in-3-20-solve-each-equation-and-check-frac34-x14-x

Question

In $3-20,$ solve each equation and check.
$$
\frac{3}{4} x=14-x
$$

Dilip Paruchuri · Accepted Answer

Okay. Hey, guys. So in this problem, we&#x27;re given the equation 3/4 x equals 14 minus X and were asked to solve this for X. And then after we saw this for ex worker and actually verified that it&#x27;s true, because when you have an equation, it&#x27;s really easy to verify. An answer is true, because you just plug it in back into the equation and see if both sides are equal. So first, let&#x27;s start off by solving this equations. We have 34 Sex equals 14 minus X, so we want to get all our exes on one side, and all are constants on one side. So in order to get all our exes on one side, let&#x27;s add X to both sides. When we add X to both sides, we get the re Ford&#x27;s X Plus X, which is seven forts X equal to 14 because these X&#x27;s cancel out over here and now we have 7/4 X equals 2 14 In order to isolate this X, we&#x27;re gonna need to multiply by 4/7 on both sides when multiplying by four sevens on both sides on the left side, we get cancellation, and so on the left side. We&#x27;re just left with X equals and on the right side we do 14 times 4/7. And before we just instead of just multiplying out, um, 14 times four, divided by seven. We could reduce our terms here. We know that 14 7 goes into 14 twice, so we could, ah, simplify and we have two times for and we get Our X is equal to eight. And so let&#x27;s verify this is the left hand side. Just so most label like this left hand side we have. So it&#x27;s bugging three Fords X. Our ex, we said, is equal to eight. So 3/4 of eight hopes reports of eight is equal to Let&#x27;s call Simplify. Four goes into twice, so 3/4 of eight is equal to six. And now let&#x27;s take a look at our right hand side. Our right hand side is 14 minus X. We set our X was eight 14 minus eight is equal to six. We see that both our left hand side and our right hand side or equal so we know that our solution is correct. And so our final answer is X equals eight. Thanks for listening eyes. And I hope this helped answer this equation

Derrick Hanson · Answer

This is another problem where we have two different options to solve it. We could go about just the general method of finding common denominator, multiply the whole problem by that to cancel out all the denominators. But what I think would be faster in this case, or at least, would be just as fast. I noticed that I have only one fraction on the left side of the equation and one fracture on the right side of the equation. One rational expression on either side. If you have exactly one rational expression, one fraction on either side of the equal son, then you are able to cross multiply here, meaning set this Dagnall of seven times X equal to this Dagnall of four times the quantity of two X minus three. You don&#x27;t have to do this. You can solve it out the other way of just finding a common denominator. In this case, your common denominator would be X times two X minus three. That would be your common denominator. And if you multiply both sides by that, you&#x27;re gonna get the same thing that we just got here anyway, So it&#x27;s truly up to you, but I figured since you know, there are other problems on the section that we&#x27;re doing with the common denominator. If we&#x27;re able to cross multiply, I had to show that option as well. We have what we have here. Either way. Seven X equals four times two x minus three is a quantity. We would need to distribute that four for stuff. So four times two X would be a X four times negative three would be negative. 12. My goal is to get all the exes on one side, all the constants on the other side and then isolate X completely. Well, since I already have the minus 12 on the right side, I might as well go ahead and subtract a X to the left. That way that constant gets isolated and the ex gets isolated at the same time. Seven. X minus eight X would be a negative one x or you could just say negative X. You don&#x27;t have to write the one there, and that is equal to negative 12. Well, if negative X is equal to negative 12 then positive X must be equal to positive 12. So our answer for this would be X equals 12. Now, on this, we do need to do a little bit of checking here because we do have exes in my denominator. So I needed to go to the original equation. Just gonna rewrite it over here. Seven over two X minus three equals four over X. And what we&#x27;re gonna do is we&#x27;re going to plug 12 in for both of those exes and make sure that we don&#x27;t accidentally get zero in either of the denominators, because then it would turn out This is actually a no solution problem. Undefined. So that means we would have 7/2 times 12. My history equals four over 12. Well, two times 12 is 24 so we&#x27;d have 24 minus three. And over here, I&#x27;ve got nothing to do with that. 24 miles three is 21 so we&#x27;ve got 7/21 is equal to 4/12. Neither of those air zeros. So this is looking good. I just need to simplify those fractions. 7/21 simplifies to 1/3 4/12 also simplifies toe 1/3. So this checks out. I did not get zero in my denominator. Didn&#x27;t get undefined or anything. So my answer truly is 12

