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Multiplying Fractions
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The Square of a Binomial
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Multiplying Binomials Using the FOIL Method
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Algebra
Order of Operations
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The Appearance of a Polynomial Equation
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Positive Integral Divisors
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Use of Parentheses or Brackets (The Distributive Law)
Multiplying and Dividing by Monomials
Solving Quadratic Equations by Graphing
Multiplying Decimals
Use of Parentheses or Brackets (The Distributive Law)
Simplifying Complex Fractions 1
Adding Fractions
Simplifying Complex Fractions
Solutions to Linear Equations in Two Variables
Quadratic Expressions Completing Squares
Dividing Radical Expressions
Rise and Run
Graphing Exponential Functions
Multiplying by a Monomial
The Cartesian Coordinate System
Writing the Terms of a Polynomial in Descending Order
Fractions
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Solving Rational Inequalities with a Sign Graph
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Solving an Equation with Two Radical Terms
Simplifying Rational Expressions
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Intercepts of a Line
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Order of Operations
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Collecting Like Terms
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Percent of Change
Powers of ten (Scientific Notation)
Comparing Integers on a Number Line
Solving Systems of Equations Using Substitution
Factoring Out the Greatest Common Factor
Families of Functions
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Multiplying and Dividing Complex Numbers
Properties of Exponents
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Adding or Subtracting Rational Expressions with Different Denominators
Expressions with Variables as Exponents
The Quadratic Formula
Writing a Quadratic with Given Solutions
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Adding and Subtracting Square Roots
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Combining Like Radical Terms
Solving Systems of Equations Using Substitution
Dividing Polynomials
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Product of a Sum and a Difference
Solving First Degree Inequalities
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Adding Fractions

The objective of this lesson is very succinct:

  • That you learn how to add fractions correctly

 

Numerator and Denominator

The number or algebraic expression that appears on the top line of a fraction is called the numerator of the fraction.

The number of algebraic expression that appears on the bottom line of a fraction is called the denominator of the fraction.

Adding Fractions

Expressed in symbols, the rule for adding fraction is as follows:

Let’s break this down to see everything that is expressed in this rule.

The numerator of the sum is a·d + b·c.

You can remember the numerator without having to memorize this particular formula by remembering the pattern of cross-multiplying. To create the numerator, you multiply each numerator by the opposing denominator, forming a “cross” pattern.

To get the denominator of the sum, you just multiply the two denominators (b and d) together.

Example

Work out each of the following sums of fractions.

Solution

(a)

Often it will be possible for you to simplify your fractional expressions by combining “like terms” just as you do when FOILing a polynomial. Although this kind of simplification is not always needed just to get the right answer, if can make your fractional expressions much easier to deal with. Remember to keep the numerator and denominator separate when combining like terms!

(b)

In Example (b), note how when the cross-multiplication is done, the “7” from the numerator of the first fraction multiplies the entire quantity (x + 1) that is in the denominator of the second fraction, not just the x. Also notice that when the two denominators are multiplied to create the denominator of the sum, the “10” from the denominator of the first fraction multiplies everything (i.e. the entire quantity (x + 1)) that appears in the denominator of the second fraction.

(c)

When simplifying fractions, simplify the numerator and denominator separately. You cannot combine like terms from the numerator with like terms from the denominator (or vice versa). Often you will need to FOIL when simplifying the numerator and denominator of fractions that involve algebraic expressions such as x.

(d)

This answer is not the simplest one that is possible. If you look closely at the middle fraction above, you can see that every single term in the numerator has at least one factor of (x + 1). The denominator also has a factor of (x + 1). These “common” factors can be factored out of the numerator and the denominator as shown below.

When you have a common factor that you have pulled out of every term in the numerator, and it matches a factor that shows up in the denominator, you can almost always cancel this factor from both the numerator and the denominator.

provided x ≠ -1.

The only situation when it is not okay to cancel the factor of (x + 1) from the top and bottom is when you have the x-value of x = -1 (i.e. the particular x-value that makes the factor of (x + 1) equal to zero).

 
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