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
.
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
and
.
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.
\
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The function intervals are
and
.
\
Choosing a number for
in each interval and evaluating
.
| Interval | \ \
| \
\
\
| \
Conclusion | \
| \
| \
\
| \
\
| \
Positive | \
| \
| \
\
| \
\
\
| \
Negative | \
| \
| \
\
| \
\
\ \ \ \
\ | \
Positive | \
| \
| \
\
| \
\
| \
Positive | \
\
\
Solution in set notation :
.
Solution in interval notation:
.
\
Solution in set notation :
.
Solution in interval notation:
.