Answer
$x=13$
Work Step by Step
For easier work, it is better to get rid of the fractions.
This can be achieved by multiplying the LCD of $12$ on both sides of the equation to obtain:
$12(\frac{x+3}{6}) = 12(\frac{2}{3}+\frac{x-5}{4})
\\\frac{12(x+3)}{6} = \frac{24}{3} +\frac{12(x-5)}{4}
\\2(x+3) = 8+3(x-5)
\\2(x) + 2(3) = 8+3(x)-3(5)
\\2x+6 = 8+3x-15
\\2x+6 = 3x-7$
Subtract $2x$ and add $7$ on both sides to obtain:
$6+7 = 3x-2x
\\13=x$
Check:
$\begin{array}{ccc}
&\frac{13+3}{6} &= &\frac{2}{3}+\frac{13-5}{4}
\\&\frac{16}{6} &= &\frac{2}{3}+\frac{8}{4}
\\&\frac{8}{3} &= &\frac{2}{3} + 2
\\&\frac{8}{3} &= &\frac{2}{3} + \frac{6}{3}
\\&\frac{8}{3} &= &\frac{8}{3}\end{array}$
