Let’s see how to solve rational inequalities.
Here the steps:
- Move everything to one side of the inequality sign
- Simplify the rational function
- Find the zeros from the numerator and undefined points in the denominator
- Derive intervals
- Find the sign of the rational function on each interval
- Select the proper inequality
Let’s go through our first example step by step to understand the concept better.
Here’s our first example (click here):
Step 1: Move everything to one side of the inequality sign
Make sure you combine everything into one rational function.
Step 2: Simplify the rational function
It’s already simplified. Nothing to cancel out.
Step 3: Find the zeros from the numerator and undefined points in the denominator
Zero Undefined point
Step 4: Derive intervals
Step 5: Find the sign of the rational function on each interval
We changed the format of the intervals to inequalities. We pick a number in the interval, plug the number in the rational function and see what sign the answer is (negative or positive).
Step 6: Select the proper inequality
We refer back to the original inequality and see which inequality satisfies the original inequality. We are looking for a number that produces a negative number. We refer back to the table and see that x<3 and x>4 satisfy this.
Alright, that was a mouthful. Let’s see some more examples now…
Second example (click here):
Last example (click here):
Make sure you double check your work for calculation errors because that is where it’s very easy to make a mistake. For more practice, check out Symbolab’s practice.
Until next time,