Intermediate Algebra Tutorial 33


Intermediate Algebra
Tutorial 33: Adding and Subtracting Rational Expressions


WTAMU > Virtual Math Lab > Intermediate Algebra  



deskLearning Objectives


 
After completing this tutorial, you should be able to:
  1. Find the least common denominator of rational expressions.
  2. Add and subtract rational expressions.




desk Introduction



Do you ever feel dazed and confused when working with fractions? If so, you are not alone.  This is your lucky day!  We have a whole other tutorial devoted to rational expressions (fractions).  In this tutorial we will be looking at adding and subtracting them.  If you need a review on simplifying, multiplying and dividing rational expressions, feel free to go back to Tutorial 32: Multiplying and Dividing Rational Expressions.  It is time to get started with this tutorial.

 

 

desk Tutorial



 

Adding or Subtracting Rational Expressions 
with Common Denominators

adding fractions

subtract


 
Step 1: Combine the numerators together.

Step 2: Put the sum or difference found in step 1 over the common denominator.

Step 3:  Reduce to lowest terms as shown in Tutorial 32: Multiplying and Dividing Rational Expressions.


 
 
  Why do we have to have a common denominator
when we add or subtract rational expressions?????

 
Good question.  The denominator indicates what type of fraction that you have and the numerator is counting up how many of that type you have.  You can only directly combine fractions that are of the same type (have the same denominator).  For example if 2 was my denominator, I would be counting up how many halves I had.  If 3 was my denominator, I would be counting up how many thirds I had.  But I would not be able to add a fraction with a denominator of 2 directly with a fraction that had a denominator of 3 because they are not the same type of fraction.  I would have to find a common denominator first, which we will cover after the next two examples.

 
 
 
notebook Example 1:  Perform the indicated operation example 1a.

 
Since the two denominators are the same, we can go right into adding these two rational expressions.

 
Step 1: Combine the numerators together

AND

Step 2: Put the sum or difference found in step 1 over the common denominator.


 
example 1b
*Common denominators
 
 

*Combine the numerators
*Write over common denominator
 

 


 
Step 3: Reduce to lowest terms.

 
example 1c

 

*Factor out a GCF of 5 in num.
 

*Divide out the common factor of 3x + 1
 

 


 

notebook Example 2:  Perform the indicated operation example 2a.

 
Since the two denominators are the same, we can go right into subtracting these two rational expressions.

 
Step 1: Combine the numerators together

AND

Step 2: Put the sum or difference found in step 1 over the common denominator.


 
example 2b
*Subtract EVERY term of second num.
 
 

 


 
Step 3: Reduce to lowest terms.

 
example 2c
*Factor the trinomial in num.
*No common factors


 

Least Common Denominator (LCD)
 
Step 1: Factor all the denominators
 
If you need a review on factoring, go to any or all of the following tutorials: Tutorial 27: The GCF and Factoring by Grouping, Tutorial 28: Factoring Trinomials, or Tutorial 29: Factoring Special Products.

 

Step 2: The LCD is the list of all the DIFFERENT factors in the denominators raised to the highest power that there is of each factor.


 
 
notebook Example 3:  Find the LCD of example 3a.

 
Step 1: Factor all the denominators.

 
The first denominator "3x" has two factors, 3 and x.

The second denominator "2" has only one factor, 2.


 
Step 2: The LCD is the list of all the DIFFERENT factors in the denominators raised to the highest power that there is of each factor.

 
Putting all the different factors together, we get (3)(x)(2) = 6x for our LCD.

 
 
 
notebook Example 4:  Find the LCD of example 4a.

 
Step 1: Factor all the denominators.

 
The first denominator has the following two factors:

 
example 4b

*Factor the diff. of two squares

 
The second denominator has the following factor:

 
example 4c

*Factor the perfect square trinomial

 
 
Step 2: The LCD is the list of all the DIFFERENT factors in the denominators raised to the highest power that there is of each factor.

 
Putting all the different factors together and using the highest exponent, we get the following LCD:

 
example 4d


 
  Adding and Subtracting Rational Expressions
Without a Common Denominator

 
Step 1: Find the LCD as shown above if needed.

 
Step 2: Write equivalent fractions using the LCD if needed.

 
If we multiply the numerator and denominator by the exact same expression it is the same as multiplying it by the number 1.  If that is the case,  we will have equivalent expressions when we do this. 

Now the question is WHAT do we multiply top and bottom by to get what we want?  We need to have the LCD, so you look to see what factor(s) are missing from the original denominator that is in the LCD.  If there are any missing factors than that is what you need to multiply the numerator AND denominator by.


 
Step 3: Combine the rational expressions as shown above.

 
Step 4: Reduce to lowest terms as shown in Tutorial 32: Multiplying and Dividing Rational Expressions.

 
 
 
notebook Example 5:  Perform the indicated operation example 5a.

 
Step 1: Find the LCD as shown above if needed.

 
The first denominator has the following two factors:

 
example 5b
*Factor the trinomial

 
The second denominator has the following factor:

 
example 5c


 
Putting all the different factors together and using the highest exponent, we get the following LCD:

 
example 5d


 
Step 2: Write equivalent fractions using the LCD if needed.

