Solve each inequality. Give the solution set in interval notation...

Question

Solve each inequality. Give the solution set in interval notation. (x-4)(x-1)(x+2)>0

Accepted Answer

Hey, everyone in this problem, we're asked to provide the solution set for the given rational inequality in interval notation. The inequality we're given is X minus 13, multiplied by X minus 11, multiplied by X plus 21 is greater than zero. We're given four answer choices A through D that each contain the union of two different sets. And we're gonna come back to these as we work through the problem. Now, when we're given an inequality like this, a rational inequality, the first thing we wanna do is get all of the values on one side have the other side equal to zero. OK? And then we want a factor. And what you'll notice is that this has already been done in this problem. OK. So this saves us a lot of work. We have X minus 13, X minus 11, X plus 21 all three of those factors multiply together. And that's greater than zero. We want to think about equality. OK? If this was an equation and we had an equal sign instead of a greater than sign, OK, we would have three different solutions X equals 13 coming from this first term X equals 11 coming from the second term and X equals negative 21 coming from this third term. OK? Setting each of those individually equal to zero. So these are the three points that we want to look at. OK? These are not solutions but they're points of interest. OK? And we can imagine that these points where each of these factors equal zero is we're gonna change over from these factors being positive or negative. And that's gonna affect whether we have a solution or whether we don't have a solution. So what we wanna do here, we're gonna draw a number line. So we have our number line and we're gonna put these three values of X on that number line. So on the far left, we have negative 21. OK? We have our zero and we're gonna have 11. So we're gonna have 13 and our number line might not be quite to scale, but these numbers are in the correct order and they've left a little bit of room to do the work. Now we're gonna look at each of these intervals. So the first one we're gonna look at is when X is less than negative 21 when X is less than negative 21 X minus 13 is gonna be negative. OK? Less than zero, the factor X minus 11 is also gonna be less than zero. The factor X plus 21 again, less than zero. So we have three negative values when we multiply a negative by a negative by a negative, we get something that is negative or less than zero. And so we're gonna say that L H S the left hand side of our inequality is also gonna be less than zero. What that means is that for values of X less than negative 21 we do not have a solution. OK. This does not satisfy your inequality because we want values greater than zero. So we're gonna draw a red X in this interval that does not satisfy our inequality. And we move to the next interval and that next interval is between negative up to 11. OK. In this interval, we have X minus 13. Now, in this interval, we can imagine being at zero, if we substitute an X equals zero, we have a negative value for factor X minus 11. This will also be negative. And for our factor X plus 21 this is gonna be positive or greater than zero. So a negative multiplied by a negative gives us a positive and then multiplied by a positive, gives us a positive. So the left hand side of our inequality or this interval is gonna be greater than zero. It's gonna be positive, which is what we want. That means that this interval is a solution to our inequality. So we're gonna give it a green checkmark. Now, what we have to do in addition to finding this interval is figure out if the end point should be included or not, what happens when X is equal to negative 21 or X is equal to 11. In those cases, the left-hand side is equal to zero. But our inequality is strictly greater then OK. So if the left hand side is equal to zero, that does not satisfy our inequality zero is not greater than zero. So we don't want to include the end points. So we're just gonna draw open circles at either of those end points to indicate that we're not gonna include them. OK? And that's just a reminder to us. So that when we go to write out these intervals in interval notation, we remember what we're including next interval between 11 and 13, X minus 13. That's gonna be less than zero X minus 11. That's gonna be greater than zero. And X plus 21 is also gonna be greater than zero, negative multiplied by positive, multiplied by positive is gonna give us a negative value. So the left hand side of our inequality is gonna be negative, which means that this inequality is not satisfied. So again, a red X on this interval that does not satisfy our inequality. And we have one interval left to look at. And that is the interval where X is greater than 13. If X is greater than 13, then X minus 13 is positive greater than zero, X minus 11 is also greater than zero and X plus 21 is also greater than zero, positive multiplied by a positive multiplied by a positive, gives us a positive value. So the left hand side is gonna be positive or greater than zero, which satisfies our inequality. And so this interval gets a green check that's gonna be included in our solution set. Now, one more thing to do is to figure out this end point of 13, OK, whether it's included or not. And again, if X is equal to 13, we get the left hand side equal to zero, which does not satisfy your inequality. So we're gonna draw an open circle there as well to indicate that we are not going to include that point. So when we write out our intervals in interval notation, we found that the intervals that satisfy our inequality are from negative 21 up to 11. OK? And we're gonna take the union of that with this second interval we found which is from 13 up to positive infinity. And we are using round brackets for all of these values because none of these end points are included. Now, if we look at our answer choices, OK, based on the first interval, we can already eliminate options A and D because they go from 21 to 11 instead of negative 21 to 11. OK? So we're left with option B or C. Option C has the second interval from negative 13 to infinity which is not correct. And so the correct answer here is option B we have the union of the two sets. The first from negative 21 to 11 with the second from 13 to positive infinity. Thanks everyone for watching. I hope this video helped see you in the next one.

Solve each inequality. Give the solution set in interval notation. (x-4)(x-1)(x+2)>0

Key Concepts

Inequalities

Factoring Polynomials

Interval Notation

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