Question

How do you solve `3/(x+6)=5/(x-2)` and find any extraneous solutions?

Answer 1

`x=-18`

Explanation:

`"multiply both sides by "(x+6)(x-2)`

`cancel((x+6))(x-2)xx3/cancel((x+6))=cancel((x-2))(x+6)xx5/cancel((x-2))`

`rArr3(x-2)=5(x+6)larrcolor(blue)"no fractions"`

`"distribute brackets on both sides"`

`3x-6=5x+30`

`"subtract 5x from both sides"`

`3x-5x-6=cancel(5x)cancel(-5x)+30`

`rArr-2x-6=30`

`"add 6 to both sides"`

`-2xcancel(-6)cancel(+6)=30+6`

`rArr-2x=36`

`"divide both sides by "-2`

`(cancel(-2) x)/cancel(-2)=36/(-2)`

`rArrx=-18`

`color(blue)"As a check"`

Substitute this value into the equation and if both sides are equal then it is the solution.

`"left "=3/(-12)=-1/4`

`"right "=5/(-20)=-1/4`

`rArrx=-18" is the only solution"`