Question

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

Answer 1

`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`

Answer 2

`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`