Multiply both sides by #4# and then #3# to cancel out the fractions. Then, use the distributive property to get like terms and isolate #x#:
#1/4(x+6)=1/6(x+8)#
#color(blue)(4*)1/4(x+6)=color(blue)(4*)1/6(x+8)#
#color(red)cancelcolor(black)(color(blue)(4*)1/4)(x+6)=color(blue)(4*)1/6(x+8)#
#x+6=color(blue)4/6(x+8)#
#x+6=color(blue)2/3(x+8)#
#color(blue)(3*)(x+6)=color(blue)(3*)color(blue)2/3(x+8)#
#color(blue)(3*)(x+6)=color(blue)(color(red)cancelcolor(blue)3*)color(blue)2/color(red)cancelcolor(black)3(x+8)#
#color(blue)3*(x+6)=color(blue)2*(x+8)#
#color(blue)3x+color(blue)3*6=color(blue)2x+color(blue)2*8#
#color(blue)3x+color(blue)18=color(blue)2x+color(blue)16#
Now, subtract #2x# from both sides, then #18#:
#3x+18=2x+16#
#3x+18color(blue)-color(blue)(2x)=2x+16color(blue)-color(blue)(2x)#
#3xcolor(blue)-color(blue)(2x)+18=2x-color(blue)(2x)+16#
#x+18=color(red)cancelcolor(black)(2x-color(blue)(2x))+16#
#x+18=16#
#x+18color(blue)-color(blue)18=16color(blue)-color(blue)18#
#xcolor(red)cancelcolor(black)(+18color(blue)-color(blue)18)=16color(blue)-color(blue)18#
#x=16-18#
#x=-2#