The graph of the equation `y + abs(y) - x - absx = 0` is represented by -

Options

A. the x-axis
B. the bisector Urie of the first quadrant
C. a pair of lines bisecting all the quadrants
D. all points of the third quadrant

Answer

To solve the equation \( y + |y| - x - |x| = 0 \) and analyze the graph, we will consider different cases based on the signs of \( x \) and \( y \).

### Step-by-Step Solution:

1. **Rewrite the Equation**:
We start with the equation:
\[ y + |y| - x - |x| = 0 \]
This can be rearranged to:
\[ y + |y| = x + |x| \]

2. **Case 1: \( x \geq 0 \) and \( y \geq 0 \)** (First Quadrant)
- Here, \( |y| = y \) and \( |x| = x \).
- The equation simplifies to:
\[ y + y = x + x \implies 2y = 2x \implies y = x \]
- This represents the line \( y = x \) in the first quadrant.

3. **Case 2: \( x < 0 \) and \( y \geq 0 \)** (Second Quadrant)
- Here, \( |y| = y \) and \( |x| = -x \).
- The equation simplifies to:
\[ y + y = x - x \implies 2y = 0 \implies y = 0 \]
- This represents the line \( y = 0 \) (the x-axis) in the second quadrant.

4. **Case 3: \( x < 0 \) and \( y < 0 \)** (Third Quadrant)
- Here, \( |y| = -y \) and \( |x| = -x \).
- The equation simplifies to:
\[ y - y = -x - x \implies 0 = -2x \implies x = 0 \]
- This means all possible values of \( y \) are valid when \( x = 0 \). Thus, this case does not restrict \( y \) and represents the entire y-axis.

5. **Case 4: \( x \geq 0 \) and \( y < 0 \)** (Fourth Quadrant)
- Here, \( |y| = -y \) and \( |x| = x \).
- The equation simplifies to:
\[ y - y = x + x \implies 0 = 2x \implies x = 0 \]
- This means all possible values of \( y \) are valid when \( x = 0 \). Thus, this case also represents the entire y-axis.

### Summary of Cases:
- In the **first quadrant**, the line \( y = x \) is present.
- In the **second quadrant**, the line \( y = 0 \) (x-axis) is present.
- In the **third quadrant**, all points satisfy the equation when \( x = 0 \).
- In the **fourth quadrant**, all points satisfy the equation when \( x = 0 \).

### Conclusion:
The graph of the equation \( y + |y| - x - |x| = 0 \) bisects the first quadrant along the line \( y = x \) and includes the x-axis in the second quadrant, while the third and fourth quadrants are represented by all points along the y-axis.