solve-each-inequality-for-x-a1-e3-x-12-quad-text-b-1-2-ln-x-3

Question

Solve each inequality for x.

$$(a)1< e^{3 x-1}<2 \quad 	ext { (b) } 1-2 \ln x< 3$$

Carlos Pinilla · Accepted Answer

um so you won&#x27;t do solo. Gina Kuwaiti for X must&#x27;ve broke the story. There is one so, huh? So in the guarantee for X, we would mean too get an equal 18 damsel picks. So Well, we have that here. Exponential. So Well, there is that natural Originally a universal explanation. Number is two. Ana. Well, uh, so you&#x27;re more, uh, actually, there is an increasing function. Uh, so do you have some values, eh? The more than eight a plane natural love it. You&#x27;ll also preserve, then quit. You got these being matter a little for service. The nobody under is that it&#x27;s been important because we&#x27;re playing Martha Louise. Well, perhaps you don&#x27;t get the integrated. Well, in this case, we do, because matter, aluminum is increasing. So is, like, what? You have something that he&#x27;s increasing. Uh, so if, uh, a this is what I&#x27;m being I mean, you play the function Well, then also a pretty This is more than me. Hey, so Well, we&#x27;re gonna play. That&#x27;s our logarithms. Over. Get natural. Oh, Reum off one. His water and natural rhythm off, But about three weeks. What mortar? That&#x27;s a little too. So for his number. Er, that is zero. He&#x27;s one turns into zero Natural. Oregon is the English of explanation so that you only get what do you see? Exponents get just that three X minus one. Then what do you have There? Something there&#x27;s of Ah, well, we can do better. We can divide by three once he&#x27;s, uh for it&#x27;s positively so. My wife we hear does nothing subzero lose them eggs my nose. 1/3. Liz Lemon, I don&#x27;t want to Anybody like three so well to get on like, um, actual anybody before for ex weakened Just up 1/3 to every side of the equation with gravy. This will give you money, liquidity and the other 1/3 many people have preserved. So you give, you get my third. That is mortar. The Knicks. These 1/3 consuls we&#x27;ve done once heard. There you have. It&#x27;s morning. Actually, two waited, like three plus 1/3. So this would be your first. What works? No. Here. These one, uh, or or B? You&#x27;re saying one of these, So, uh, well, it&#x27;s let&#x27;s have to matter a living will, thanks to both sides of the equation. So here I&#x27;m there. So are you. Toe my turn. Local banks. Both sides do not give you two or the So I give you one, they to matter a little back schools away and then always decide We will help that three plus two after a little eggs. And Dana Well, it comes Obstruct three recital equation, but this goes away and then go in there we would have I was three was one that&#x27;s minus two when it still would be smaller boys eggs. So well, we&#x27;re gonna divide my till. So that was away. Hey, someone&#x27;s wonder. So then you would get, uh But, uh, really, the more face must be greater minus one. So or you raised the bar off equation? Well, you it&#x27;s also increasing. He looks like So is he gonna preserve? It&#x27;s increasing, so it preserves its quality from the then we&#x27;ll get He did the miners one. This is Moeller, either Natural living. Well, thanks Very six. Well, he can write that us one away. So your ex was b big eared on one word

Heather Zimmers · Answer

okay to solve these inequalities. I&#x27;m going to start with the first inequality and I&#x27;m going to take the natural log of all three parts. So we have natural log of one is less than natural law gov to the three x minus. One is less than natural log of to now in the middle natural log of Ito. A power is just going to be that power because natural log and each of the power or in vs and they cancel. So in the middle, we just have three X minus one Now on the left side. We know that natural log of one is zero, right, because each of the zero is one and then on the right, we just have natural log of to Okay, let&#x27;s go ahead and add one to all three parts and we get one is less than three. X is less than natural log to. And then we&#x27;re going to divide all three parts by three and we get one. Third is less than X is less than natural log of two divided by three. Now let&#x27;s work on the other one. So let&#x27;s start by subtracting one from both sides and we have negative to natural. Log X is less than two now. We divide both sides by negative to and remember with inequalities. When you divide by a negative, you have to reverse the signs. So now it&#x27;s a greater than Okay, finally, let&#x27;s exponentially eight. Which means each side becomes an exponents. Let&#x27;s make each side and exponents on E so we hav e to the natural log of X is greater than e to the negative one e to the natural log of X is just X. Since inverse functions cancel so X is greater than e to the negative one. And if we want to, we can write that as one over e.

