Simplify the LHS:
[(x+7)2-(x-7)2]/[x(x2-49)]=(x+7+x-7)(x+7-x+7)/[x(x2-49)]=
(2x)(14)/[x(x2-49)]=28/(x2-49).
So we have:
28/(x2-49)=7/(x2-73),
4/(x2-49)=1/(x2-73),
4(x2-73)=x2-49,
4x2-292=x2-49,
3x2=243,
x2=81, so x=9 or -9.
CHECK
The original equation becomes:
16/18-2/144=7/8,
64/72-1/72=63/72=7/8 which is true so the solution is correct for x=9.
(-2)/((-9)(-16))-(-16)/((-9)(-2))=-2/144+16/18=-1/72+64/72=63/72=7/8, which is true so the solution is correct for x=-9.