 
Since the first rational expression already has the LCD, we do not need to change this fraction.

 
example 5e

 

*Rewriting denominator in factored form

 


 
Rewriting the second expression with the LCD:

 
example 5f

*Missing the factor of (x - 1) in the den.

*Mult. top and bottom by (x - 1)

 


 
Step 3: Combine the rational expressions as shown above.

 
example 5g

 
 

*Subtract EVERY term of (   ).
 
 
 

 


 
Step 4: Reduce to lowest terms as shown in Tutorial 32: Multiplying and Dividing Rational Expressions.

 
This rational expression cannot be simplified down any farther.  So, our answer is 

 
example 5h.


 
 
 
 
notebook Example 6:  Perform the indicated operation example 6a.

 
 
Step 1: Find the LCD as shown above if needed.

 
The first denominator has the following factor:

 
example 6b


 
The second denominator has the following two factors:

 
example 6c

*Factor the sum of cubes

 
Putting all the different factors together and using the highest exponent, we get the following LCD:

 
example 6d


 
Step 2: Write equivalent fractions using the LCD if needed.

 
Rewriting the first expression with the LCD:

 
example 6e

*Missing the trinomial factor in the den.

*Mult. top and bottom by the trinomial


 
Since the second rational expression already has the LCD, we do not need to change this fraction.

 
example 6f

 

*Rewriting denominator in factored form

 


 
Step 3: Combine the rational expressions as shown above.

 
example 6g
*Add the two numerators together
*Write over the LCD
 

 


 
Step 4: Reduce to lowest terms as shown in Tutorial 32: Multiplying and Dividing Rational Expressions.

 
example 6h

 

*Divide out the common factor of x + y

 


 
 
 
notebook Example 7:  Perform the indicated operation example 7a.

 
 
Step 1: Find the LCD as shown above if needed.

 
The first denominator has the following factor:

 
example 7b


 
The second denominator, 7 - x, looks like the first denominator except the signs are switched.  We can rewrite this as

 
example 7c

*Factor out a -1

 
Putting all the different factors together and using the highest exponent, we get the following LCD:

 
example 7d


 
Note that I did not put the -1 that was in front of the second denominator's (x - 7) factor.  In step 3, I will put the negative into the problem by placing it in the numerator of that second fraction.

 
Step 2: Write equivalent fractions using the LCD if needed.

 
Now the two fractions have a common denominator, so we do not have to rewrite the rational expressions.

 
 
Step 3: Combine the rational expressions as shown above.

  example 7e
 

*Combine the two num.
*Write over the common den.
 
 

 


 
Step 4: Reduce to lowest terms as shown in Tutorial 32: Multiplying and Dividing Rational Expressions.

 
This rational expression cannot be simplified down any farther.  So, our answer is 

 
example 7f.


 
 
 
notebook Example 8:  Perform the indicated operation example 8a.

 
 
Step 1: Find the LCD as shown above if needed.

 
The first denominator has the following factor:

 
a


 
The second denominator has the following two factors:

 
example 8a

*Factor out a GCF of 2

 
 
The third denominator has the following two factors:

 
example 8c

*Factor out a GCF of a

 
Putting all the different factors together and using the highest exponent, we get the following LCD:

 
example 8d


 
Step 2: Write equivalent fractions using the LCD if needed.

 
Rewriting the first expression with the LCD:

 
example 8e

*Missing 2(a + 2) in the den.
 

*Mult. top and bottom by 2(a + 2)
 


 
Rewriting the second expression with the LCD:

 
example 8f

*Missing a in the den.
 
 

*Mult. top and bottom by a
 


 
 
Rewriting the third expression with the LCD:

 
example 8g

*Missing 2 in the den.
 

*Mult. top and bottom by 2
 


 
Step 3: Combine the rational expressions as shown above.

 
example 8h
*Combine the two num.
*Write over the common den.
 

 


 
Step 4: Reduce to lowest terms as shown in Tutorial 32: Multiplying and Dividing Rational Expressions.

 
example 8i
*Factor out a GCF of a in the num.
 
 

*Divide out the common factors of a and (a + 2)


 

 

 
desk Practice Problems


  
These are practice problems to help bring you to the next level.  It will allow you to check and see if you have an understanding of these types of problems. Math works just like anything else, if you want to get good at it, then you need to practice it.  Even the best athletes and musicians had help along the way and lots of practice, practice, practice, to get good at their sport or instrument.  In fact there is no such thing as too much practice.

To get the most out of these, you should work the problem out on your own and then check your answer by clicking on the link for the answer/discussion for that  problem.  At the link you will find the answer as well as any steps that went into finding that answer.

 

pencil Practice Problems 1a - 1c: Perform the indicated operation.

 

1a. problem 1a
(answer/discussion to 1a)
1b. problem 1b
(answer/discussion to 1b)

1c. problem 1c
(answer/discussion to 1c)

 

 

 

desk Need Extra Help on these Topics?



 

Go to Get Help Outside the Classroom found in Tutorial 1: How to Succeed in a Math Class for some more suggestions.


 


WTAMU > Virtual Math Lab > Intermediate Algebra


Last revised on July 17, 2011 by Kim Seward.
All contents copyright (C) 2001 - 2011, WTAMU and Kim Seward. All rights reserved.