If the line `y = 3x - 15` intersects the line `y = mx + 8` in the third quadrant, which of the follwing must be true ?

Options

A. m is positive
B. m is negative
C. m = 0
D. `0 lt m lt 1`

Explanation

To solve the problem, we need to determine the conditions under which the lines \( y = 3x - 15 \) and \( y = mx + 8 \) intersect in the third quadrant.

### Step-by-Step Solution:

1. **Set the equations equal to each other**:
Since both equations represent \( y \), we can set them equal to find the intersection point:
\[ 3x - 15 = mx + 8 \]

2. **Rearrange the equation**:
Move all terms involving \( x \) to one side and constant terms to the other side:
\[ 3x - mx = 15 + 8 \]
This simplifies to:
\[ (3 - m)x = 23 \]

3. **Solve for \( x \)**:
Now, we can express \( x \) in terms of \( m \):
\[ x = \frac{23}{3 - m} \]

4. **Determine the sign of \( x \)**:
For the intersection to occur in the third quadrant, both \( x \) and \( y \) must be negative. Since \( y = mx + 8 \) must also be negative, we first analyze \( x \):
- For \( x \) to be negative, the denominator \( (3 - m) \) must be positive (because 23 is positive). Therefore:
\[ 3 - m > 0 \implies m < 3 \]

5. **Determine the sign of \( y \)**:
Next, we substitute \( x \) back into one of the equations to find \( y \):
Using \( y = 3x - 15 \):
\[ y = 3\left(\frac{23}{3 - m}\right) - 15 \]
Simplifying this gives:
\[ y = \frac{69}{3 - m} - 15 \]
To ensure \( y < 0 \):
\[ \frac{69 - 15(3 - m)}{3 - m} < 0 \]
Simplifying the numerator:
\[ 69 - 45 + 15m < 0 \implies 15m < -24 \implies m < -\frac{24}{15} \implies m < -1.6 \]

6. **Combine the inequalities**:
We have two inequalities:
- \( m < 3 \)
- \( m < -1.6 \)

Since \( m \) must be less than both values, the more restrictive condition is:
\[ m < -1.6 \]

### Conclusion:
The line \( y = mx + 8 \) intersects the line \( y = 3x - 15 \) in the third quadrant if \( m < -1.6 \).