How do you solve #\frac{ 3}{ x } + \frac{ x }{ x + 2} = \frac{ - 2}{ x + 2}#?

2 Answers
May 17, 2017

#x=-3#

Explanation:

Since both sides of the equation have #x+2# as a common denominator, we can subtract #x/(x+2)# from both sides

#3/x+x/(x+2)=(-2)/(x+2)#

#3/x+cancel(x/(x+2)color(red)(-x/(x+2)))=(-2)/(x+2)color(red)(-x/(x+2))#

#3/x=(-2-x)/(x+2)#

Multiply both sides by #-1#

#3/xcolor(red)((-1))=(-2-x)/(x+2)color(red)((-1))#

#(-3)/x=((-2-x)(-1))/(x+2)#

#-3/x=(2+x)/(x+2)#

#-3/x=(x+2)/(x+2)#

#-3/x=1#

Multiply both sides by #x#

#-3/cancelxcancelcolor(red)((x))=1color(red)((x))#

#-3=x#

#x=-3#

May 17, 2017

#color(purple)(x=-3#

Explanation:

#3/x+x/(x+2)=(-2)/(x+2)#

#:.(3(x+2)+(x)(x)=-2(x))/(x(x+2))#

multiply L.H.S AND R.H.S. by #color(purple)( x(x+2)#

#:.3x+6+x^2=-2x#

#:.3x+2x+x^2=-6#

#:.5x+x^2=-6#

#:.x^2+5x+6=0#

#:.(x+3)(x+2)=0#

#:.x+3=0# or #x+2=0#

#:.color(purple)(x=-3# or #color(purple)(x=-2#

substitute #color(purple)(x=-3#

#:.3/color(purple)(-3)+color(purple)(-3)/(color(purple)(-3)+2)=(-2)/(color(purple)(-3)+2)#

#:.-1+(-3)/-1=(-2)/(-1)#

#:.-1+3=2#

#:.2=2#

substitute #color(purple)(x=-2#

#:.3/color(purple)(-2)+color(purple)(-2)/(color(purple)(-2)+2)=(-2)/(color(purple)(-2)+2)#

#:.-1.5+(-2)/0=(-2)/0#

extraneous solution, so # x # can't be #-2#