Lesson 12: Polynomial and rational inequalities

Hi everyone! Read through the material below, watch the videos, and follow up with your instructor if you have questions.

Question of the Day: What does it mean to “solve” an inequality (as opposed to “solving” an equation?)

Introduction to polynomial inequalities

Definition. A polynomial inequality is an inequality (which means it uses one of these: $<,\leq,>,\geq$ instead of an equals sign) with a polynomial on each side

Example 1: $x^2-3x-4\geq 0$

We’re interested in solving these inequalities, which means answering the question: “For which real numbers x is the inequality true?”

Now let’s look at the same example, and see how to solve it without looking at the graph:

Example 2: Solve $x^{4}-x^{2}>5\left(x^{3}-x\right)$

Example 2, concluded:

What about the Magic?

In the video above (Example 2 – conclusion), there was a place where we needed to solve the following equation:

$x^4-5x^3-x^2+5x=0$

In the video, I used magic to find the roots — in the next video, I’ll actually go through the steps:

Rational inequalities

What happens if we allow rational functions instead of just polynomials?

Example 3: Solve $\frac{x^{2}-5 x+6}{x^{2}-5 x} \geq 0$

Good job! You are now ready to practice on your own – give the WeBWorK assignment a try. If you get stuck, try using the “Ask for Help” button to ask a question on the Q&A site.