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 will analyze the expression for two cases: when \( x \) is positive and when \( x \) is negative.

### Step 1: Solve for \( x > 0 \)
1. Start with the inequality:
\[ \frac{4}{x} < \frac{1}{3} \]
2. Multiply both sides by \( 3x \) (since \( x > 0 \), this does not change the inequality):
\[ 3 \cdot 4 < 1 \cdot x \]
This simplifies to:
\[ 12 < x \quad \text{or} \quad x > 12 \]

### Step 2: Solve for \( x < 0 \)
1. Start with the same inequality:
\[ \frac{4}{x} < \frac{1}{3} \]
2. Multiply both sides by \( 3x \) (since \( x < 0 \), this will reverse the inequality):
\[ 3 \cdot 4 > 1 \cdot x \]
This simplifies to:
\[ 12 > x \quad \text{or} \quad x < 12 \]
However, since we are considering \( x < 0 \), we combine this with the condition \( x < 0 \):
\[ x < 0 \]

### Step 3: Combine the results
From the two cases, we have:
- From \( x > 0 \): \( x > 12 \)
- From \( x < 0 \): \( x < 0 \)

Thus, the possible range of values for \( x \) is:
\[ x > 12 \quad \text{or} \quad x < 0 \]

### Final Answer:
The possible range of values for \( x \) is:
\[ x < 0 \quad \text{or} \quad x > 12 \]

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