The inequality is .
Step 1: Determine the real zeros ( -intercepts of the graph )of
and the real numbers for which
is undefined.
The zeroes of the function are the values of for which
.
THe function is .
The zeroes of are
.
A rational function is undefined when denominator is zero.
\ is undefined for
and
.
Step 2: Use the zeros and undefined values found in Step 1 to divide the real number line into intervals.
\Denominator of the function should not be zero.
\ and
and
.
The function is defined for all values of except at
and
.
The function intervals are .
Step 3: Select a number in each interval, evaluate at the number, and determine whether
is positive or negative.
If is positive, all values of
in the interval are positive. If
is negative, all values of
in the interval are negative.
.
The real zero of numerator is and the real zeros of denominator
and
.
\
So the real zeros are divide the - axis into three intervals.
\
The function intervals are .
\
Choosing a number for in each interval and evaluating
.
Interval | \ \
| \
\
\ | \
Conclusion | \
\
| \
\
| \
\
| \
Positive | \
\
| \
\
| \
\
\ | \
Positive | \
\
| \
\
| \
\
\ \ \ \ \ | \
Negative | \
\
| \
\
| \
\
| \
Positive | \
\
\
The solution of the inequality in set notation is
.
The solution of the inequality in interval notation is
.
\
;
.