6. On the coordinate plane, the area of the plane region bounded by the conditions $\left\{\begin{array}{l}y \geqslant-|x|-1, \\ y \leqslant-2|x|+3\end{array}\right.$ is
6. 16 .

For $\left\{\begin{array}{l}y \geqslant-|x|-1, \\ y \leqslant-2|x|+3\end{array}\right.$, discuss for $x \geqslant 0, x<0$.
When $x \geqslant 0$, it is $\left\{\begin{array}{l}x \geqslant 0, \\ y \geqslant-x-1, \\ y \leqslant-2 x+3 .\end{array}\right.$
When $x<0$, it is $\left\{\begin{array}{l}x<0, \\ y \geqslant x-1, \\ y \leqslant 2 x+3 .\end{array}\right.$
On the coordinate plane, draw the lines $y=-x-1, y=-2 x+3$ and $x=0$, intersecting to form $\triangle A B C, A(4,-5), B(0,-1)$, $C(0,3)$. Then draw the lines $y=x-1, y=2 x+3, y=0$, intersecting to form $\triangle A^{\prime} B C$, where $A^{\prime}(-4,-5)$. Clearly,
$\triangle A B C \simeq \triangle A^{\prime} B C$.
The region defined by $\left\{\begin{array}{l}y \geqslant-|x|-1, \\ y \leqslant-2|x|+3\end{array}\right.$ is shown as the shaded area in the figure.

Its area is $2 S_{\triangle A B C}=2 \times 8=16$.