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Homework: Section 4.7 Polynomial & Rational Inequalities Score: 0 of 1 pt 12 oi 12 (11 complete) 4.7.47Solve the inequality algobralcally:The solulion (Simplily...

Question

Homework: Section 4.7 Polynomial & Rational Inequalities Score: 0 of 1 pt 12 oi 12 (11 complete) 4.7.47Solve the inequality algobralcally:The solulion (Simplily your answer Type your answer intorval nolatlonUse inlegers (racuons for any numbers Ina axpression )

Homework: Section 4.7 Polynomial & Rational Inequalities Score: 0 of 1 pt 12 oi 12 (11 complete) 4.7.47 Solve the inequality algobralcally: The solulion (Simplily your answer Type your answer intorval nolatlonUse inlegers (racuons for any numbers Ina axpression )



Answers

Solve each polynomial inequality in Exercises $1-42$ and graph the solution set on a real number line. Express each solution set in interval notation.
$$
(x+1)(x-7) \leq 0
$$

All right. So we already have this when in fact, it form the first thing we need to find our critical points or what we can use. Get each of these factor stick reserve through a negative one. There you have positive seven. And those should be the two points that were you to do our interval attack. Syria From negative infinity to negative one. We have from negative 1 to 7. We have seven to infinity. Right from negative. Infinity to negative one are choosing negative to negative two plus one Negative. Negative. Two minus 17. Negative. Very coarse. Positive. And we're looking for here. Last in record to zero. You were looking for a negative. Did not work, huh? That the network It's a negative on the seven. I do zero. Is there a place you want to be positive? Is there a minus 73 negatives? We combine them, we get a negative. So we're like that one. I use it for the next 28 plus one is positive. Eight minus seven spices. We end up getting a Peiser. If we don't like that one, it should travel We like It's from negative 17 Here and because they left their equal to, we can include both of these. And then our number line will look like this. We had a negative one. We have positive. Seven. We can fill in everything. Um, close third works because we can include them and our solution stated everything in between. There's two numbers.

So in this problem, we are as to find the solution set to this inequality. So first, we must find the critical points, or basically, when this expression is equal to zero. So this happens when X equals seven and X equals negative three. So that sets up our main intervals that we need to test, which will go from, uh, negative infinity to negative three negative, 3 to 7 and seven to infinity. So, from negative infinity to negative three, we contest the point X equals negative for which would equal s. So if we evaluate this expression, we get negative 11 times negative one which is equal to 11 which is not less than or equal to zero. So that part of the interval doesn't work or that interval doesn't work from negative 3 to 7. We can track X equals zero. So that would evaluate to, um, negative seven times three, which is evil, the negative 21 which is less than or equal to zero. So that would work. Um, And from for the interval from seven to infinity, we contest the point X equals eight s, so that would be eight minus seven equals one and eight plus three equals 11. So that would be not less than or equal to zero. So that would not work. So now we know that the only interval that works would be from negative 3 to 7 inclusive. So that is our answer. And if we were to sketch this on the number line, we would have our critical points. Negative three and seven. And our range will be from negative three inclusive. Which is why I have the clothes circles to seven inclusive. Thank you.

Okay, so we have, um, fat in factor form already. So the first thing we need to do, so we need to find our critical point. Essentially. What can we plug in fact equals there for Beth. Elise, you're released One. We get seven for this when we get negative. Three to the work of the two critical points that we have and that we should the two points that we want to make for our intervals when we do our interval. So we have from negative infinity the negative three. We have a negative three to positive. Seventh. We have from seven to infinity. You know, we just pick a number and plug it in there. I remember we're looking for this one to be less than or equal to reserves. They were looking for a negative number here. All right. From negative. Infinity and negative. Three. I'll use negative four. Negative for minus seven will give me a negative Negative for minus. Plus three will give me a negative. Combined to give me a positive. I guess we're looking for a negative. We don't like that one. Ah, you're zero for the next interval. Is there a minus? 77 Negative seven. Is there a place stories they positive become bound Then we get a negative here. We like that interval from seven to infinity. I use a eight. Minus seven is positive. Eight plus three is positive. We end up getting a positive. We don't like that one. The only never that works your negative 32 positive seven Because it elected are equal to we can include the negative three and the positive seven. And then when I would make my Leinart Here we have negative three. We have positive seven. We can include them both. We have close circle and everything in between. Negative 37 including mega during seven.

So we're going to be solving this inequality. Excuse for seven X squares. One expects 17 is less than zero. Uh, can I factories? But I see a seven here on the seven hit. So I'm gonna try working with that guy taken X squared. That's my busty terms. I get experts seven. And yet if I take a minus one out my 2nd 2 terms, I get expose seven as well. So that's great. I can't back to his office into X squared minus one times X plus seven on dhe. So it's you find my boundary points. I'm gonna pretend this is equal to zero instead of less than which gives me my boundary points here as being X equals 17 and for this one ISS minus one as well. So I have three boundary points. I'm going to keep the prince sees version the factory explosion. That's gonna be, uh, useful later. The numbers throughout my real number line and now looking more run of the line, I have minus seven minus one and one. So I have four. Interval, sir longer and see which of these four intervals it's this inequality of less than zero. So let's do some test values up. For my 1st 1 of the news minus 10 anything less than minus seven will work. So if I look at this, I get positive on this. Prince sees on negative on this prince sees. So overall, I get negative Tip this one I'm gonna use minus two. What? His guns give me four months when it's positive. And this is all supposed to be Something's gets overrule a positive result. Yeah, well, you zero. So I get monos one is negative and seven is positive. So, like next result. But to get positive on this, Prince sees Onda posted on this parentheses so ever. I guess, of course she'll have done have not actually calculated. I've just because almost replying to Princess Cavor, I'm just looking at the signs. But now I can see which of minds fools Lesson zero It's this one. This one's negative and small second, because it's less about it's not less than and equal. These are open sets. This boundary points aren't included in the solution. Me Just get rid of the unnecessary information here so you can see here a solution number line or if I write out the solution sets, I get it minus have been seen to my seven and this is an open set on. I also get minus 1 to 1. Also an open set. That's my solution.


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