High School Math Solutions – Inequalities Calculator, Rational Inequalities

Last post, we talked about solving quadratic inequalities. In this post, we will talk about rational inequalities. Let’s recall rational functions are an algebraic fraction that contain polynomials in the numerator and denominator. Solving rational inequalities is a little different than solving quadratic inequalities. Both share the concept of a table and testing values.

Let’s see how to solve rational inequalities.

Here the steps:

1. Move everything to one side of the inequality sign
2. Simplify the rational function 
3. Find the zeros from the numerator and undefined points in the denominator
4. Derive intervals
5. Find the sign of the rational function on each interval
6. 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):

\frac{x}{x-3}<4

Step 1: Move everything to one side of the inequality sign

\frac{x}{x-3}<4
\frac{x}{x-3}-4<0
\frac{-3x+12}{x-3}<0

Make sure you combine everything into one rational function.

Step 2: Simplify the rational function

\frac{-3x+12}{x-3}<0
\frac{3(-x+4)}{x-3}<0

It’s already simplified. Nothing to cancel out.

Step 3: Find the zeros from the numerator and undefined points in the denominator

-3x+12=0                                                        x-3=0
x=4                                                                     x=3

          Zero                                                                     Undefined point

Step 4: Derive intervals

Step 5: Find the sign of the rational function on each interval

Table Header	x\lt3	3\ltx\lt4	x\gt4
\frac{-3x+12}{x-3}	\frac{-3(0)+12}{(0)-3}=-4	\frac{-3(3.5)+12}{(3.5)-3}=3	\frac{-3(5)+12}{(5)-3}=-\frac{3}{2}
Sign	\quad\quad\quad\quad-	\quad\quad\quad\quad+	\quad\quad\quad\quad-

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

\frac{-3x+12}{x-3}<0
x<3\quad\:and\quad\:x>4

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