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.

Question

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.

Knowee AI · Accepted Answer