If `P` is a point `(x ,y)` on the line `y=-3x` such that `P` and the point (3, 4) are on the opposite sides of the line `3x-4y=8,` then `x >8/(15)` (b) `x >8/5` `y<-8/5` (d) `y<-8/(15)`

Options

A. `x gt 8//15`
B. `x gt 8//5`
C. `x lt -8//5`
D. `y lt -8//15`

Explanation

To solve the problem step by step, we need to analyze the given conditions and derive the inequalities for \( x \) and \( y \).

### Step 1: Understand the given lines and points
We have:
1. The line \( y = -3x \) which represents the point \( P(x, y) \).
2. The line \( 3x - 4y = 8 \) which separates the points.
3. The point \( Q(3, 4) \).

### Step 2: Rewrite the line equation
We can rewrite the line \( 3x - 4y = 8 \) in the form:
\[ 3x - 4y - 8 = 0 \]
This will help us evaluate whether points are on opposite sides of the line.

### Step 3: Evaluate the position of point \( Q(3, 4) \)
Substituting \( Q(3, 4) \) into the line equation:
\[ 3(3) - 4(4) - 8 = 9 - 16 - 8 = -15 \]
Since the result is negative, point \( Q \) lies on one side of the line.

### Step 4: Determine the condition for point \( P(x, y) \)
For point \( P(x, y) \) to be on the opposite side of the line, we need:
\[ 3x - 4y - 8 > 0 \]

### Step 5: Substitute \( y \) from the line equation
Since \( P \) lies on the line \( y = -3x \), we substitute \( y \):
\[ 3x - 4(-3x) - 8 > 0 \]
This simplifies to:
\[ 3x + 12x - 8 > 0 \]
\[ 15x - 8 > 0 \]

### Step 6: Solve for \( x \)
Rearranging gives:
\[ 15x > 8 \implies x > \frac{8}{15} \]

### Step 7: Determine the condition for \( y \)
Next, we need to find the condition for \( y \). Substitute \( y = -3x \) into the inequality:
\[ 3x - 4(-3x) - 8 > 0 \]
This leads to:
\[ 3x + 12x - 8 > 0 \implies 15x - 8 > 0 \]
Now substituting \( y \):
\[ 3x - 4y - 8 > 0 \implies 3x - 4(-3x) - 8 > 0 \]
This simplifies to:
\[ 3x + 12x - 8 > 0 \]
\[ 15x - 8 > 0 \implies x > \frac{8}{15} \]
Now substituting \( x = -\frac{y}{3} \) into the inequality:
\[ 3(-\frac{y}{3}) - 4y - 8 > 0 \]
This simplifies to:
\[ -y - 4y - 8 > 0 \implies -5y - 8 > 0 \implies 5y < -8 \implies y < -\frac{8}{5} \]

### Conclusion
Thus, we have derived:
1. \( x > \frac{8}{15} \)
2. \( y < -\frac{8}{5} \)

### Final Answer
The correct options are:
(a) \( x > \frac{8}{15} \) and (c) \( y < -\frac{8}{5} \).