in-3-38-solve-each-equation-for-the-variable-check-and-write-the-solution-set-2sqrt3-x-23-x

Question

In $3-38,$ solve each equation for the variable, check, and write the solution set.
$$
2+\sqrt{3 x-2}=3 x
$$

Nick Johnson · Accepted Answer

here we have the equation. Two plus the square root off three X minus two is equal to three X. So we&#x27;re gonna do is first subtract the two over. Just isolate the square root here. Square three X minus two That gives us the square root of three X minus two is equal to three X well minus two. So if we square both sides, we then eliminate the square root so we get while three x minus two squared It just takes minus two is equal to three X minus two squared but three X minus two squared is three x minus two times three x minus two which gives us well a three x times three x is nine x squared and then a minus six x minus six X is a minus 12 x and then plus four. So three of minus two squared is nine x squared, minus 12 X plus four. So therefore, we square the left side. We just get three X minus two. Many square the right side. We get nine X squared minus 12 ax plus four. So then we take our quadratic Here we subtract the three X term and try and add to both sides to set our quadratic equal to zero. So we get nine X squared minus 15 x, because minus 12 x minus three X is minus 15 x and then we&#x27;re adding to to both sides. So four plus two or six So get nine X square minus 15 X plus X is equal to zero. And we could try and back to this, and we might come into sexism. Well, first, let&#x27;s go ahead and that we see we can factor out a common factor of three. Right? Let&#x27;s pull out a common factor of three. So we get three times while three X squared gives us in nine X squared and in three times minus five. X gives us a negative 15 x and in three times to use a six. So plus two o k. Is equal to, um, zero, right? That&#x27;s nine X squared minus 15 X plus 60 0 Now we could try and maybe trying factor this so we might run into some problems. Elves. Let&#x27;s go ahead, and that&#x27;s try and use the quadratic formula on this inner part Reddick&#x27;s we Now we know that either is a +300 which streets never equal to zero. So have even need that. We need three X squared, minus five acts plus two to be equal to zero. But you can go ahead and use the quadratic formula to solve for X, so we know that X is equal to or negative. B. Zaheer B is indicative. Five. So we have negative negative five. We have five plus or minus the square root of B squared native. Five squared. That&#x27;s negative. Five. Squared um, so native B plus or minus the square root of B squared minus four times a time. Sees or four times a his three I&#x27;m see, which is to all divided by +28 divided by two Times Street. OK, so this gives us while five plus or minus the square root of what while native five square is 25 square root of 25 minus four times three times to four times three is 12. 12 times two is 24 25 minus 24 all over six. Okay, three times to a six, but 25 miles. 24. That&#x27;s one. We have five plus or minus the square root of one, which is just one right, so we have five plus or minus one all over six. So therefore, Acts writes X is equal to five plus or minus 1/6. Therefore, X is either equal to while five um plus 1/6. That&#x27;s 6/6, but six over sex is just even though one for exercise with one or exercise equal to five, minus 1/6 or five minus one over sex. That&#x27;s four over sex, right? But for over 65 both numbers by to hear and forward six is equal to 2/3, which two thirds so therefore X is equal to either one or two thirds. Now go ahead and check both the solutions and make sure they&#x27;re actually valid solutions to our original equation. OK, way up here, X is equal to one. Well, let me just get two plus, while the square root of three times one, which is just three minus two, should be equal to three times one or three. So therefore, a to be equal to with three minus two is just one screwed of one, just one, so you have two plus one is equal to three. That&#x27;s true, right? Three is equal to three for that checks out, so therefore one is a balanced solution. How about two thirds? That&#x27;s a two thirds. You would have to plus the square root of three times two thirds minus two. And that&#x27;s what the equal to three times two thirds okay or three times two thirds is just equal to write because, such as 3/1 threes cancel out. Just get two or two minus two is equal to zero. So we get that two plus two squared 00 So that, too, is equal to while three times two thirds is also equal to write because can three is cancel out become one and the the two is equal to which checks out. That&#x27;s also true. So therefore, both solutions here are valid. X is equal to one or act is equal to two thirds. So therefore, our solution set is the second can be these two values one and two thirds or two thirds and one remember order you put him in so we have the solution set. Here it&#x27;s set containing two thirds and one. All right, take a

