If `x+4|y| = 6y` then y as a function of x is

Options

A. defined for all real x
B. continous at x=0
C. derivable at x=0
D. `(dy)/(dx) =1/2` for `x gt 0`.

Answer

To find \( y \) as a function of \( x \) from the equation \( x + 4|y| = 6y \), we will consider different cases based on the value of \( y \).

### Step 1: Rewrite the equation
We start with the equation:
\[ x + 4|y| = 6y \]

### Step 2: Case 1 - \( y \geq 0 \)
In this case, \( |y| = y \). Thus, the equation becomes:
\[ x + 4y = 6y \]
Rearranging gives:
\[ x = 6y - 4y \]
\[ x = 2y \]
From this, we can express \( y \) as a function of \( x \):
\[ y = \frac{x}{2} \quad \text{for } y \geq 0 \]

### Step 3: Case 2 - \( y < 0 \)
In this case, \( |y| = -y \). Thus, the equation becomes:
\[ x + 4(-y) = 6y \]
This simplifies to:
\[ x - 4y = 6y \]
Rearranging gives:
\[ x = 6y + 4y \]
\[ x = 10y \]
From this, we can express \( y \) as a function of \( x \):
\[ y = \frac{x}{10} \quad \text{for } y < 0 \]

### Step 4: Combine results
We have two expressions for \( y \):
1. \( y = \frac{x}{2} \) for \( y \geq 0 \)
2. \( y = \frac{x}{10} \) for \( y < 0 \)

### Step 5: Determine the intervals
- For \( y = \frac{x}{2} \) to be valid, we require \( y \geq 0 \), which implies \( x \geq 0 \).
- For \( y = \frac{x}{10} \) to be valid, we require \( y < 0 \), which implies \( x < 0 \).

### Final Result
Thus, we can summarize \( y \) as a function of \( x \):
\[ y = \begin{cases} \frac{x}{2} & \text{if } x \geq 0 \\ \frac{x}{10} & \text{if } x < 0 \end{cases} \]