Rational Inequalities
There is one step added to the process of solving rational inequalities because a rational function can also change signs at its vertical asymptotes or at a break in the graph. For instance, look at the graph of the function below.
If we want to solve the inequality , then we need to use the following critical points: and . is the solution of setting the numerator equal to 0, and this gives us the only root of the function. are the vertical asymptotes, the coordinates that make the function undefined because putting in 3 or -3 for will cause a division by zero.
Testing the intervals between each critical point to see if the values in that interval satisfy the function gives us:
Interval | Test Point | Positive/Negative? | Part of Solution set? |
---|---|---|---|
-4 | - | no | |
(-3, 0) | -2 | + | yes |
(0, 3) | 2 | - | no |
4 | + | yes |
Thus, the solutions to are .
Guided Practice
Questions
1)
2)
Solutions
1) To identify the graph of the inequality , first treat it as if it were the equality
- For
- To find the critical points, identify the value(s) which make the denominator = 0:
- That gives us a vertical asymptote of
- The horizontal asymptote becomes apparent as x becomes truly huge and the "+5" and "-1" no longer matter. At that point, we have So the horizontal asymptote is
- Now that you know the shape of the graph, simply shade the area above the lines, since the original function was f ( x ) is greater-than function, and leave the lines solid since it was a greater-than or equal to .
- The final graph should look like:
2) To graph first treat it as if it were
- To identify domain limitations, find value(s) which make the denominator = 0: In this case, where , the only variable in the denominator, is added to 0 , any value for x will be positive. So the domain is all real numbers.
- With no limitations on the domain, there are no vertical asymptotes.
- The horizontal asymptote: becomes apparent as x becomes huge and the "+2" and "+1" no longer have an effect, giving: So the horizontal asymptote is 0
- Now that you know the shape of the graph, simply shade the area below the lines, since the original function was f ( x ) is less-than function, and leave the lines solid since it was or equal to .
- The final graph should look like: