If `4/x < -1/3`, what is the possible range of values for x?

Answer

To solve the inequality \( \frac{4}{x} < -\frac{1}{3} \), we need to find the possible range of values for \( x \).

### Step-by-step Solution:

1. **Start with the given inequality**:
\[ \frac{4}{x} < -\frac{1}{3} \]

2. **Identify the cases for \( x \)**:
- Case 1: \( x > 0 \) (x is positive)
- Case 2: \( x < 0 \) (x is negative)

3. **Case 1: \( x > 0 \)**:
- Since \( x \) is positive, we can multiply both sides of the inequality by \( 3x \) without changing the inequality sign:
\[ 4 \cdot 3 < -1 \cdot x \]
- Simplifying gives:
\[ 12 < -x \]
- Multiplying both sides by -1 (which reverses the inequality):
\[ -12 > x \quad \text{or} \quad x < -12 \]
- However, this contradicts our assumption that \( x > 0 \). Therefore, there are no valid solutions in this case.

4. **Case 2: \( x < 0 \)**:
- Since \( x \) is negative, we multiply both sides by \( 3x \) (which is negative) and reverse the inequality:
\[ 4 \cdot 3 > -1 \cdot x \]
- Simplifying gives:
\[ 12 > -x \]
- Multiplying both sides by -1 (which reverses the inequality):
\[ -12 < x \quad \text{or} \quad x > -12 \]

5. **Combine the results**:
- From Case 2, we have \( -12 < x \) and from our assumption \( x < 0 \). Therefore, we can combine these results:
\[ -12 < x < 0 \]

6. **Final answer**:
- The possible range of values for \( x \) is:
\[ x \in (-12, 0) \]
- In interval notation, this is expressed as:
\[ x \text{ belongs to } (-12, 0) \]