#### Answer

The solution is $[-4,\infty)$

#### Work Step by Step

$-\dfrac{2}{3}x-\dfrac{1}{6}x+\dfrac{2}{3}(x+1)\le\dfrac{4}{3}$
Multiply the whole inequality by $6$:
$6\Big[-\dfrac{2}{3}x-\dfrac{1}{6}x+\dfrac{2}{3}(x+1)\le\dfrac{4}{3}\Big]$
$-2(2)x-x+2(2)(x+1)\le4(2)$
$-4x-x+4x+4\le8$
Simplify the left side:
$-x+4\le8$
Take $4$ to the right side:
$-x\le8-4$
$-x\le4$
Rearrange:
$x\ge-4$
Expressing the solution in interval notation:
$[-4,\infty)$