Solve the following.−3nn+2+2n4n+8=68n+16-\frac{3n}{n+2}+\frac{2n}{4n+8}=\frac{6}{8n+16}

n = − 3 10 n=-\frac{3}{10} n = − 10 3

Solve the following. − 3 n n + 2 + 2 n 4 n + 8 = 6 8 n + 16 -\frac{3n}{n+2}+\frac{2n}{4n+8}=\frac{6}{8n+16}

this problem, we are asked to solve the following equation for n. And for part a of this problem, what we need to do is figure out how to find a solution for what this n is. Now the strategy for solving problems where you have these rational expressions that make up the equation is to first find the least common denominator. This is going to help you a lot. Now the way I can find the least common denominator is by factoring the denominator in each of these terms that I see. Now n plus two is already factored. So negative three over n plus two is going to stay the same. Now for this two n over four n plus eight, I can factor a four out of this denominator. So that's going to be four. And then dividing each of these terms by four, we're going to get n plus two. Now all of this is going to equal six over eight n plus 16. And I can pull an eight out of each of these terms, giving me eight times n plus 16 divided by eight, which is two. So notice that this is the expression that we end up with. Now it turns out I can make this expression even more simple, because four is the same thing as two times two, and eight is the same thing as two times two times two. Two times two is four, times another two would give us eight. And writing it like this allows me to simplify these by putting exponents on them. So two times two is two squared, and two times two times two is two cubed. Now we have everything as factored as it can get. Now I noticed that we have an n plus two in common with each of these denominators. So listing the unique prime factors, we're going to have n plus two. Now I can also see we have a two squared, but we also have a two cubed. When you run into situations where you have the same base, you wanna look for the highest exponent, which in this case is two cubed. So now we have found the least common denominator. Now from here, what I'm gonna do is take my least common denominator, and I'm going to multiply it on both sides of our equation. So it's gonna be two cubed times n plus two that I'm multiplying on both sides. Now what I need to do from here is distribute this n cubed times n plus two into the parentheses for each of these terms on the left and for this term on the right. Now as I do this, I'm going to cancel the common terms that I see. So if I multiply two cubed times n plus two, I take that whole thing and I multiply it by three n over n plus two, notice the n plus two will cancel completely. So all we're going to end up with is two cubed times negative three n. Now if we go ahead and distribute this into the next term here, we'll notice once again the n plus two cancels completely. So all we're gonna end up with is two cubed, two cubed times two n over two squared, so times two n over two squared. And this is all going to be equal to and then in this case, multiplying this into the parentheses here, notice that the two cubed times n plus two entirely cancels. Those denominators completely go away, so all we end up with is six. So this is what we get. Now there's one more thing that I can do because notice we have a two cubed and we have a two squared. What I can do is I can cancel the two twos with two of these twos, leaving me with just a two. So, ultimately, we're going to get two cubed times negative three n plus two times two n equals six. And notice how this is a lot more simple than where we started with since we've cleared all the denominators. Now I'm not at my final solution yet because I still need to solve for this n. But to solve for it, well, two cubed, that's going to give me eight. So we'll have eight times negative three n, and we're going to have plus, and then we have two times two, which is four. N is equal to six. And then eight times negative three, that's gonna give me negative 24 n plus four n equals six. Negative 24 plus four is going to give me negative 20, and that's gonna be negative 20 n equals six. And I divide that negative 20 on both sides of the equation. The negative twenties cancel there, giving me that n is equal to negative six over 20. Now what I can do is break this down further because six is the same thing as two times three, and twenty is the same thing as two times 10. So if I cancel those twos right about there, that's going to give me that n is equal to negative three over 10. So this right here is our solution for n, and that's the answer to this problem. So this is how we can solve for a variable in a rational equation. The main step that you need to know is first finding the LCD and then multiplying it by both sides of the equation. That will significantly simplify the equation that you are dealing with and give you something straightforward that you can solve. So this is how you can find the solution. Hope you found this video helpful, and let's move on and get some more practice. See you in the next one.

Solve the following. − 3 n n + 2 + 2 n 4 n + 8 = 6 8 n + 16 -\frac{3n}{n+2}+\frac{2n}{4n+8}=\frac{6}{8n+16}

n = − 3 10 n=-\frac{3}{10} n = − 10 3

n = − 1 2 n=-\frac12 n = − 2 1

n = 3 10 n=\frac{3}{10} n = 10 3

8. Rational Expressions and Equations

Rational Equations

Beginning & Intermediate Algebra

8. Rational Expressions and Equations

Rational Equations

Struggling with Beginning & Intermediate Algebra?

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Multiple Choice

Solve the following.
$- \frac{3 n}{n + 2} + \frac{2 n}{4 n + 8} = \frac{6}{8 n + 16}$

$n=-\frac12$

$n=\frac{3}{10}$

$n=-\frac{3}{10}$

$n=1$

Verified step by step guidance

Start by factoring the denominators where possible to simplify the equation. Notice that \$4n + 8$ and $8n + 16$ can be factored as $4(n + 2)$ and $8(n + 2)$ respectively.

Rewrite the equation using the factored denominators: $-\frac{3n}{n+2} + \frac{2n}{4(n+2)} = \frac{6}{8(n+2)}$.

To eliminate the denominators, multiply every term in the equation by the least common denominator (LCD), which is \$8(n+2)$, to clear the fractions.

After multiplying, simplify each term carefully and combine like terms to form a linear equation in terms of $n$.

Solve the resulting linear equation for $n$ by isolating the variable on one side, and then simplify the solution to find the value(s) of $n$.

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