Nick Johnson · Answer

three plus the square of four x minus street is equal to two X, so go ahead. And to isolate the radical term here, that thing I knew need the square root. We can go ahead and subtract three X from both sides Equation that&#x27;s gonna give us Well, the square root of four X minus three is then going to be equal to out two X minus street. Okay, then we could go ahead and square both sides of the equation because the square root of four X minus three squared is just gonna give us what&#x27;s under the radical is gonna give us for X minus three, but squaring two x ministry with a square. We have two terms here, so we&#x27;re squaring two X minus three. So that means we take two X minus three times two X minus three and we have to distribute here. So this is gonna give us Wow, This is equal to two x times two x That&#x27;s four x squared and get two x times negative. Three such a negative six x and then well minus another six x so minus six x minus six X is gonna B minus 12 acts and then we have minus three times minus three is gonna be a plus night. So two X minus three squared is four X squared minus 12 X plus nine. So therefore, if we square about side or the left side, we just get again. Which underneath the square root So the square root of four X minus three squared is just equal to four X minus three. And then that&#x27;s equal to what two X minus three squared, which we see here is four X squared, minus 12 acts plus nine. Okay, now we can go ahead and weaken. Basically, subtract these two terms to the other side to then have a quadratic which is equal to zero. So we&#x27;re gonna have Well, we we have a four x squared term. So I have four x squared. But then we have native 12 x subtracting for X so minus 12 exit minus four x Well, that&#x27;s 12 13 14 15 16 minus 60. Next, we have four x squared, minus 16 acts, and then we have while nine and then we&#x27;re adding three to both sides. So nine plus three is 10. So we have four x squared minus 60. Next 10 is 12 right? Nine plus three is 12. So nine months we get plus 12. Right? Adding three. The both sides for nine plus tree is 9, 11 12 guests trouble. So now we have this being equal to zero. Okay, so? Well, um, well, we can definitely factor out a four year, right, because four goes into four. X squared goes in 16 and goes into 12 by factor out a four. We have four times Well x squared, and then four times on minus four. X gives us in 1916 x and then four plus three is 12 7 plus three. Okay. And that&#x27;s equal to zero. Okay, so now I can go ahead and factor. Um, the inside here is a quadratic. So we have four times trying. Factor this quadratic. So that&#x27;s gonna factor as where x squared is x times x three. That&#x27;s three times one. I need the same signs right at the edge with negative for X off. Both of these are negative, right? That seem to be an X squared minus one x minus three x minus four X plus three K is now equal to zero So now I have, well, the product of what? Actually three things being equal is there. Also, at least one of the factors have to be zero or four is never zero right. So therefore either X minus 30 or X minus 10 X minus three is zero. That implies that X is he, won&#x27;t you? Three or if x minus 10 that implies that X is equal to one. Okay, so we have two solutions we found. But now let&#x27;s go ahead. And that&#x27;s check and make sure that both of these solutions are actually valid solutions. So if we check X equals three, we have three plus the square root of four times three minus streets. So that would be, um they were checking checking over here now. So we have three plus the square root of while four times we are checking. Executed 34 times three is 12 so 12 minus three. That should be equal to two X. So the two times three, which we go to six. Is this true? All we have three plus The square root of 12 minor street or 12 minus three is 12 11 10 9 the square root of nine. Is he gonna three? We have three. Plus three is equal to sex, which is true, but we&#x27;re six. A secret sex, therefore, that checks out and therefore X is equal to three is a valid solution. How about X? Be equal till one X is equal. No one would have three plus the square root of while four times one minus street. There will be four, Um, while four minus three. Which is right. Rex is one. Yet four times one is four for minus three is one square root of one is one for the three. Plus the square root of one is equal. Chew. Um, two times. Times one. Okay. Is this true? Well, it&#x27;s good of one is one you have three plus one would be equal to two. That would mean that four is equal to two, and that&#x27;s not true. So we go ahead and we throw out X is equal to one, and we see that the only right that we have an extraneous solution X equals one. The only valid solution is X equals three. So therefore, our solution set would be the set to make a set brackets here. So the set containing Justin number three there is our solution set. All right, take it.

Martha Richards · Answer

We&#x27;re gonna solve the solution here Checker answer and then provide the solution set for X. So here, when I have this cute, rude, isolated Aiken, just Q both sides. So then the left becomes X and two cubed is eight. So we&#x27;re gonna check her answer here. So when I put this back in the original equation, cubed root of eight is too. So this checks out. So our solution set for this equation here is for when acts is equal to eight.

