We need

#(1/x)'=-1/x^2#

The domain of #y# is #D_y=RR-{0}#

We calculate the first derivative

#y=x+4/x#

#dy/dx=1-4/x^2#

To find the critical points, we calculate the values of #x# when #dy/dx=0#

when

#1-4/x^2=0#

#1=4/x^2#

#x^2=4#

Therefore, #x=-2# and #x=2#

We can build the chart

#color(white)(aaaa)##x##color(white)(aaaa)##-oo##color(white)(aaaa)##-2##color(white)(aaaaaaaa)##0##color(white)(aaaaaaa)##2##color(white)(aaaaa)##+oo#

#color(white)(aaaa)##x+2##color(white)(aaaaaa)##-##color(white)(aaaa)##+##color(white)(aaaa)##||##color(white)(aaa)##+##color(white)(aaaa)##+#

#color(white)(aaaa)##x-2##color(white)(aaaaaa)##-##color(white)(aaaa)##-##color(white)(aaaa)##||##color(white)(aaa)##-##color(white)(aaaa)##+#

#color(white)(aaaa)##dy/dx##color(white)(aaaaaaaa)##+##color(white)(aaaa)##-##color(white)(aaaa)##||##color(white)(aaa)##-##color(white)(aaaa)##+#

#color(white)(aaaa)##y##color(white)(aaaaaaaaaa)##↗##color(white)(aaaa)##↘##color(white)(aaa)##||##color(white)(aaa)##↘##color(white)(aaaa)##↗#

The intervals of increasing are #x in ]-oo,-2[uu]2,+oo[#

The intervals of decreasing are #x in ]-2,0[uu]0,2[#