#### Answer

$x=-4$ and $x=3$

#### Work Step by Step

$\dfrac{2}{x+1}-\dfrac{1}{x-2}=-\dfrac{1}{2}$
Multiply the whole equation by $2(x+1)(x-2)$:
$2(x+1)(x-2)\Big(\dfrac{2}{x+1}-\dfrac{1}{x-2}=-\dfrac{1}{2}\Big)$
$2(2)(x-2)-2(x+1)=-(x+1)(x-2)$
Evaluate all the products:
$4x-8-2x-2=-x^{2}+x+2$
Take all terms to the left side of the equation:
$x^{2}-x-2+4x-8-2x-2=0$
Simplify the equation by combining like terms:
$x^{2}+x-12=0$
Solve this equation by factoring:
$(x+4)(x-3)=0$
Set each factor equal to $0$ and solve each individual equation:
$x+4=0$
$x=-4$
$x-3=0$
$x=3$
The initial equation is not undefined for neither of these two values of $x$, so the solutions are:
$x=-4$ and $x=3$