Question 962771
Let x be the time for Josh alone to paint the fence.


PERSON__________RATE
Josh____________1/x
Danny___________1/(x-9)
Josh&Danny______1/7


The rate of both people working together will be the sum of the individual rates.


{{{1/x+1/(x-9)=1/7}}}


Use ordinary fraction skills.  LCD is {{{7x(x-9)}}}.
Multiply both members by {{{7x(x-9)}}} and simplify before solving for x.