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Use the Newton'$ method to find the all solutions to the following equations:1=1+2Hint: One of the solution is in the range -1.5 < x < -0.5 and the other...

Question

Use the Newton'$ method to find the all solutions to the following equations:1=1+2Hint: One of the solution is in the range -1.5 < x < -0.5 and the other solution is in the range 0.5 < x < 1.5_

Use the Newton'$ method to find the all solutions to the following equations: 1=1+2 Hint: One of the solution is in the range -1.5 < x < -0.5 and the other solution is in the range 0.5 < x < 1.5_



Answers

Solve each inequality numerically. Write the solution set in set-builder or interval notation, and approximate endpoints to the nearest tenth when appropriate. $$ -5<2 x-1<15 $$

We need to solve the following. So we need to get the X by itself. So we're going to multiply both sides by five to get rid of the denominator, The fires cancel out. And we're left with three X -1 is less than 2. 15 times five 75. We're gonna add our one to both sides And we get three X is less than 76. We're going to divide both sides by three When we get 76 divided by three. So acts is less than 76 divided by three. And that doesn't become a nice hole number. So we're going to change that to the nearest 10 so it's going to be 25.3. So if we grab this, We would have 25.3, 25.3 is not included. So it has an open circle X has to be less than that. So that means everything to the left of that would be part of the solution. So using interval notation, that would mean from negative infinity all the way to 25.3, and we would have parentheses for both of them. Using set builder notation would have X such that X is less than 25.3.

In this problem, the inequality which is given to us is one less than model X less than five. Let us break it down into two parts first as one lesson monarchs. This using the second property of the absolute value can be broken us either X Less than -1, or X is greater than one, so we get x- Infinity Still -1. Union 12 Okay. Second part in which this Brokenness monarchs less than five. By using the first property of absolute value. This will open like this or X belongs to minus five 25 So this is one and this is mm So the X which satisfies our original inequality, Lying between 1-5 will be the X that satisfies Both one and 2 or which is an intersection of one and 2. So let us find the intersection of one and 2. In perfectionist X belongs to -5 till -1 and the unions 125 These are the values of X always the inequality goods.

Already. So number eight here's asking us again to determine which elements of s and they give us to set s satisfies this inequality. Here again, the first step is to solve the inequality. Ah, this one. As I said in my previous video, dividing by a negative can't be troublesome because it leaves room for you to make a mistake. But in this case, there's not really we don't really have an option. So the first step is to actually subtract three from every single expression. Here, subtract three and we get negative. Five is less than or equal to negative X, which is less than able to negative one. And once again we've isolated everything. Are we at least the middle term in terms of one term of X, which is used to the goal as it is in regular algebra o r. Regular questions. So that rule also applies the inequalities, and then we divide everything by negative one. And remember that when you divide or multiply both sides of an inequality by negative number, you have to flip the sign. So here we're gonna make the signs flip. So as you can see here, sign flips and we've got less unable to civil go greater than equal to We have X one and five and then this looks kind of awkward, right? Because we're going from of we're going from large to small, smaller numbers rather than smaller and larger. So just to keep our conventions, we're going to flip this around Flip, we're going to go. One is less than X, which is less than or equal to, but ah, now we have to determine which elements of s set s satisfies this inequality. This one is pretty straightforward. Negative one is tricky. I know everyone. Never negative one. It's not tricking my apologies. One is tricky because it looks like it does. But remember, this is not less than or equal to, because if it was less than or equal to, one would work. However, one does not work. We're going to vaccinate this. Anything that's less than one also doesn't work. Sorry. Thats why I almost confused with negative one. A cz. We can see anything. That's lesson one is nowhere because one if it's one that's it is less than one. It doesn't fit inside this this space here, right, cause it X has to be greater than one. So all of these guys, all these guys do not work. And then we're left here, left with these guys. Route five. Ah, again, I don't have a calculator, but I know that Ah, two squared is for and three squared is nine. So it's going to be somewhere between two and three, which is obviously somewhere between one and five. Route five works the square to five. My apologies. Using maths Ling here three obviously works is between one and five and then five works because it's less than or equal to fight. This right here tells us that five can work because it is less than or equal to five so square with a five, three and five all work.

All right. So we need to solve the inequality. The absolute value of X is less than five. So, um, first, we're gonna look at something a little bit easier to deal with. We're just gonna look at the up. The Inequality X is less than five is the absolute value. Just complicate things just a little bit. So looking at X is less than five. This becomes a lot easier exits, just anything that is smaller than five. So plugged in one that would make this true. It's so if X was one, I would have one is less than five. And that's a true statement. If I wanted to plug in negative 10 Brits, I would get negative tennis less than five. And that's also a true statement, so I can plug in anything smaller than five. If I were to try to plug in five, I would get five is less than five, and that is not true statement. So that's not a solution. If I would've plugging anything bigger than five like, let's say, 10 ever get 10 is less than five, and we know that's not true. Turn is definitely bigger than so. That is not a true statement. So looking at this inequality, I can see I can plug in any number that it's smaller than far and things will work out. Now let's take a look back at the absolute value of X is less than five. So when we're talking about the absolute value, remember, all that means is the distance from zero. So they're saying that the distance from zero of ex needs to be less than by. So let's take a look back at the things that were working or X is less than five only played one in that worked. Will that still work for this? Uh, um, absolute value inequality. But club one in for X I would get the absolute value of one is less than five. Well, what is the absolute value of one? The absolute value is just how far away zero is It? That's only one and one is less than five. So there's still works. I can definitely flood one in. I also loved a negative 10. Let's check and see if that would work well, the absolute value of negative 10. It's just how how many spaces away from zero is negative 10. It shouldn't 10 spaces, and then I get the inequality. 10 is less than five, but that's not true. This is not a true statement. I can't flood negative tenant, so let's think back to the meaning behind the absolute value. It's, uh, ex can be less than five spaces away from zero. So if we look at a number of line here zero If I started a venture to write 12345 spaces, I land on five and I know that X must be less than five spaces away. So anything in here the war in the to the three before anything in between those numbers. All of this is less than five spaces away from zero. Everything except for the five five is the only thing that is not less than parks faces away because it is far spaces away. And then I can also go on the left as well, though I can start it zero when I can move 123 or five spaces way that takes me to negative five, and then anything that is between negative five and zero all will be use our old Liston bar spaces away zero everything except for negative five, so that it seems like our solution is that X has to be bigger than negative five. It can't be equal to negative. Five has to be bigger than negative five. And it also has to be smaller than positive by Campbell's Think of negative five and positive five as walls, and our solution is just everything inside of that of those walls. So their directness is an inequality. As we've said in words, X must be bigger than negative five. So try that as an inequality. I can say negative vibes less the next, because that means the exact same thing as X is bigger than negative, all right, and then I know that X has to be smaller in positive five. So that's exactly what this says. I can see that X is bigger than negative five and X is smaller than positive body. So this is our solution


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