Compare numerical fractions against rational expressions. What is common and what is different about them, and how do we make sure that a rational expression does not get undefined?
Compare fractions to rational expressions:
(1) To simplify either you factor and then cancel any factor that appears in numerator and denominator.
Note that `21/15=7/5` ; on the other hand `(x^2+2x+1)/(x^2+4x+3)=(x+1)/(x+3)` for all x except x=-1, thus the two expressions are not equivalent.
(2) To add/subtract you get a common denominator then add/subtract the numerators:
(3) To multiply you multiply numerators and denominators:
(4) Both will have additive inverses found by multiplying by -1
(5) Both will have multiplicative inverses (it will be the reciprocal) except when they are 0.
`2/3*3/2=1` but `0/7` does not have a multiplicative inverse.
The multiplicative inverse of `(x+1)/(x+3)` is `(x+3)/(x+1)` except when x=-1
(6) Numerical fractions exist except when the denominator is zero.
Rational expressions are defined as long as the denominator is not zero.