Solve each polynomial inequality. Give the solution set in int...

Question

Solve each polynomial inequality. Give the solution set in interval notation. See Examples 2 and 3. (2x - 1)(5x - 9)(x - 4) < 0

Accepted Answer

Hey, everyone in this problem for the given inequality we're asked to solve and to write the solution set in interval notation inequality we're given is four, X minus three, multiplied by six X minus seven, multiplied by X minus three is less than zero. And we're given four answer choices. Option A and B are the union of two intervals. Option C is the entire real line from negative infinity to positive infinity. And option D is that there is no solution. So we're gonna start by rewriting what we were given or X minus three multiplied by six X minus seven, multiplied by X minus three is less than zero. And we have zero on one side, we have everything else on the other side. And on this left hand side, everything is factored nicely. OK. So we don't need to do that stuff that's already been done. It saves us a little bit of work. And this is the format we want this problem to be in. Now, when we have it in this format, what we want to do is identify our points of interest and our points of interest are gonna be any point where the left hand side is equal to zero. OK? We only have a numerator. We don't have a numerator and the denominator. So we don't need to worry about when the denominator is zero. In this case, we're just looking where this left hand side is equal to zero. Now, these are factors that are multiplied together. So if any of those factors are equal to zero, the left hand side will be equal to zero. So we get that X is equal to, from the first factor we have four, X minus three. If that is equal to zero, then X must be equal to three quarters. The second factor six, X minus seven, if that is equal to zero, then X is equal to 76. And finally X minus three, if that is equal to zero, then X is equal to three. OK? So we have these three X values of interest. And why are they of interest to us? OK. Well, think about on either side of zero, our function could be positive or it could be negative. OK? But at that value of zero is where that's going change from being positive to negative. Which means that that's where the left hand side is gonna change from satisfying the inequality to not satisfying the inequality. And so these are those points that's gonna happen. And so these points are gonna break up our number line into different intervals. So let's go ahead and draw a number line. And this is not necessary for this problem. But I think it's a really great way to organize our work and to kind of visualize what's going on here. And we're gonna put our points of interest on this number line. So we have three quarters, we have 76 and we have X is equal to three. And this might not be perfectly to scale, but we have those values on there in the correct order from left to right. And now we're gonna work from left to right. And we have our first interval where X is less than three quarters, we could choose a test point, say X is equal to zero. And we want to see how each of our factors behaves. So four X minus three, if X is less than three quarters is going to be less than zero to negative six, X minus seven is also going to be negative or less than three. And so is X minus three. So the left hand side of our equation, which we're gonna call L H S which is four, X minus three, multiplied by six, X minus seven, multiplied by X minus three is gonna be a negative multiplied by a negative multiplied by a negative, which gives us a negative value. So the left hand side is less than zero and that satisfies our inequality. That's exactly what we want. We want the left hand side less than zero. And so this interval is going to be a part of our solution set. It is gonna be included. We're gonna indicate that with the green checkmark on this interval. Now we move over to the right and we have the X volume of three quarters. OK. This is the end point for interval. And we need to decide whether that end point will be included or not. If X is equal to three quarters, the left hand side will be equal to zero. But this does not satisfy our inequality. Zero is not less than zero. And we have strict inequality here. We don't have less than our equal to. So zero is not going to satisfy this inequality and three quarters will not be in our solution set. So what we're gonna do is leave an open circle around three quarters to indicate that we do not include that end point. And now we move to the next interval and that next interval is from three quarters to 7/6. So we get, and you can choose a test point in there. We have the four X minus three is going to be positive. Now, it's greater than zero, six, X minus seven is still going to be negative or less than zero. And same with X minus three that is gonna be negative or less than zero. So we have a positive multiplied by a negative multiplied by a negative. The left hand side will therefore be a positive value, it's gonna be greater than zero. This does not satisfy your inequality. We want it to be less than zero. So we're gonna give a red X to this interval. This is not going to be included in our solution set. Moving to the right. Again, we get the 0.76 and just like for the other 0.3 quarters, the left hand side will equal zero that does not satisfy the inequality. So we want to exclude this point and we draw that as an open circle to indicate that moving to the right again, we have the interval from 76 up to three, choose a test point, say X is equal to two, four, X minus three is going to be greater than zero six, X minus seven is also going to be greater than zero and X minus three is going to be negative or less than zero. So we have a positive multiplied by a positive multiplied by a negative. That's gonna give us a negative value. The left hand side will be less than zero. OK. That's what we want that satisfies our inequality. So this interval is included in our solution set and we give it a green check mark. Now we need to check the other end point. We've already done the 76. Let's check the end point at X equals three and it's gonna be the exact same as the other two. OK? The left hand side will be equal to zero, that does not satisfy our inequality. So we use an open circle to indicate that we do not include this point in our solution set. And finally, our last interval when X is greater than three, again, you can choose any test point, say X is equal to four, four, X minus three is going to be positive greater than zero six, X minus seven also positive and X minus three is also going to be positive. So we have three positive values multiplied together. That's gonna give us that the left hand side is greater than zero and it's positive, which is not what we want does not satisfy the inequality. So we give this interval A red X, it is not going to be included in our solution set. So we're gonna go ahead and write this out in interval notation. But first, we're gonna eliminate the two obvious choices that we can eliminate. Option C is the entire real line from negative infinity to infinity. We know that that is not correct because we have these intervals with the red XES that we said were not included. OK. So we were not including the entire real line. So we can eliminate option C, we can also eliminate option D because we did find two intervals where the inequality was satisfied. So there is a solution and so we're left with option A and B. Let's go ahead and write up the interval notation. To figure out which one it should be. Now, the first interval we have is everything to the left of X equals three quarters. So that's gonna be from negative infinity all the way up to three quarters. And we're gonna use round brackets because we do not include the end points. Ok. All right. So, option A and B both have the same interval for their first interval in the union. And so we can't eliminate either one yet. We're gonna take the union with the second interval we found and that second interval goes from 76 up to three, not including the end points. So round brackets again, 76 up till three. And now we can see that this is gonna be option A, option B has that second interval going all the way to infinity, which is not correct. And so the solution to this inequality was the union of the interval from negative infinity to three quarters. The interval from 76 to 3 where none of the end points are included, which corresponds with answer choice. A thanks everyone for watching. I hope this video helped you in the next one.

Solve each polynomial inequality. Give the solution set in interval notation. See Examples 2 and 3. (2x - 1)(5x - 9)(x - 4) < 0

Key Concepts

Polynomial Inequalities

Critical Points and Sign Analysis

Interval Notation

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