Answer
$x=-19$
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 $24$ on both sides of the equation to obtain:
$24(5+\frac{x-2}{3}) = 24(\frac{x+3}{8})
\\24(5) + \frac{24(x-2)}{3} = \frac{24(x+3)}{8}
\\120+8(x-2) = 3(x+3)
\\120+8x-16 = 3x+9
\\8x+104=3x+9$
Subtract $3x$ and $104$ on both sides to obtain:
$8x-3x=9-104
\\5x=-95
\\\frac{5x}{5} = \frac{-95}{5}
\\x = -19$
Check:
$\begin{array}{ccc}
&5+\frac{-19-2}{3} &= &\frac{-19+3}{8}
\\&5+\frac{-21}{3} &= &\frac{-16}{8}
\\&5+(-7) &= &-2
\\&-2 &= &-2\end{array}$
