__Complex Fractions__Chapter 7

**What is a Complex Fraction?**- A complex fraction is a rational expression whose numerator and denominator are themselves rational expressions.
- Since the numerator and denominator of the complex fraction are fractions themselves, the numerator and denominator are each called
**secondary fractions**.

**Simplifying Complex Fractions**- The steps for 2 methods of simplifying complex fractions are given here.
**Method 1: Multiplying by a Common Denominator**

- Find the lowest common denominator of the two secondary fractions.
- Multiply both the numerator and denominator of the complex fraction by this lowest common denominator.
- Simplify (reduce to lowest terms) the resulting rational expression.

**Method 2: Simplifying Numerator and Denominator**

- Simplify (reduce to lowest terms) each secondary fraction.

__Solving Rational Equations__

**What is a Rational Equation?**- A
**rational equation**is an equation that contains at least one rational expression.

**To Solve a Rational Equation......**- Determine the lowest common denominator of all rational expressions in the equation.
- Multiply both sides of the equation by the lowest common denominator.
- Simplify by removing parentheses and combining like terms.
- Solve the equation resulting from step 4.
- Check the solution in the original equation. This step is essential since it is here that you will eliminate extraneous solutions (see below).

**What are Extraneous Solutions?**- An
**extraneous solution**is a value that you think is a solution because it was found by the solving process but, this value does not solve the*original*equation. - The way that you determine that a value is an extraneous solution is by checking that value in the
*original*equation:

- If a value that you think is a solution does not solve the
*original*equation, then that value is an extraneous solution.

- If a value that you think is a solution does not solve the
- The check step is where extraneous solutions are found.
- Always check all solutions in the
*original*equation to make sure they solve the*original*equation.