Question 541124
{{{1/x + 1/(x+9)= 1/6}}} is correct for an equation
I prefer to say that I am going to multiply both sides of the equation by the expression by {{{6x(x+9)}}}. (The calculations are the same, but avoid having to write all those denominators). On multiplying, we get
{{{(1/x + 1/(x+9))6x(x+9)= (1/6)6x(x+9)}}}
{{{6x(x+9)/x + 6x(x+9)/(x+9)= (1/6)6x(x+9)}}}, which simplifies to
{{{6(x+9) + 6x= x(x+9)}}} --> {{{6x+54+6x=x^2+9x}}} --> {{{12x+54=x^2+9x}}}
Subtracting 12x+54 from both sides, we get
{{{0=x^2+9x-(12x+54)}}} --> {{{x^2+9x-12x-54=0}}} --> {{{x^2-3x-54=0}}}
That expression can be factored
{{{(x-9)(x+6)=0}}}
The solutions to that equation are {{{x=9}}} and {{{x=-6}}}, but since x must be positive, {{{x=9}}} is the only solution to the problem.
It takes Juanita 9 hours to complete the job alone.