## Task B-1.4.

For which $a \in \mathbb{R}$ does the equation ||$x+2|-|2-x|+2|=2-a$ have exactly one solution?
## First Solution.

Let's sketch the graph of the function $f(x)=|| x+2|-| 2-x|+2|$.

The graph can be sketched piecewise by using the expression without the absolute value signs:

||$x+2|-| 2-x|+2|=\left\{\begin{aligned} 2, & x<-2 \\ -2 x-2, & -2 \leqslant x<-1 \\ 2 x+2, & -1 \leqslant x<2 \\ 6, & 2 \leqslant x\end{aligned}\right.$

![](https://cdn.mathpix.com/cropped/2024_05_30_aa38f3c63b8013e55583g-05.jpg?height=759&width=782&top_left_y=1365&top_left_x=180)

The given equation will have exactly one solution if the horizontal lines $y=2-a$ intersect the drawn graph at only one point. This will happen if:

1) $y=2-a=0$, for $a=2$
2) $2<y<6$, that is, $2<2-a<6$, which is for $-4<a<0$.

Finally, for $a \in\langle-4,0\rangle \cup\{2\}$, the given equation has one solution.