`x/(x^2-2)=-1/x` Solve the equation by cross multiplying. Check for extraneous solutions.

`x/(x^2-2)=(-1)/x`

Cross multiply,

`x*x=-1(x^2-2)`

`x^2=-x^2+2`

`x^2+x^2-2=0`

`2x^2-2=0`

Factorize,

`2(x^2-1)=0`

`2(x+1)(x-1)=0`

Use the zero product property,

`x+1=0`  or  `x-1=0`

`x=-1`  or  `x=1`

Now let's check the solutions by plugging them in the original equation,

For x=-1,

`(-1)/((-1)^2-2)=(-1)/(-1)`

`(-1)/(1-2)=1`

`(-1)/(-1)=1`

`1=1`

It's true.

For x=1,

`1/(1^2-2)=(-1)/1`

`1/(1-2)=-1`

`1/(-1)=-1`

`-1=-1`

It's true.

S, the solutions are -1 and 1.