Julian Wong · Answer

problem. We want ah to expand the left side so usually two x minus three e to the X minus two e to the X plus six less than two e to the X. So grouping all the like terms will haveto the two X um minus five minus seven. You to the ex What? Six less than zero? Uh, so maybe simplify the problem a little bit. Let why equal e to the X then we&#x27;re really trying to solve a quadratic. So, uh, think about this quadratic we would have. Why? Minus six. Why minus one less than zero. So that&#x27;s telling us that we&#x27;ve got a quadratic with X intercept or in this case, why intercept of Ah, really? The X intercept? I&#x27;ve just called it. Why one and six? This is a problem that opens up. So we want the stuff in between because that&#x27;s where it&#x27;s less than zero. So, in terms of why we would be talking about in between one and six. But why is actually e to the X? So we know e to the X. They all write it down in between. So one has to be in between each of the access to be in between one and six. We&#x27;re not solving for each of the extra solving for X, so we gotta sell of each case he to the excess to be greater than one. So when we think about the curve here, E to the X is an exponential curve. Greater than one means X has to be greater than zero. So this is going to be part of our answer. Also, the second case is e to the X less than six again using the curb six would be somewhere over here. Whatever the X value is, we want to go to the left of that. So, um, consider ah, you to the X equal six to find the X value first. So, um, this would be X times The natural log of e equals natural August 6 betweens, Exes, eagles of the natural log of six. So this value here is X, which is natural log of six. We want the X values that are less than this. Why value of six. So we want everything to the left of it, which means X has to be less than natural. Log of six. So putting these two solutions together ah ex would have to be in between zero and the natural log of six

Heather Zimmers · Answer

I&#x27;m going to solve both of these inequalities algebraic Lee and then confirmed them graphically. So first of all, with natural log of X is less than zero. I&#x27;m going to use a process called exponentially ating, which means that each side of the equation or each side of the inequality will be an exponents on the same base. And I&#x27;m going to use based E. So we hav e raised to the left side of the inequality is less than e raised to the right side of the inequality. And we should know, based on inverse functions that e to the natural log of X simplifies to just be X, and each of zero is one, so we have X is less than one. There&#x27;s one other thing we need to consider, though. Remember the domain of along with Mick Function X has to be greater than zero. So we do have X is less than one, and we have X is greater than zero. When you combine those together, you get zero is less than X is less than one now. Graphically, if you think about what the natural log of X function looks like, it looks like this with the next intercept at one. Where is it greater than zero Down here. We&#x27;re sorry. Less than zero down here that that&#x27;s where the Y values are. Less than zero. And that was between X equals zero and X equals one. Okay, now let&#x27;s do something similar to the other inequality. And I&#x27;m going to take the natural log of both sides. So we have natural log of e to the X is greater than natural law. Go five because of inverse functions. Natural log of each of the X simplifies to just be x so x is greater than natural log five.

