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