#### Answer

$[-1,\infty)$

#### Work Step by Step

Step 1: $-2x-2\leq1+x$
Step 2: Adding $2$ to both sides, $-2x-2+2\leq1+x+2$
Step 3: $-2x\leq3+x$
Step 4: Subtracting $x$ from both sides, $-2x-x\leq3+x-x$
Step 5: $-3x\leq3$
Step 6: Dividing both sides by -3 (this reverses the direction of the inequality symbol):
$\frac{-3x}{-3} \geq \frac{3}{-3}$
Step 5: $x\geq-1$
According to the inequality, the interval includes $-1$ and all values greater than $-1$. Since $-1$ is part of the interval, a square bracket is used on its side. On the other hand, we represent all values greater than $-1$ by the symbol $\infty$. Therefore, a parenthesis is used on its side.
Therefore, the interval notation for this inequality is written as $[-1,\infty)$.