Simplify the inequality, we cannot do crossing over.
#x/(x-2) > -1/(x+3)#
#x/(x-2)+1/(x+3)>0#
#(x(x+3)+(x-2))/((x-2)(x+3))>0#
#(x^2+3x+x-2)/((x-2)(x+3)) >0#
#(x^2+4x-2)/((x-2)(x+3)) >0#
The roots of the numerator
#x^2+4x-2=0#
are
#x=(-4+-sqrt(16-4(1)(-2)))/(2)#
#=-2+-sqrt6#
#x_1=-2-sqrt6=-4.45#
#x_2=-2+sqrt6=0.45#
Let
#f(x)=((x-x_1)(x-x_2))/((x-2)(x+3))#
We can build the sign chart
#color(white)(aaaa)##x##color(white)(aaaa)##-oo##color(white)(aaaa)##x_1##color(white)(aaaa)##-3##color(white)(aaaaa)##x_2##color(white)(aaaaa)##2##color(white)(aaaaa)##+oo#
#color(white)(aaaa)##x-x_1##color(white)(aaaa)##-##color(white)(aaaa)##+##color(white)(aaaa)##+##color(white)(aaaa)##+##color(white)(aaaa)##+##color(white)(aaaa)###
#color(white)(aaaa)##x-3##color(white)(aaaaa)##-##color(white)(aaaa)##-##color(white)(aa)##||##color(white)(a)##+##color(white)(aaaa)##+##color(white)(aaaa)##+#
#color(white)(aaaa)##x-x_2##color(white)(aaaa)##-##color(white)(aaaa)##-##color(white)(aa)####color(white)(aa)##-##color(white)(aaaa)##+##color(white)(aaaa)##+#
#color(white)(aaaa)##x-2##color(white)(aaaaa)##-##color(white)(aaaa)##-##color(white)(aa)####color(white)(aa)##-##color(white)(aaaa)##-##color(white)(aa)##||##color(white)(a)##+#
#color(white)(aaaa)##f(x)##color(white)(aaaaaa)##+##color(white)(aaaa)##-##color(white)(aa)##||##color(white)(a)##+##color(white)(aaaa)##-##color(white)(aa)##||##color(white)(a)##+#
Therefore,
#f(x) >0# when #x in (-oo,-4.45] uu(-3, 0.45] uu (2, +oo)#
graph{(x^2+4x-2)/((x-2)(x+3)) [-10, 10, -5, 5]}