1. What values of x are excluded for a rational
expression?
What is the domain of a rational function?
Look at denominator only !
Denominator cannot equal zero. So, set denominator ≠ zero
Factor if possible; Eliminate x-values that give a 0 denominator
Example: 
Denominator can’t equal 0.
Expression is defined for all real numbers , x, except
x = 4 , x = –5
Domain of the rational function is all real numbers x such that x ≠ 4, and x ≠
–5.
D: {x | x ≠ 4, x ≠ –5}
2. Simplify a rational expression:
Look for factors in numerator and denominator.
Cancel identical factors.
Example:
3. Multiply rational expressions:
DO NOT NEED COMMON DENOMINATOR
Factor all numerator (s) and denominator(s)
Cancel identical factors as in number 2
Example:
4. Divide rational expressions:
First rewrite problem as a multiplication problem
Do this by inverting the divisor.
Then proceed as you did for multiplication ( number 3)
Example:
5. Add or subtract rational expressions:
*** YOU MUST HAVE A LEAST COMMON DENOMINATOR (LCD) ***
Example:
When you have found the LCD, add or subtract the
numerators .
*Remember to group (put parentheses around) ex pressions you subtract .
Be careful! Distribute the minus sign throughout when
subtracting!
Combine like terms in numerator :
If numerator can be factored, do so and see if you can
reduce.
*Many times the numerator won’t factor, or if it does,
the factors won’t cancel into the
denominator.
Comparison and Examples
(I) Adding or Subtracting Rational Expressions
(II) Solving Equations Containing Rational Expressions
You must find a LCD for both types of problems, I and II.
I. To add/subtract, multiply each term by a form of 1:
Example:
This problem just asks us to add the two expressions. The
denominators are different so
we need to find the LCD. LCD = 18x2y
We have a denominator of 6xy. Determine what factors are necessary to multiply
and get
the LCD. You need 3x as a multiplier so that 6xy becomes 18x2y.
Multiply
Multiply
Add:
(can’ t be simplified )
NOTE: There will still be a denominator in your answer!
Example: 
Add LCD = 4x
(Note: 
and
)
*** There is still a denominator in your answer!
THESE FIRST TWO EXAMPLES DIDN’T ASK YOU TO SOLVE FOR X.
THEY DID NOT INVOLVE AN EQUATION to be solved.
II. NOW: SOLVE AN EQUATION FOR X.
We still need a common denominator. The common denominator
is 4x.
If we change each term to an equivalent term, but with the LCD, we would have
(as we did in the previous example):
How do we write “2” as
a fraction with LCD = 4x?
Remember that we are allowed to multiply both sides of an
equation by the same value.
Therefore, it’s legal to multiply both sides by the LCD, 4x.
Why do we want to do this? So we can get an equation that does not contain
fractions, just like
we did earlier in the semester.
Multiplying each term by 
gives us:
This reduces to x 2 + 12 = 8x, which is an equation you can
solve by factoring.
So in the kind of problem we just solved, the denominator is eliminated
before you solve for your answer!
Finally, we will look at a shorter way to do this same problem.
Solve for x:
Remember, we can multiply both sides of an equation by the
same value.
We’ll use the LCD as the multiplier, as before.
This time, we are merely canceling like factors in each
term:
See that we get the same equation to solve as before: x2 + 12 = 8x.
x2 – 8x + 12 = 0
(x – 6)(x – 2) = 0 Check: Substitute 6 for both x’s in original problem.
x = 6 or x = 2 Then substitute 2 for both x’s in original problem.
Example: Solve for x.
Find the LCD: the denominators are 1, 2x, and (x+1).
These denominators have NO factors in common except for 1. Notice that the
denominator (x + 1) can’t be factored. Therefore the “x” in the “2x” factor
isn’t related or
included in the (x + 1) factor.
The LCD is 2x(x+1). Multiply each term by the LCD. (This removes the
denominator.)
4x (x + 1) – 5(x + 1) = 2x (2x)
4x2 + 4x – 5x – 5 = 4x2
– x – 5 = 0
x = –5
Once again, you can check your answer by substituting
x = –5 into the original equation.
