Area of region bounded by inequalities

The given inequalities are:

1. $y \ge |x| - 1$
2. $y \ge 0$
3. $y \le 1$

Let's find the boundaries of the region in the $xy$-plane:

• At $y = 0$: $|x| \le 1 \implies -1 \le x \le 1$. Length of base $= 2$.
• At $y = 1$: $|x| \le 2 \implies -2 \le x \le 2$. Length of top side $= 4$.

The region bounded between $y = 0$ and $y = 1$ is a symmetric trapezoid with parallel sides $a = 2$ and $b = 4$, and height $h = 1$.

$\text{Area} = \frac{1}{2} \times (a + b) \times h = \frac{1}{2} \times (2 + 4) \times 1 = 3$