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# Help with algebra

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

Apr 13, 2022

#1
+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
+1

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

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Apr 13, 2022
#3
+1

deleted

Apr 13, 2022
edited by qjin27  Apr 13, 2022
#4
+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   Apr 13, 2022
edited by CPhill  Apr 13, 2022
#5
-3

It -4

Apr 13, 2022