Find the area of the region determined by the system: \begin{align} y & \ge |x| \\ y & \le -|x+1| +4 \\ \end{align} Find the area of the region determined by the system: \begin{align} y
 &amp; \ge |x| \\ y &amp; \le -|x+1| +4 \\ \end{align}


My attempt


Assuming $x&gt;0$


I have the system 
$$\begin{cases}\begin{align}
y &amp;\ge x \\
y &amp; \le -x+3 \\
\end{align}\end{cases}$$


Assuming $x&lt;0$


$$\begin{cases}\begin{align}
y &amp;\ge -x \\
y &amp; \le x+5 \\
\end{align}\end{cases}$$

A relevant interval is $-1&lt;x&lt;0$ because in this interval $y \le -|x+1| +4 $ is still positive while $y \ge |x|$ is negative.

How do I combine now this information to solve the problem ?

I don't see how I can  get something of the form $y \ge a$ and $y \le b$ which give clear bounds about  area... in this problem it seems more complicated.
If you can use graphs, then draw the graphs of the functions. The graph of $y=|x|$ is a pretty basic one. You can use transformation to get the graph of the other one. The following is the resulting picture:



You can see that it forms a rectangle. Can you proceed from here?

By the way, if you use algebraic way, you have to discuss three cases:
$$x\leq-1\\
-1\leq x \leq 0\\
x\geq 0$$
Hint:You have 4 equations of form $y=mx+c$ so find $4$ points of intersection $2$ each with opposite slopes also its a polygon with $2$ pair of lines parallel . Can you proceed further.