Algebra - Rational Inequalities - Pauls Online Math Notes

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.

\[{\mbox{numerator : }}x = - 1\hspace{0.5in}{\mbox{denominator : }}x = 5\]

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 \(x = - 1\). Notice that we will also need to avoid \(x = 5\) since that gives division by zero.

The solution for this inequality is,

\[ - 1 \le x < 5\hspace{0.25in}\left[ { - 1,5} \right)\]