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We are given the inequality:

\[
\frac{3x - 1}{x + 4} \leq 0
\]

### Step 1: Find critical points
The critical points come from setting the numerator and denominator equal to zero.

#### 1.1: Solve \( 3x - 1 = 0 \) (numerator):
\[
3x - 1 = 0 \implies x = \frac{1}{3}
\]

#### 1.2: Solve \( x + 4 = 0 \) (denominator):
\[
x + 4 = 0 \implies x = -4
\]

Thus, the critical points are \( x = \frac{1}{3} \) and \( x = -4 \).

### Step 2: Analyze intervals
The critical points divide the number line into three intervals:

1. \( (-\infty, -4) \)
2. \( (-4, \frac{1}{3}) \)
3. \( (\frac{1}{3}, \infty) \)

We will test each interval by choosing a sample value from each.

#### Interval \( (-\infty, -4) \), test \( x = -5 \):
\[
\frac{3(-5) - 1}{-5 + 4} = \frac{-15 - 1}{-1} = \frac{-16}{-1} = 16 \quad (\text{positive})
\]

#### Interval \( (-4, \frac{1}{3}) \), test \( x = 0 \):
\[
\frac{3(0) - 1}{0 + 4} = \frac{-1}{4} = -\frac{1}{4} \quad (\text{negative})
\]

#### Interval \( (\frac{1}{3}, \infty) \), test \( x = 1 \):
\[
\frac{3(1) - 1}{1 + 4} = \frac{3 - 1}{5} = \frac{2}{5} \quad (\text{positive})
\]

### Step 3: Determine the solution set
We are looking for where the expression is less than or equal to zero. From the analysis:

- The expression is negative in the interval \( (-4, \frac{1}{3}) \).
- At \( x = \frac{1}{3} \), the expression equals zero.
- At \( x = -4 \), the expression is undefined due to division by zero.

Thus, the solution is:

\[
(-4, \frac{1}{3}]
\]

### Step 4: Graph on the number line
- Open circle at \( x = -4 \) (since \( x = -4 \) is excluded).
- Closed circle at \( x = \frac{1}{3} \) (since \( x = \frac{1}{3} \) is included).
- Shade the region between \( -4 \) and \( \frac{1}{3} \).

Would you like further details on how to visualize this on a number line? Here are some related questions to explore:

1. How do we solve rational inequalities in general?
2. Can rational inequalities have more than two critical points?
3. What does it mean for a point to make the expression undefined?
4. Why is the boundary point \( x = \frac{1}{3} \) included in the solution?
5. How can we use sign charts to solve inequalities like this?

**Tip:** When solving rational inequalities, always check for points where the denominator is zero to identify undefined values in the domain.

Solve the inequality \( \frac{3x - 1}{x + 4} \leq 0 \) and graph it on a number line.

Learn how to solve the rational inequality \( \frac{3x - 1}{x + 4} \leq 0 \) using critical points and sign charts. This solution is suitable for high school students studying algebra and covers rational expressions, inequalities, and number line graphing.

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