What is the area of the region represented by |x| + |y| + |x + y| = 8 (in sq. units)?

To find the area of the region represented by the equation $|x| + |y| + |x + y| = 8$, we need to analyze the equation by considering different cases based on the signs of $x$ and $y$.

Step-by-Step Solution:

1. Case 1: $x \geq 0$ and $y \geq 0$

   In this case, $|x| = x$, $|y| = y$, and $|x + y| = x + y$. The equation becomes: $x + y + x + y = 8 \implies 2x + 2y = 8 \implies x + y = 4$ This represents a line segment in the first quadrant from $(0, 4)$ to $(4, 0)$.

2. Case 2: $x \geq 0$ and $y \leq 0$

   In this case, $|x| = x$, $|y| = -y$, and $|x + y| = x + y$ (since $x + y$ can be positive or negative, but in this case, it will be non-negative because $x \geq 0$ and $y \leq 0$). The equation becomes: $x - y + x + y = 8 \implies 2x = 8 \implies x = 4$ This represents a vertical line segment in the fourth quadrant from $(4, 0)$ to $(4, -4)$.

3. Case 3: $x \leq 0$ and $y \geq 0$

   In this case, $|x| = -x$, $|y| = y$, and $|x + y| = -(x + y)$ (since $x + y$ will be non-positive). The equation becomes: $-x + y - (x + y) = 8 \implies -2x = 8 \implies x = -4$ This represents a vertical line segment in the second quadrant from $(-4, 0)$ to $(-4, 4)$.

4. Case 4: $x \leq 0$ and $y \leq 0$

   In this case, $|x| = -x$, $|y| = -y$, and $|x + y| = -(x + y)$. The equation becomes: $-x - y - (x + y) = 8 \implies -2x - 2y = 8 \implies x + y = -4$ This represents a line segment in the third quadrant from $(0, -4)$ to $(-4, 0)$.

Combining the Cases:

The lines $x + y = 4$, $x = 4$, $x = -4$, and $x + y = -4$ form a diamond (or rhombus) centered at the origin with vertices at $(4, 0)$, $(0, 4)$, $(-4, 0)$, and $(0, -4)$.

Calculating the Area:

The rhombus can be divided into four right triangles, each with legs of length 4. The area of one such triangle is: $\text{Area of one triangle} = \frac{1}{2} \times 4 \times 4 = 8 \text{ square units}$

Since there are four such triangles, the total area of the rhombus is: $\text{Total area} = 4 \times 8 = 32 \text{ square units}$

Therefore, the area of the region represented by $|x| + |y| + |x + y| = 8$ is $32$ square units.