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

Question

Solve each polynomial inequality. Give the solution set in interval notation. (x - 4)(2x + 3)(3x - 1) ≥ 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, we have X minus three multiplied by four, X minus three, multiplied by six X plus five is greater than, or equal to zero. We have four answer choices. Option A and option B are the union of two different sets. Option C is the interval from negative infinity to infinity. And option D is that there is no solution. So we're gonna start by writing at our inequality, we have X minus three multiplied by four, X minus three, multiplied by six, X plus five is greater than, or equal to zero. We're gonna notice that this is already factored. We have zero on one side, we have a function on the other side that is completely factored. OK. So that saves us a little bit of work. This is the format we want it in and now we're left with finding the points of interest and the points of interest for this function. We're gonna be where the function on the left-hand side is equal to zero. Hm So these are factors multiplied together if any single, one of them is equal to zero. The entire left hand side will be equal to zero. And so we can take them one at a time. Looking at the first one, if X minus three is equal to zero, then X will be equal to three next one. If four, X minus three is equal to zero. OK, X will be equal to three quarters. And finally, if six, X plus five is equal to zero, X will be equal to negative +56. And we move that five to the right hand side by subtracting and then divide by six. So we have these three X values of interest. And why these are of interest is because that's where the function on the left hand side will be equal to zero. And we know that on either side of zero, the function could switch from being positive to being negative. OK? So these are the points where that function could switch from positive to negative, which is going to change whether the interval is a solution or not. So we're gonna go ahead and draw a number line and these three points are gonna break that number line up into four intervals. So on the left of our number line, we have X equals negative 56. OK? Remember numbers further to the left are nega more negative or less and moving to the right, we have three quarters and moving further to the right, we get X is equal to three and this is not drawn to any particular scale, but we just want to indicate those values on our number line. OK? And now we're gonna just take interval by interval and figure out whether this is a solution or not. Now, what we're gonna do first is we're gonna take the left hand side of our inequality which is X minus three multiplied by four, X minus three, multiplied by six X plus five. And we're gonna call this L H S. This is the left hand side of our inequality. So we're gonna label it L H S just so it's easier to refer to. Now, starting on the left of our number line, we have values X values that are less than negative 56, we can take a test point, say negative one. OK. If we substitute that into each factor X minus three is going to be negative less than zero or X minus three will also be less than zero and six X plus five will also be less than zero. OK? So the left hand side is gonna be a negative multiplied by a negative multiplied by a negative, which will give us a negative value. So the left hand side will be less than zero. Now, our inequality wants it to be greater than or equal to zero. So we can eliminate this interval. This does not satisfy the inequality. So we're gonna draw a red X there. Now he moved to the right, we're gonna get to the point negative 56 A X is equal to negative 56. When we substitute that into the left hand side of our equation, we get that the left hand side is equal to zero. OK. Now our inequality is greater than or equal to zero. So having this equal to zero does satisfy our inequality. So this is going to be a solution. So we're gonna draw a closed circle over that point to indicate that we are including that. And it's just a reminder when we go to write this in interval notation, not to forget these points moving to the right. Again, we have the interval from negative 56 to 3 quarters. We can use a test point say X is equal to zero. If we do that X minus three is going to be less than zero, negative four, X minus three will also be less than zero six X plus five. However, is going to be positive. Now. So the left hand side will be a negative multiplied by a negative multiplied by a positive. This is gonna give us a positive value. The left hand side is gonna be greater than zero, which is what we want. This satisfies our inequality. So we're gonna give a green check mark to this interval. Now we move to the right. Again, we have the point X is equal to three quarters and just like with X is equal to negative 56. This makes the left hand side equal to zero, which satisfies the inequality. So we're going to do a closed circle here to indicate that we include this point moving to the right again, X values between three quarters and three, you can choose a test point, say X equals one whoops X minus three. It's going to be less than zero. Still, in this case, four, X minus three is going to be positive and be greater than zero and six X plus five is also going to be greater than zero. So now we have a negative multiplied by a positive multiplied by a positive. The left hand side will be less than zero. And so we're gonna give a red X to this interval as well. This does not satisfy our inequality. Moving further to the right. We have the point X equals three and just like the other two points X equals three causes the left-hand side to be equal to zero that satisfies your inequality. So this is included and we're gonna do a closed circle there to indicate that. And now we're on to our final interval. We have X is greater than three. If X is greater than three, then X minus three will be greater than zero four, X minus three will be greater than zero and six X plus five will also be greater than zero. So we're multiplying three positive values together, that's gonna give us a left hand side that is greater than zero, which satisfies the inequality. And so this interval is included as well. We give it a green check mark. All right. So we can already eliminate two answer choices based off of what we've just done. OK. We know that we can eliminate option D, no solution because we have found solutions. And we can also eliminate option C that says that we have uh the solution from negative infinity to positive infinity because we've had two intervals that we've had to exclude. And so we're left with A and B, both of them are two intervals. So let's go ahead and write out the intervals that we've found. And then we can compare those with the answer choices. The first interval in our solution set is from negative up to three quarters. And we're gonna use square brackets because we include the end points. OK. Those end points are solutions. They're included. We use square brackets. We're gonna take the union of this interval with the second interval we found which is the interval from three, up to infinity. OK. Anything greater than, or equal to three. And we use a square bracket on the three to indicate that we include that value as well. And so this is going to be our solution set for the problem. We have the union of the interval from negative 56 to 3 quarters with the interval from three to infinity where we include those end points and this corresponds with answer choice B thanks everyone for watching. I hope this video helped see you in the next one.

Solve each polynomial inequality. Give the solution set in interval notation. (x - 4)(2x + 3)(3x - 1) ≥ 0

Key Concepts

Polynomial Inequalities

Critical Points and Sign Analysis

Interval Notation

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