College Algebra & Trigonometry, 2018a

104 CHAPTER 2. POLYNOMIAL AND RATIONAL FUNCTIONS
2.4 Solution of Rational Inequalities by Graphing
In the previous section, we saw how to solve polynomial inequalities by graphing.
In this section, we will use similar methods to solve rational inequalities.
Rational inequalities involve ratios of polynomials or fractions. Because these
types of problems involve fractions, the graphs of the functions that we work
with will have what are known as asymptotes. This word comes from a Greek
root having to do with two lines that come very close to each other but never
meet.
The vertical asymptotes of a graph will appear at places where the original expression
has a zero denominator. This means that the function is not defined at
those x values and so, rather than having a y value at that point, the graph has an
asymptote.
Example
Below is a graph of the function y = x +2
x − 1
5
−8 −6 −4 −2 2 4 6 8
−5
Rather than having a y value at the point where x =1, the dotted line indicates
the asymptote where the function is not defined. In the previous section, we were
interested in finding the roots of the function because these are the places where
y =0, and can be the dividing points between where the y values are greater than
zero (y >0) and the where the y values are less than zero (y