Polynomial Inequalities Made EASY | Interval Testing Explained!

Added: 2026-05-27

[00:00:00]Everyone, welcome to the channel. It's Professor Kartush, and in today's video, we are going to deal with the polynomial inequalities, which is a very important topic in math. So, let's first start by taking a beautiful example. The example is as follows. We have x squared minus 5x plus 6 is greater than 0. So, in this case, everything is on one side of the inequality, so we have nothing to do.

[00:00:15]Else, we have to move everything to one side of the inequality. Next is to what?

[00:00:18]Is to factorize. Now, we have to factorize it. This is a second-degree polynomial. So, I did show you the method in previous videos. So, we need to find two numbers that if you multiply them, will give me 6, and if we add them, they will give me minus 5. Those two numbers are minus 2 and minus 3. So, the factorization become as x minus 2, x minus 3 is greater than or equal to 0.

[00:00:37]Next step is to what? Is to find the critical values, which is very important when you deal with polynomial inequalities. The critical values in this case will be x minus 2 is equal to 0. Then, our first critical value is x is equal to 2. And the second one is x minus 3 is equal to 0. Then, you have x is equal to 3 as the second critical value. So, critical values does uh separate the line. So, we have two right here. We have three right here.

[00:00:58]So, everything that is less than 2. So, where x is less than 2.

[00:01:02]This is the first part, where we have x is between 2 and 3, which is the second part. And the third part is where x is greater than 3, which is here. So, these critical values separate the line in this way. Now, we have to test the equa- the answers for each one of the intervals. Okay? So, how can we test this? We take each part of the interval and we try it. So, let's say we have x is less than 2. We take x is equal to 0, and we test it out. So, 0 minus 2 times uh 0 minus 3. It will give me minus 2 times minus 3. It will will give me 6. 6 is positive, then it is true because we have here that it should be positive.

[00:01:36]Next, we test it out between 2 and the 3. Okay? So, between 2 and 3, we can take the number 2.5. So, 2.5 minus 2 is greater times 2.5 minus 3. Okay?

[00:01:47]Doing that will give me uh minus uh is equal to minus 0.5 plus 0.5 times minus uh 0.5, which will give me something negative. That's why the second one is not. So, we have this out of the case. So, if we have x is greater than three, we can take, let's say, four. So, for x is equal to four, x is equal to four here, x equal to three.

[00:02:06]Here we took x is equal to 2.5. Now, we do So, it's 4 - 2 4 - 3. So, it will lead to 2 * 1, which is 2, which is positive. Then we have it. So, we have this interval and this interval, okay?

[00:02:21]So, the middle interval doesn't work.

[00:02:23]So, the final answer is what? The final answer for this polynomial is these two values, which are We have that x is less than two and x is greater than three.

[00:02:33]So, the solution is what? The solution is x is less than two and x is greater than three. So, quickly for your exam, if you have polynomial inequality, never stop after factorizing. You must test the intervals to know where the expression is positive, where is it negative based on your initial expression. If you enjoyed these types of videos, please consider liking this video, sharing it with your friend, commenting down below down below what you'd like me to cover in next videos.

[00:02:54]It was Professor Kartush with you. We'll see you in the next one. Bye-bye.