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Solve for n: (2-n/n+1) + (2n-4/2-n)

 
 Aug 2, 2022
 #2
avatar+262 
+1

Answer: \(\mathbb{R} - (-1, 2)\) (with what I have, anyway...)

I assume this is what you mean: \((\frac{2-n}{n+1})+(\frac{2n-4}{2-n})\).

Of course, n could be literally anything other than -1 or 2 because there isn't an equality present (maybe you forgot it????? Or maybe I went blind????)

 
 Aug 2, 2022
 #3
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2-n/n+1 + 2n-4/2-n = 1

 
 Aug 2, 2022
 #4
avatar+2270 
+1

We have the equation: \({2 - n \over n+1} + {2n - 4 \over 2 - n} = 1\)

 

Make a common denominator: \({(2 -n)(2-n) \over (n + 1)(2-n)} + {(2n-4)(n+1) \over (2-n)(n+1)} = 1\)

 

Simplify: \({n^2−4n+4 \over −n^2+n+2} + { 2n^2−2n−4\over -n^2 + n + 2} = 1\)

 

Add the fractions: \({3n^2 - 6n \over -n^2 + n + 2} = 1\)

 

Cross multiply: \(3n^2 - 6n = -n^2 + n + 2 \)

 

Move everything to the left-hand side: \(4n^2 - 7n - 2 = 0\)

 

Factor: \((4n+1)(n−2)=0\)

 

So, the solutions are \(-{1 \over 4} \) and \(2\).

 

But, to verify their validity, we need to plug them back into the original equation. 

 

Doing so, eliminates \(n = 2\), so \(n = \color{brown}\boxed{-{1 \over 4}}\)

 
 Aug 2, 2022

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