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**Review of Addition and Subtraction of fractions and Introduction to Simplifying Complex Fractions**

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**Review of Addition, Subtraction of Fractions**

To add or subtract rational expressions, a common denominator is necessary... Example: Simplify Find the LCD: 6x Now, rewrite the expression using the LCD of 6x Simplify... Add the fractions... = 19 6x

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**Let’s try one with polynomials as denominators...**

Example: Simplify Find the LCD: (x + 2)(x – 2) Rewrite the expression by multiplying the top and bottom of each fraction by whatever is required to get the LCD of (x + 2)(x – 2) Simplify... (watch out for the negative!) = –5x – 22 (x + 2)(x – 2)

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**Complex Fraction – a fraction with a fraction in the numerator and/or denominator.**

Such as: How would you simplify this complex fraction? Multiply the top by the reciprocal of the bottom!

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**Method 1 For simplifying Complex Fractions**

Work on the numerator and denominator separately. Find the common denominator of the fractions on the top and combine them. Find the common denominator of the fractions on the bottom and combine them. Invert the bottom and multiply by the top. Simplify where possible.

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Example: Invert the bottom and multiply

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Another Example:

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Yet another example:

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**Method 2 For simplifying Complex Fractions**

To simplify complex fractions, find the LCD of all the little fractions Multiply every term by the LCD of all the little fractions ... Simplify Divide out where you can ...

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**Example: Simplify 12 + 18 4 + 9 = 30 13 Using Method 2**

To simplify complex fractions, find the LCD of all of the little fractions: 12x Multiply every term by the LCD... • 12x 1 • 12x 1 Simplify (divide out where you can )... • 12x 1 • 12x 1 12 + 18 4 + 9 = 30 13

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**Here is a complex fraction with polynomials using method 2 ...**

Example: Simplify LCD of all of the little fractions : x(x + 2) Multiply every term by the LCD... • x(x + 2) 1 Simplify (divide out where you can )... • x(x + 2) 1 • x(x + 2) x 6x2 + 12x + 4x + 8 = x 6x2 + 16x + 8

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The final example explores a problem that has a fraction as one of its terms in a “deep” layer. The method in question solves the equation from the innermost fractions to the outer layer, by finding the LCD’s of that layer.

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**Example : Simplify Step 1. Group and expand each fractional term.**

Step 2. Find the LCD of the innermost fraction. Step 3. Simplify. Step 4. Add fractions Step 5. Invert and multiply.

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Example (continued): Step 6. Rewrite with LCD in both terms. Step 7. Multiply and combine the fractions. At the end , check to see if there are common factors in the numerator and denominator that can be used to reduce. This is the final solution because there are no common factors in the problem.

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