Rational Expressions
A quotient of two integers,
, where 0, is called a rational expression.
Some examples of rational expressions are
,
,
, and
. When 4, the denominator of the
expression
becomes 0 and the expression is meaningless. Mathematicians state this fact by saying that
the expression
is undefined when 4. One can see that the value
, makes the expression
undefined. On the other hand, when any real number is substituted into the expression
, the answer is
always a real number. There are no values for which this expression is undefined.
EXAMPLE Determine the value or values of the variable for which the rational expression is defined.
a)
b)
Solution a) Determine the value or values of x that make 2x – 5 equal to 0 and exclude these. This can
be done by setting 2x – 5 equal to 0 and solving the equation for x.
2 5 0
2 5
Do not consider
when considering the rational expression
. This expression is
defined for all real numbers except
. Sometimes to shorten the answer it is written as
.
b) To determine the value or values that are excluded, set the denominator equal to zero and
solve the equation for the variable.
6 7 0
7 1 0
7 0 or 1 0
7 1
Therefore, do not consider the values 7 or 1 when considering the rational
expression
. Both 7 and 1make the denominator zero. This is defined for
all real numbers except 7 and 1. Thus, 7 and 1.
SIGNS OF A FRACTION
Notice:
Generally, a fraction is not written with a negative denominator. For example, the expression
would be
written as either
or
. The expression
can be written
since 4 4 or 4.
