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Find all solutions to (x + 4)/(x + 5) = (x - 4)/(2x) + (x + 3)/(x + 5).

 Apr 13, 2022
 #1
avatar+578 
+3

\(\frac{\left(x+4\right)}{\left(x+5\right)}=\frac{\left(x-4\right)}{\left(2x\right)}+\frac{\left(x+3\right)}{\left(x+5\right)}\)

 

Multiply by LCM :

 

\(2x\left(x+4\right)=\left(x-4\right)\left(x+5\right)+2x\left(x+3\right)\)

 

\(2x^2+8x=3x^2+7x-20\)

 

Using quadratic formula to solve, 

 

\(x_1=\frac{-\left(-1\right)+9}{2\cdot \:1},\:x_2=\frac{-\left(-1\right)-9}{2\cdot \:1}\)

 

\(x = 5, -4\)

 Apr 13, 2022
edited by Vinculum  Apr 13, 2022
 #2
avatar+61 
+1

\(\frac{\left(x+4\right)}{\left(x+5\right)}=\frac{\left(x-4\right)}{\left(2x\right)}+\frac{\left(x+3\right)}{\left(x+5\right)}, 2x\left(x+4\right)=\left(x-4\right)\left(x+5\right)+2x\left(x+3\right), x=5, x=-4\)

.
 Apr 13, 2022
 #3
avatar+61 
+1

deleted

 Apr 13, 2022
edited by qjin27  Apr 13, 2022
 #4
avatar+124592 
+2

Thanks guys.....another slightly different approach

 

We can subtract the second term from the right side from both sides  and we get

 

(1) / (x + 5)  =(x -4) /(2x)

 

Cross-multiply

 

2x = (x + 5) ( x - 4)

 

2x = x^2 + x -20

 

x^2  -x - 20   =   0

 

Factor

 

(x -5) (x + 4)   = 0

 

Set each factor to 0 and solve for x and we  get  that   x= 5  or x  = -4

 

cool cool cool

 Apr 13, 2022
edited by CPhill  Apr 13, 2022
 #5
avatar+114 
-3

It -4

 Apr 13, 2022

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