Example 1 Solve
.
Solution
Before we get into solving these we need to point out that
these DON’T solve in the same way that we’ve solve equations that contained
rational expressions. With equations
the first thing that we always did was clear out the denominators by
multiplying by the least common denominator.
That won’t work with these however.
Since we don’t know the value of x we can’t multiply both sides by anything that contains an x.
Recall that if we multiply both sides of an inequality by a negative
number we will need to switch the direction of the inequality. However, since we don’t know the value of x we don’t know if the denominator is
positive or negative and so we won’t know if we need to switch the direction
of the inequality or not. In fact, to
make matters worse, the denominator will be both positive and negative for
values of x in the solution and so
that will create real problems.
So, we need to leave the rational expression in the
inequality.
Now, the basic process here is the same as with polynomial
inequalities. The first step is to get
a zero on one side and write the other side as a single rational
inequality. This has already been done
for us here.
The next step is to factor the numerator and denominator
as much as possible. Again, this has
already been done for us in this case.
The next step is to determine where both the numerator and
the denominator are zero. In this case
these values are.
Now, we need these numbers for a couple of reasons. First, just like with polynomial
inequalities these are the only numbers where the rational expression may change sign. So, we’ll build a number line using these
points to define ranges out of which to pick test points just like we did
with polynomial inequalities.
There is another reason for needing the value of x that make the denominator zero
however. No matter what else is going
on here we do have a rational expression and that means we need to avoid
division by zero and so knowing where the denominator is zero will give us
the values of x to avoid for this.
Here is the number line for this inequality.
So, we need regions that make the rational expression
negative. That means the middle
region. Also, since we’ve got an “or
equal to” part in the inequality we also need to include where the inequality
is zero, so this means we include . Notice that we will also need to avoid since that gives division by zero.
The solution for this inequality is,
