If the points `(x, -3x)` and (3, 4) lie on the opposite side of the line `3x-4y=8`, then

Options

A. `x gt (8)/(15), y lt(-8)/(5)`
B. `x gt(8)/(5), y gt(-8)/(15)`
C. `x lt (8)/(15), y gt(-8)/(5)`
D. `x=(8)/(15), y=(-8)/(5)`

Answer

To solve the problem, we need to determine the interval of \( x \) such that the points \( (x, -3x) \) and \( (3, 4) \) lie on opposite sides of the line given by the equation \( 3x - 4y = 8 \).

### Step 1: Rewrite the line equation in standard form
The line equation is given as:
\[ 3x - 4y - 8 = 0 \]
Here, \( a = 3 \), \( b = -4 \), and \( c = -8 \).

### Step 2: Substitute the first point \( (x, -3x) \) into the line equation
We will substitute \( (x, -3x) \) into the line equation:
\[ 3(x) - 4(-3x) - 8 = 0 \]
This simplifies to:
\[ 3x + 12x - 8 = 0 \]
Combining like terms gives:
\[ 15x - 8 \]

### Step 3: Substitute the second point \( (3, 4) \) into the line equation
Now we substitute \( (3, 4) \):
\[ 3(3) - 4(4) - 8 = 0 \]
This simplifies to:
\[ 9 - 16 - 8 = -15 \]

### Step 4: Determine the conditions for opposite sides
For the points to lie on opposite sides of the line, the results from the substitutions must have opposite signs. We found:
- For point \( (x, -3x) \): \( 15x - 8 \)
- For point \( (3, 4) \): \( -15 \)

Thus, we need:
\[ (15x - 8) \cdot (-15) < 0 \]
This simplifies to:
\[ 15x - 8 > 0 \]

### Step 5: Solve the inequality
Now we solve the inequality:
\[ 15x > 8 \]
Dividing both sides by 15 gives:
\[ x > \frac{8}{15} \]

### Step 6: Write the final interval
The interval for \( x \) is:
\[ x \in \left( \frac{8}{15}, \infty \right) \]

### Final Answer
The interval of \( x \) such that the points \( (x, -3x) \) and \( (3, 4) \) lie on opposite sides of the line \( 3x - 4y = 8 \) is:
\[ \boxed{\left( \frac{8}{15}, \infty \right)} \]