Adding and Subtracting Rational (Fractional) Expressions
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OKAY! ... so Algebraic Fractions are challenging.  But you CAN DO THESE problems!  The secret is to work slowly and show all of your work. 


Fractions are fractions!
 It doesn't matter if the fractions are made
 up of numbers, or of algebraic variables,
 the rules are the same!  

The Basic RULE for Adding and Subtracting Fractions:

Get a Common Denominator!


Examine the basic process:

            Add:   

Get a common denominator - the smallest number that both denominators can divide into without remainders.  In this case, the number is 12.
To change the denominator of 3 into 12 requires multiplying by 4.  To change the denominator of 4 into 12 requires multiplying by 3.
With each fraction, whatever is multiplied times the bottom must ALSO be multiplied times the top.

You CANNOT reduce (cancel) those 3's!!!!

Note:  The smallest common denominator is called the "least common denominator" or LCD.


 

 

Do not add the common denominators.  Add only the numerators (tops).

Multiplying the top and bottom by the same number is multiplying by
the multiplicative identity element (which is = 1) and therefore does not change the fraction.

 

Now, apply this basic process to algebraic fractions:

1.

 

 

The Least Common Denominator (LCD) is 8.  Multiply BOTH the top and bottom of the first fraction by 2 to create the common denominator.  Remember, when adding "like" variables, add only the numbers in front of the variables (the coefficients).
2.

 

The Least Common Denominator is 12xIf the denominator contains monomials with variable(s), the common denominator will need to contain the variable(s) with the largest power(s). Since the largest power of x in this problem is 1, the common denominator must contain an x to the first power, multiplied by a common denominator for the 6, 3 and 4 (which is 12).

 

3.

The Least Common Denominator is .   The largest power needed for both a and b in this problem is a power of 2.

Remember, when working with binomials, like
(
a - b) and (a + b), think of each binomial as ONE entity.  For example, in (a - b), the a cannot go anywhere without his buddy (-b) tagging along.  You can never reduce (cancel) only the a or only the b in (a - b).

Notice that when we add in this problem, the ab terms are eliminated.

The final answer cannot be reduced further. 

 

4.

 

The Least Common Denominator is a(a - 5).  Notice that one of the denominators now contains a binomial,  (a - 5). 

Remember that the  a  in (a - 5) cannot go anywhere without his buddy (-5) tagging along.  You cannot separate the a from the -5, as they must come as a pair.  In addition, you can never reduce (cancel) only the a or only the 5 in
 (a - 5).

 

 

The final answer may be expressed in several ways.

5.

Be careful when combining fractions under subtraction!  Be sure to subtract the ENTIRE numerator value behind the subtraction sign.  In this problem, when "subtracting" (x - 6) it is necessary to distribute the negative (subtraction) sign across the parentheses, creating -x + 6.

The Least Common Denominator is
 x(x - 6)(x + 6)
When working with polynomial denominators, always consider the possibility of "factoring" to find the LCD.

Instead of just multiplying the two denominators together to find the common denominator, first look to see if it is possible to find the least common denominator (LCD) by factoring and looking at those results.  Factoring may lead to a simpler and faster solution for the LCD.

Both of these denominators can be factored.
  and
Wow!  They were each hiding a factor of (x + 6).
 

 
 

  Factoring is your friend!

(Factoring lets you find the smallest common denominator quickly, thus making your work easier.)

 

6.







 
Tricky one!!!
At first glance it appears that we will need to multiply both denominators to get the common denominator - but look more closely!

If you multiply one of the denominators by (-1), you will create a factor that matches the other denominator.
2x - 6 = (-1)(6 - 2x) = (-6 + 2x) = 2x - 6
(one denominator is the additive inverse of the other)

When multiplying one of the denominators by
(-1), be sure to multiply its numerator by (-1) also!

This technique can also be accomplished by "factoring out a -1" from one of the denominators.

Whew!!!