Answer
Solution set = $\{1\}.$
Work Step by Step
Multiply both sides with the LCD, 12
$ 12 \left[\displaystyle \frac{x+1}{4}\right]=12 \left[ \dfrac{1}{6}+\dfrac{2-x}{3}\right]\quad$ ... distribute and simplify
$ 3(x+1)=2+4(2-x)\quad$ ... distribute and simplify
$3x+3=2+8-4x$
$3x+3=10-4x \quad$ ... add $4x-3$ to both sides
$3x+4x=10-3$
$7x=7$
$x=1$
Solution set = $\{1\}.$
