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 it represents, we will consider different cases based on the signs of \( x \) and \( y \).

### Step 1: Case 1 - Both \( x > 0 \) and \( y > 0 \)
In this case, both \( |x| \) and \( |y| \) can be replaced by \( x \) and \( y \) respectively.

\[ y + y - x - x = 0 \]
This simplifies to:
\[ 2y - 2x = 0 \quad \Rightarrow \quad y = x \]

**Hint:** When both variables are positive, you can remove the absolute value signs.

### Step 2: Case 2 - \( x < 0 \) and \( y > 0 \)
Here, \( |x| = -x \) and \( |y| = y \).

\[ y + y - x + x = 0 \]
This simplifies to:
\[ 2y = 0 \quad \Rightarrow \quad y = 0 \]

**Hint:** When \( x \) is negative and \( y \) is positive, the equation simplifies to \( y = 0 \).

### Step 3: Case 3 - Both \( x < 0 \) and \( y < 0 \)
In this case, both \( |x| \) and \( |y| \) can be replaced by \( -x \) and \( -y \) respectively.

\[ y - y - x + x = 0 \]
This simplifies to:
\[ 0 = 0 \]
This means that any point in the third quadrant satisfies the equation.

**Hint:** When both variables are negative, the equation holds true for all points in that quadrant.

### Step 4: Case 4 - \( x > 0 \) and \( y < 0 \)
Here, \( |x| = x \) and \( |y| = -y \).

\[ y - y - x - x = 0 \]
This simplifies to:
\[ -2x = 0 \quad \Rightarrow \quad x = 0 \]
However, this case does not yield any valid points since \( x \) cannot be zero when we assumed \( x > 0 \).

**Hint:** When \( x \) is positive and \( y \) is negative, the equation leads to a contradiction.

### Conclusion
From the analysis of the four cases:
- Case 1 gives the line \( y = x \) in the first quadrant.
- Case 2 gives the line \( y = 0 \) along the x-axis in the second quadrant.
- Case 3 indicates that all points in the third quadrant satisfy the equation.
- Case 4 does not yield any valid points.

Thus, the graph of the equation \( y + |y| - x - |x| = 0 \) is represented by:
- The line \( y = x \) in the first quadrant,
- The x-axis in the second quadrant,
- All points in the third quadrant.

### Final Answer
The graph is represented by the lines and regions as described above.