The area bounded by the lines y = |x| – 1 and y = - |x| + 1 is

The correct option is "2 sq. units".

To find the area of the region bounded by the given lines, let us first write down the equations of the boundary lines:
Line 1: $$y=\left|x\right|-1$$
Line 2: $$y=-\left|x\right|+1$$

We can rewrite these equations by analyzing the behavior of the absolute value function $\left|x\right|$ for different intervals of $x$:

For $x\ge 0$ (the right half of the coordinate plane):
Line 1 becomes: $y=x-1$
Line 2 becomes: $y=-x+1$

For $x<0$ (the left half of the coordinate plane):
Line 1 becomes: $y=-x-1$
Line 2 becomes: $y=x+1$

Now, let us find the intersection points of these boundary lines:
1. Intersection of $y=x-1$ and $y=-x+1$ (for $x\ge 0$):
Equating the two expressions for $y$:
$$x-1=-x+1\Rightarrow 2x=2\Rightarrow x=1$$
Substituting $x=1$ into either equation gives $y=0$. So, one vertex is $(1,0)$.
2. Intersection of $y=-x-1$ and $y=x+1$ (for $x<0$):
Equating the two expressions:
$$-x-1=x+1\Rightarrow 2x=-2\Rightarrow x=-1$$
Substituting $x=-1$ into either equation gives $y=0$. So, another vertex is $(-1,0)$.
3. Along the y-axis where $x=0$:
For Line 1, $y=\left|0\right|-1=-1$, giving the vertex $(0,-1)$.
For Line 2, $y=-\left|0\right|+1=1$, giving the vertex $(0,1)$.

Plotting these four points $(1,0)$, $(-1,0)$, $(0,1)$, and $(0,-1)$ on the coordinate plane and connecting them shows that the bounded region is a square (or a rhombus) whose diagonals lie along the coordinate axes.

Let us calculate the lengths of the diagonals:
Diagonal 1 (along the x-axis, from $-1$ to $1$):
$$d_1=1-(-1)=2\text{units}$$
Diagonal 2 (along the y-axis, from $-1$ to $1$):
$$d_2=1-(-1)=2\text{units}$$

The area of a quadrilateral with perpendicular diagonals is given by the formula:
$$\text{Area}=\frac{1}{2}\times d_1\times d_2$$
Substituting the diagonal lengths into this formula:
$$\text{Area}=\frac{1}{2}\times 2\times 2=2\text{sq. units}$$

Thus, the area bounded by the given lines is 2 sq. units.