Nick Johnson · Answer

plus three is equal to the square root of one minus three X. So we&#x27;re gonna go ahead with a square both sides of the equation to get rid of the radical but squaring expose three. Let&#x27;s expose Story Square is exposed three times X plus three. So squaring left side we get X plus three times X plus three Okay, that&#x27;s gonna be equal to an X squared plus three x plus three X plus six acts and then plus three times three plus nine. So x squared plus six x plus nine is X plus three squared and that&#x27;s gonna be equal to while the square root of one minus three X squared just gives us what&#x27;s under the radical, namely one minus three acts. Okay, so now we&#x27;re gonna go. We&#x27;re gonna go in, um, get all our terms on one side of the equation, but have a quadratic equal to zero. So first, let&#x27;s go ahead and add three extra both sides. So we get X squared. Plus, while six x plus three x is plus nine acts and then we&#x27;re going to subtract one from both sides, so nine minus one is eight to get plus eight is gonna be than equal to zero so that we can We have a quadratic here in standard form setting with a zero. But you can go ahead and try and factor Mr Factor as, um, x times X is x squared it. Well, that&#x27;s eight times one, right. We need everything positive. Here. Is that add to a nine X in the middle. Okay, this is equal to zero. This means that either X is equal to or negative eight or access is equal to negative one. All right, let&#x27;s go ahead and out. Chuck these solutions and make sure they work in our original equation. The first score headed check, maybe exit was negative. One X equals negative one. We get negative one plus three. So native one plus three is equal to we have to is equal to the square root of one minus three times a negative one. So we have two is equal to the square root of one minus three times a negative one. Um, is a negative three. If you have one minus and negative three. That&#x27;s one plus three was just four. You have to is equal to the square root of four, which is equal to two. Right? We have two is equal to four is to have two is equal to two, Which checks out. So therefore, um negative one is a solution. How about negative? Eat? If X is negative eight, we have negative eight plus three. That would be 345678 A. B negative five would have Negative five is equal to the square root of one minus three times. Um A Did you did you do three times a negative, right? Which are gonna get a great so three times a negative. It&#x27;s all right. You have negative. Five is equal to the square. Root off while one minus three times a debate is a negative. 24. We have one minus and native 24. That&#x27;s one plus 24 which is 25. If you have negative five is equal to the square root of 25. Is that true? Well, we have negative five right is equal. Choose grew. 25 is well five. So negative five is equal to five is not true. Everything go ahead and can throw out X equals negative eight and therefore our solution set is just a set containing negative one. So we get, um the solution set here is the set containing negative one. All right, take her.

Martha Richards · Answer

you were asked to solve this equation, check for it and then provide the solutions that so when the radical is isolated, we can just square both sides. So I get one plus X on the left is equal to nine, cause three squared is nine. And then I subtract one on both sides, and I end up here with X equal to eight. So now let&#x27;s check this in the original equation. So I replaced acts with eight. And here, when I have eight plus one, that&#x27;s the square root of nine and the spirit of nine is three. So this does check out. So our solutions that for the equation above here is for X equal to eight.

Nick Johnson · Answer

having equation here. One plus the square root of X is equal to three. So on the left side of the equation, I have two terms, right? I have one in the square Durex. So I have an ex term and have a constant. Let&#x27;s go ahead and move. All are constants toe one side of the equation by just subtracting one from both sides. Track one from both sides. One minus 10 I just get zero plus bread of expert justice. Square root of acts is equal to three minus one squared of access equal to two and then the south for X because I have the square root of X Aiken Square, both sides to undo the square root is the square root of X squared is equal to X, and then we square both sides so excessively could the two square or two times two, which is equal to four so X secretive form, meaning that our solution set contains just the number of four. There&#x27;s our solutions said All right,

Problem

In $3-38,$ solve each equation for the variable, …

Problem 29 Hard Difficulty

In $3-38,$ solve each equation for the variable, check, and write the solution set.
$$
2+\sqrt{3 x-2}=3 x
$$

Answer

$\left\{\frac{2}{3}, 1\right\}$

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$\left\{\frac{2}{3}, 1\right\}$

Algebra 2 and Trigonometry

Ann Xavier GantertAlgebra 2 and Trigonometry

Ann Xavier Gantert

Algebra 2 and Trigonometry