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Questionof 14 StepFind the solution of the following polynomial inequality: Express your answer in interval notation;Jxe > 2 +5AnswerPoints...

Question

Questionof 14 StepFind the solution of the following polynomial inequality: Express your answer in interval notation;Jxe > 2 +5AnswerPoints

Question of 14 Step Find the solution of the following polynomial inequality: Express your answer in interval notation; Jxe > 2 +5 Answer Points



Answers

Solve the polynomial inequality and express the solution set in interval notation. $$ y^{2}<-4 $$

So we want to solve this inequality and we'll use the zeros to help us do that and I'll make sure I wrote this down correctly. I'm looking at this. Yeah babe, I thought that was a cube there a second as I'm saying that's not going to be very good to solve but there's our inequality so let's set it equal to zero. Oh and that won't change the sense of the sign. And now we're going to factor and let's find what the zeros are. So we have two X squared times X squared. And let's see we need a negative five for miller terms. Let's put a three there on a one there and negative and a positive. And this will factor appropriately. Now this factor can never equal 02 X squared plus one can never equal zero. So the only place where we can have an equal zero is when this equals zero. And so that is going to happen at plus or minus the square root of three. And we know that those values can't be used because we want it to be strictly greater than zero. But let's get our little sign chart here and let's plug into values. Well let's just do this middle one, let's plug in zero by plug zero in. I'm going to end up getting something positive and if I plug zero in here I get a negative and so the product of those two will be negative. So this whole region, anything input between these values will result in a negative value. Those are not solutions. Now let's plug in something that is down here like let's say like a negative two is smaller than that. So if I plug negative two in here I'm going to get something positive here. And if I plug in negative to hear it gets squared and it becomes something positive so that will result in a product of something positive. So all these results will be positive and now if we plug in likewise something that's greater than two, Excuse me? Greater than square 23. Like to buy plug to in here, I get something positive and if I plug to in here I also get something positive and so regions here and here will give us a solution set that has a value greater than zero. So we know that negative infinity up to but not including that negative square root of three because that's what makes it zero, Also positive square root of three, not including it must be bigger up to positive infinity and there is our solution set. Yeah, yeah.

In this activity we're being asked to find the solutions to an expression B squared plus four P. Is less than -3. This is um asking what are all the values of P. That would make this whole expression truth. But this expression the way that's written right now is going to be a little awkward to figure out. So let's just move the three to the other side of the inequality so that I have P squared plus four. P plus three is less than zero. I've just added three to both sides. That way I can figure out where P will make this whole expression equal to zero. And then it can figure out what intervals between those values. Make the original expression true or make this expression positive or negative. So the first thing I notice is That I can factor this two. p plus three times he plus one. Mhm. Let's first of all figure out Where this is equal to zero. Well it's a product P plus three Times P Plus one. To make a product equal to zero. I just need one piece to be zero. So if P plus three equals zero The whole expression will equal zero. Or if P plus one equals zero The whole expression will equal zero. So if P equals either negative three or negative one then it would make this expression equal to zero. It would make this expression equal to -3. I don't want when the expression is equal to negative three. I want when the expression is less than negative three or when mhm P squared plus four. P plus three is less than zero. So let's quit Where we know the expression equal zero on a number line Equals 0 -3 Or at -1. And we're going to test values less than negative three and see if that makes the expression negative. We're going to test values between -3 and -1. See if that Makes the expression less than zero. We're going to test values that are greater than negative one and see if that makes this expression positive or negative. And figure out what values of P. Give me Valid solutions. So starting with values less than -3. I can use any number so an easy number would just be negative for he is a whole number that's in that interval. If I plug negative for in or P here I will get negative four plus three is a negative value -4 Plus one. Is a negative value a negative times a negative is a positive that's greater than zero. Yeah I want values that are less than zero. So that interval less than -3 is not part of my solution. Yeah Values between -3 and -1. Let's try negative too. If I plug negative too in here for p negative two plus three is a positive one that's positive. Yeah negative two Plus one is a -1. That's negative. A positive times a negative is negative. That's less than zero. I forgot to write my zero here. Yeah that's less than zero. That's what I want values that make this expression less than zero. So that is true. That will be part of my solution values that are between -3. That's I don't know what I drew. Their values that are between a negative three and negative one but not including negative three or negative one are part of my solution. And I know that this is a problem. So I know that the last interval isn't going to be part of the solution but let's check anyway. A number that's greater than negative one that's easy to use. 00 plus three is three that's positive. Yeah. zero Plus 1 is one that's positive. A positive times A positive still gives me a positive that is greater than zero. Greater than zero is not part of my solution. Yeah. So only values that are between negative three and negative one But not including -3 or -1 will be solutions to this inequality. Mhm.

Here we are given with the inequality as X scare plus three x less than equal to six, we can rewrite this equation as X care plus three X minus six less than equal to zero. Now, to find out the solution off this equation, we're using quadratic formula bitches. X is equal to minus B plus X is equal to minus be less minus on the road. Be scared minus four. Is he upon the way so we can find out the values off ABC by comparing the give any question. By general Equation, which is a XK plus, BX plus e is equal to zero. Now I'm just substituting the values and finding out the solution for X X will vehicle dough minus three less minus B squared, minus four. A civil vehicle toe 33. I'm just writing out 33 over here born toe. So there will be two values for X, which are ex physical do ministry, minus through 33 over two earned minus three less Route 33 over to these tools. Value are estimated as minus 4.37 earned fun 0.37. So we're putting these numbers on number line. But these boundary points here we have minus 4.37 or we can save minus three minus rule 33. But go on, be here. We have one burned 37 which can be served as ministry plus room 33 to now we have three intervals were taking one point from each interval were taking minus six from first interval. We are taking minus four from second interval on. We're taking toe from the third interval. Now we're going to check the inequality on these three points. So let's start with minus six substituting minus six. In the equation, we will get 36 minus 18 minus six less than equal to zero. So 36 minus 13 minus 26 minus six will be equal to 12. So 12 less than equal to zero, which is false. Completely not taking it. X is equal to minus four, substituting minus four in the equation here. So we get 16 minus 12 minus six less than 1/4 0 so minus 13 plus 16 which will be called minus two as an equal to zero. So the question is true here, not taking the third guest X is equal do such that you're going to in the question we get to scare really four plus six minus six as the nickel to zero. So we are left with four less than equal to zero, which is foods. So we have a inequality. A true only in the interval from minus 41 37 to 1.37 or we can say it from minus three, minus rotor fatally by Tau Tau minus three plus through 33. Where to now, as the boundary points followed, the in question or these boundary points holds the inequality True. So we're going to include these boundary points into the interval so we can write out the solution centers minus three minus Route 33. But I do minus three plus road 33 by two. We're showing that these points are included in the set. We have used scared records. That's it

Given to X -1 times three X minus two times four X plus seven is less than zero. We're going to solve the polynomial inequality and give the solution, set an interval notation. So to do this, we're going to take each one of these factors. So for example, to explain this one and set it equal to zero. Because each of these zeros is where this function is going to change direction. If we were to graphic, I have two, x minus one equals zero, 3, -2 equals zero and four, X plus seven equals zero. I'm going to start over here with the two X -1. We had two x equals one And x equals 1/2 On this one, I get three, X equals to divide both sides by three. I get X equals two thirds on this one. Subtract seven from both sides. I get four X equals negative seven, Divide both sides by four, and I get x equals negative 7 4th. So for this one, our interval notation is gonna be negative infinity, comma negative 7/4 negative one half and two thirds. And this is where this function is less than zero. When x is negative infinity to negative 7/4 and then again, when it's between negative one half and two thirds


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