Derrick Hanson · Answer

So we&#x27;re trying to solve this problem for X first thing notice here is that obviously we do have some rational expressions here. Some fractions. I don&#x27;t want to leave those those fractions. So the first thing I want to do is I want to find a common denominator so that I can multiply the entire problem by that number to get rid of all of my denominators. Right now, I see a denominator of three and denominator of four. So the question is, what is the least what is the lowest number that both three and four can multiply up to? In this case, I would say that is 12. Okay, so we&#x27;re going to multiply this entire problem then by 12. Understand? This is 12/1, meaning 12 is being multiplied to the numerator. Okay, I have to multiply every single part of the equation by 12 so that I&#x27;m not actually changing the value of this equation. Meaning I&#x27;m going to take 12 times two x over three. Well, 12 times two x would be 24 x over three. Then I have to take 12 times one pretty straightforward. 12 times one is just 12 right, And then finally, I have to take 12 times three x over 4 12 times three x would be 36 x over four. So now, assuming that we chose something that a number that truly was a common denominator, all of my fraction should be able to reduce well. 24 x over three could reduce down to eight. X 12 Doesn&#x27;t need to reduce its on a fraction anyways. And 36 x over four should reduce to nine x 36 divided by four is nine so good? Looks like we did do the right thing here We did pick a common denominator cause all my fractions are gone. So now I simply just need to solve this equation out to get X by itself and figure out what happens equal. Ah, I would say they use this way to do that would be dis attract eight x to the right side because that goes ahead and isolates the Constance as well as the exes at the same time. So that would leave me with just 12 on the left side of the equation. Nine X minus eight X would be one x or just x so looks like our answers. X equals 12. That&#x27;s it.

Dilip Paruchuri · Answer

All right. Hey, guys, showing this problem, We have the equation X over two equals three x three over two X plus one trying to solve for acts and verifying if our exes correct. So we&#x27;re going to do first is we&#x27;re going to multiply by two X plus one on both sides of the equation. And we&#x27;re doing this so we could get our ex out of the denominator on the right hand side. So on the right hand side of the two X plus ones, cancel out we&#x27;re left with the re on the right hand side on the left. Inside we&#x27;re left with X times two x, less one all over, too. And we&#x27;re gonna do something similar, like the first step. We&#x27;re gonna try to get rid of the denominator of the left hand side. Not did not. The denominator on the left hand side is just a constant, but it&#x27;s just nicer when the denominators, when it&#x27;s just normal members and we don&#x27;t have any fractions, it&#x27;s easier to deal with the equation. So we&#x27;re gonna multiply by two on both sides on the left hand side, the twos kids allowed, and you&#x27;re left with X plus two x plus one x times the two whole quantity of to us This one. My bed on the right hand side, you&#x27;re left with six. You see that? This is a quadratic equation. We&#x27;re gonna We&#x27;re gonna get a quadratic equation. So on the left hand side, if we distribute, we get to explore close X. Well, six. If you subtract six on both sides, you get Q X squared plus X minus six. Now we just factor. You could use the quadratic formula to solve this. Just plug in Shug on is gonna factor this out. We know are two X squared plus X minus six is gonna be equal to two X squared plus four X minus three X minus six. Uh, this is just a simple factoring trick. I&#x27;m just gonna factor really quickly. And in order to do this, we&#x27;re gonna group these two terms together and these two terms together, we&#x27;re gonna have to x times x plus two x plus two minus three times X plus two and we&#x27;re gonna factor that out again. Oh, let&#x27;s ah, rewrite this really quickly. Let&#x27;s go back a sec. So we have this equation will go back to this step because I think I wrote something incorrectly. So we have our two x squared plus four X minus three expected sticks. We said That&#x27;s equal to zero. So let&#x27;s go up here to solve the equation. If we factor these two terms that I just grouped up together, these two sets of terms will get to X explodes too. Minus three X plus two equals zero. And on the left, basically on the left hand side. If we distribute if you we factor out once again if you factor out the Expos to we get to X minus three, I&#x27;m expose to equal zero and solving this. We know either the two X minus three equals zero or the experts to equal zero. So the solution says gonna be X equal to three halves and X is equal to minus two. And so let&#x27;s verify. Ah, this equation. So let&#x27;s go to another screen right here. So the equation we&#x27;re trying to verify his X over two is equal to three over to X plus one. Let me see if I copy that. Correct. And I did. And so the first hole that we&#x27;re trying to verify his X equals the re half. So on the left hand side, we have to re have divided by two because we just it&#x27;s x over to our X. We&#x27;re saying is three halves we have three has divided by two so three halves times 1/2 is equal to three Fords. And on the right hand side, we have three over two times, three over two plus one, just plugging in our value of three halves into to expose one. And we have three over the twos. Cancel out. Still three plus one that&#x27;s equal to three over four A. CZ. You see the left hand side on the right hand side of the same. So X equals three have is a correct solution. Now let&#x27;s try. X equals minus two. X equals minus two on the left hand side. If you plug that into X over two, you&#x27;re gonna have negative to over two, which is equal to negative one. And on the right hand side, you&#x27;re gonna have the re over two times. Negative two plus one two times in 32 is negative force. You have three over negative four plus one negative four plus one is equal to 93. So you simplify this to three overnight of three, which is equal to negative one, and negative one is equal to negative one. So X equals minus two is also a valid solution to this equation. Let&#x27;s get a check mark and then let&#x27;s go back to our first screen. And we can easily and definitively say that X equals three halves and and X equals minus to our solution set to the equation X over two equals three over to expose one. Thanks for listening eyes, and I hope this hope.

