The area of the bounded region enclosed by the curve $y=3-\left|x-\frac{1}{2}\right|-\left|x+1\right|$ and the $x$-axis is

Plotting the graph of the function $y=3-\left|x+1\right|-\left|x-\frac{1}{2}\right|$ and $x$-axis, we get

Now applying the formula of area of trapezium which is $\frac{1}{2}\times \text{height}\times \left(\text{sum of parallel sides}\right)$ we get,

Required area of $\mathrm{Trap}\left(ABCD\right)=\frac{1}{2}\times \frac{3}{2}\times \left(3+\frac{3}{2}\right)=\frac{27}{8}$