#(2x)/(x^2-25)-1/(x-5)=1/6#
multiply both sides by#(x^2-25)#
Note that #color(red)(x^2 - 25= (x+5)(x-5))#
#2x-color(red)((x+5)cancel((x-5)))/cancel((x-5))=(x^2-25)/6#
#2x-(x+5)=(x^2-25)/6#
multiply both sides by 6
#12x-6x-30=x^2-25#
#6x-30=x^2-25#
#6x-x^2+25-30 =0" "# (make a quadratic equal to 0)
#-x^2+6x-5=0#
#x^2-6x+5=0" "# make #+x^2#
#(x-5)(x-1)=0" "# find factors
#x=5 or x=1#
Check:
#x=5# will give : #(2x)/(x^2-25) rarr 10/0#
Division by 0 is not defined, so #x!=5#
#x = 1#
#(2(1))/(1^2-25)-1/(1-5)=1/6#
#(2(1))/(1-25)-1/(1-5)=1/6#
#cancel2^1/cancel(-24)^-12-1/(-4)=1/6#
#1/(-12)-1/(-4)=1/6#
#- 1/12+1/4=1/6#
#(-1+3)/12=1/6#
#cancel2^1/cancel12^6=1/6#
#1/6=1/6#