Joshua Eastwood · Answer

lawn of X is less than zero. That is the inequality you want to try to solve. So when it says this solving a call whenever you see that instruction, regardless of what the inequality is you&#x27;re trying to isolate X, you&#x27;re trying to get it by itself. And so in order to do that, we have to remove this lawn. So what we have to think about is what is the inverse operation of lawn? How do how do I remove the lawn? And the way you do that is by using this rule of law algorithms, which says that if you have log of some base B of a number, let&#x27;s say a and it&#x27;s equal to some number. See that you can rewrite that statement. So if you have a log on one side and a number on the other, you can rewrite as an exponential, and what you do is you take the base, so whatever that base happens to be, we&#x27;re gonna call it base is be for now. But it&#x27;s always gonna be some number, and in our case, the natural log rhythm is based E, and then you raise it to the exponents So you say B to the sea. You kind of think about it this way is equal to a So in order to remove the lager them in this question. What we&#x27;re gonna do is we&#x27;re just gonna rewrite it as e, which is our base raised to that exponents zero, and that&#x27;s going to give us the number. And then we simply figure out what is eating zero on evens. Here was one. So our answer from removing the logger them is axe is less than one. But that&#x27;s not quite the final answer because we have to consider lawn or any log rhythm is not allowed to be negative. And the reason is not a levee negative. You think graphically, longer them&#x27;s looked like this. They have a vertical ash into their they don&#x27;t ever cross zero. So you kind of think photographically you can see why negatives might be a problem. And if you think about it in this context, in this rule that we just talked about, you can also see why negatives to be a problem, I think, because if you take the base and it&#x27;s positive and all the bases are that we use longer than is for. And no matter what you raise it to, then it&#x27;s an exponents. You can never get a negative number from a positive base, Raceman exponents. You can&#x27;t do that mathematically, and so we can&#x27;t have negative. So we have to add in a stipulation, saying X has to also be greater than zero. And that&#x27;s my solution. The lawn X is less than zero. It&#x27;s remove the wog rhythm by writing as an exponential men solving. Now, the second part of this question, part beast, is e to the X is greater than five, and I want to know the solution to that. So it&#x27;s a slightly different take on the same rule we know we just wrote down that log base B of A is equal to see means that hey is beat of the sea. So in the first part, in part A. We used this to this. We went from the logger them to the exponential. In this question, we&#x27;re actually gonna go from the exponential to the logger them. So the base is E. This is my exponents X, and that&#x27;s the number that I end up with. That&#x27;s the value so what I&#x27;m gonna do. He is. I&#x27;m gonna rewrite this now as a log rhythm. So this is the exponents of X becomes greater than log of the base, which is e and then of that number five. So we just then can write. This log base is written as a natural longer than it has its own statement as lawn five. So my answer is X has to be greater than lawn five. So we&#x27;re using the same rule in the first case just to quickly recap We took the log rhythm and wrote it as an exponential to solve this. And in the second case, we took the exponential and Rodas of lager than to solve it.

Amy Jiang · Answer

okay so as to solve the full on crack suicide by taking immutable shots here so we can solve for X. That&#x27;s X is Austin E to party girl, and that&#x27;s people to one another apart being Let&#x27;s take the natural audible sides so we get X is greater than Elena fly.

Problem

Dialysis time Hemodialysis is a process by which …

Problem 52 Hard Difficulty

Solve each inequality for x.

$$(a)1< e^{3 x-1}<2 \quad \text { (b) } 1-2 \ln x< 3$$

Answer

(a) $\frac{1}{3}< x <\frac{1+\ln 2}{3}$
(b) $x >\frac{1}{e}$

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James Stewart

Biocalculus Calculus for the Life Sciences

Anna Marie V.

Heather Z.

Samuel H.

Joseph L.

Precalculus Review - Intro

Functions on the Real Line - Overview

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(a) $\frac{1}{3}< x <\frac{1+\ln 2}{3}$(b) $x >\frac{1}{e}$

Biocalculus Calculus for the Life Sciences

Anna Marie V.

Heather Z.

Samuel H.

Joseph L.

Precalculus Review - Intro

Functions on the Real Line - Overview

James StewartBiocalculus Calculus for the Life Sciences

Anna Marie V.

Heather Z.

Samuel H.

Joseph L.

(a) $\frac{1}{3}< x <\frac{1+\ln 2}{3}$
(b) $x >\frac{1}{e}$

James Stewart

Biocalculus Calculus for the Life Sciences