College Algebra & Trigonometry, 2018a

2.4. SOLUTION OF RATIONAL INEQUALITIES BY GRAPHING 105
The importance of the asymptotes in analyzing rational functions is that, like the
roots, these represent x values that can be the dividing points between where
y>0 and where y 0
First we examine the graph:
5
−8 −6 −4 −2 2 4 6 8
−5
Notice that the asymptote for this graph occurs at the value x =3, because this is
the x value that creates a zero denominator. Also notice that the y values switch
from being negative to being positive across the asymptote.
There are no roots for this function because there are no x values that make y =0.
For a fraction to be zero, the numerator must equal zero. In this example the
numerator is 2 and no value of x will make it equal zero. Therefore the only
possible dividing point on the graph is x =3, and the solution to the inequality
is x>3.