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

Table Header | |||
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Sign |

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,

Leah