Answer
$x=1$
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:
$12(\frac{x+1}{4}) = 12(\frac{1}{6}+\frac{2-x}{3})
\\\frac{12(x+1)}{4} = \frac{12}{6} +\frac{12(2-x)}{3}
\\3(x+1) = 2+4(2-x)
\\3(x)+3(1) = 2+4(2) - 4(x)
\\3x+3=2+8-4x
\\3x+3=10-4x$
Add $4x$ and subtract $3$ on both sides to obtain:
$3x+4x=10-3
\\7x=7
\\\frac{7x}{7}=\frac{7}{7}
\\x = 1$
Check:
$\begin{array}{ccc}
&\frac{1+1}{4} &= &\frac{1}{6}+\frac{2-1}{3}
\\&\frac{2}{4} &= &\frac{1}{6}+\frac{1}{3}
\\&\frac{1}{2} &= &\frac{1}{6} + \frac{2}{6}
\\&\frac{1}{2} &= &\frac{3}{6}
\&\frac{1}{2} &= &\frac{1}{2}\end{array}$