Dilip Paruchuri · Answer

all right. Hey, guys. So in this problem, we have the rational equation X over experts. Five equals 2/3. Now, the first step in this problem is to make it easier for us for solve this by a kn fraction ing everything so we don&#x27;t want we don&#x27;t really want to deal with the fraction. So what we&#x27;re gonna do is we&#x27;re gonna first multiply by X plus five on both sides of the equation. So first step, we&#x27;re gonna multiply by expose five on the left hand side, and we&#x27;re gonna most by by experts five on the right hand side. So on the left hand side of the to expose fives here, cancel out and we&#x27;re left with X on the left hand side. When the rights hand side, we&#x27;re left with 2/3 times X plus five. And now, in order to make things a little bit more easier for ourselves, what we can do is we can cross multiply, not cross month by, but in a sense, divide out the 2/3 from the right hand side. So we&#x27;re just left with the X plus five. So what we&#x27;re gonna do is we&#x27;re going to multiply by three halves on the right hand side and three halves on the left hand side. The goal of this basically this step right here is just so we could make this equation just little bit easier to solve. Make out, make the algebra nicer for us. And so what happens on the right hand side is the 2/3 cancels out with it that we have so on the left hand side, you&#x27;re just left with experts five, which is nice, which is really nice to work with. And on the left hand side, you&#x27;re left with the rehabs times X, which is also a nice basically a nice function to work with. And so we want to get all our variables to one side now. So let&#x27;s subtract X on both sides, tracking acts on both sides. We get 1/2 X equals five and in order to get her ex by itself, we&#x27;ve multiplied by two on the left hand side and two on the right hand side. Then we get that X is equal to 10. Now we plug in, shrug and see whether the left hand side equals the right hand side. We have our left hand side is equal to X over explode. Oh, that&#x27;s right. That better is equal to X over X plus five. We plug in our value of 10. So we have 10 over templates. Five at 10 over. Tempus five is the same thing as 10 over 15 which is the same thing as 2/3. And as you noticed, our right hand side is also equal to 2/3. So we know that our left inside right to inside are equal. So value X equals 10 is the correct solution for this equation, and we have verified it. So thanks for listening, guys. And I hope this help solve this rational.

Derrick Hanson · Answer

When you have a problem like this, you actually have two different ways that you can choose to solve this out. You can go the normal route that we would use for any rational equation that we want to solve, which would be to find a common denominator and then multiplied by that common denominator to get everything. Um, all our denominators gone, all of them cancelled out, right. However, in the case of this problem, there is another option. Now, this Onley works if you have exactly one fraction on each side of the equation. But if you have only one fraction on each side of the equal sign, this would be something that you can treat like a proportion meaning. If you remember probably a long time ago, when you&#x27;ve done proportions, you are able to cross multiply here, meaning we can set this diagonal of three terrible three of three times X plus two equal to this diagonal of five times X minus two. Again, this is the same thing you would end up getting if you multiplied by a common denominator, right? If you look for a common denominator here, the common denominator between five and three would be to multiply by 15. So if we multiplied, everything here by 15 will think about it. 15 over five out of you put 15 here. 15/5 would cancel 23 right, 15 divided by three for the second fraction would cancel to five. So it works the same way. Either way, you look at it. It really doesn&#x27;t matter. And I don&#x27;t know that either one saves you a whole bunch of time. But I didn&#x27;t want to point out that since there&#x27;s only one Fraction, one rational expression on each side of the equation, you can just cross multiply. It only works if you have one each side that either way, whichever method you choose to go about, you would end up with where we&#x27;re here. Three times the quantity of explosive two equals five times the quantity of X minus two. So we would distribute the 33 times X is three x three times to it&#x27;s six. Then we distribute the 55 times X is five x five times Negative too, is negative. 10. So now our goal is to get all the exes on one side of the equation. Doesn&#x27;t matter which and get all our constants on one side of the equation. I am going to choose to subtract three X to the other side because I like to keep my ex positive, if possible. And five x minus three X would give me a positive two X. All right. And so then because of that, if I&#x27;m gonna put on LAX is on the right side. I need to move that negative 10 over to the left side so that all my constance air on the opposite side. So the opposites attraction is addition. So I would need to add 10 over here, which would give me 16 equals two X. I need to then divide by two to solve out for X. And that would give me that eight is equal to X or X equals eight. Whichever way you prefer to read it.

Problem

In $3-20,$ solve each equation and check. $$ \f…

Problem 4 Medium Difficulty

In $3-20,$ solve each equation and check.
$$
\frac{3}{4} x=14-x
$$

Answer

$x=8$

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$x=8$

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Absolute Value - Example 1

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Ann Xavier GantertAlgebra 2 and Trigonometry

Anna Marie V.

Alayna H.

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Michael J.

Ann Xavier Gantert

Algebra 2 and Trigonometry