLVING EQUATIONS CONTAINING RATIONAL EXPRESSIONS

HW # 29 ¨C 57

Recall the difference between an equation and an expression. We solve an equation and simplify

expressions.

A rational equation is an equation that contains one or more rational expressions.

Examples of rational equations:

2

x 3 x

? ?

2 6 3

1?

36

2x

?1 ?

x ?9

x?3

3

?4

y

2

TO SOLVE A RATIONAL EQUATION

Step 1.

Step 2.

Step 3.

Step 4.

Step 5.

Determine the LCD of all fractions in the equation (Factor all denominators

completely to determine the LCD).

Multiply both sides of the equation by the LCD (This requires multiplying

EACH term in the equation by the LCD).

Simplify (cancel) on each side of the equation (All denominators should

cancel.)

Determine whether the equation is linear, quadratic or higher degree and solve

accordingly.

You MUST check your proposed solution(s) in the ORIGINAL equation to

insure that your solution does not make any of the rational expressions

¡°undefined¡± (division by zero).

Example 1

x 3 x

Solve ? ?

2 6 3

Example 2

36

2x

? 1?

x ?9

x?3

2

3

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