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The inequality you have provided is:

\[
\frac{x - 4}{2x + 1} < 5
\]

### Step-by-Step Solution:

1. **Isolate the fraction**:
   Subtract 5 from both sides:
   \[
   \frac{x - 4}{2x + 1} - 5 < 0
   \]
   To combine the terms, express 5 as a fraction with the same denominator:
   \[
   \frac{x - 4}{2x + 1} - \frac{5(2x + 1)}{2x + 1} < 0
   \]
   \[
   \frac{x - 4 - 5(2x + 1)}{2x + 1} < 0
   \]

2. **Simplify the numerator**:
   Distribute the 5 in the second term:
   \[
   x - 4 - 10x - 5 = -9x - 9
   \]
   So, the inequality becomes:
   \[
   \frac{-9x - 9}{2x + 1} < 0
   \]
   Factor the numerator:
   \[
   \frac{-9(x + 1)}{2x + 1} < 0
   \]

3. **Critical points**:
   The inequality \(\frac{-9(x + 1)}{2x + 1} < 0\) changes sign when the numerator or denominator equals 0. These give the critical points:
   - From the numerator: \(x + 1 = 0\), so \(x = -1\)
   - From the denominator: \(2x + 1 = 0\), so \(x = -\frac{1}{2}\)

4. **Test intervals**:
   The critical points divide the real number line into three intervals: 
   - \( (-\infty, -1) \)
   - \( (-1, -\frac{1}{2}) \)
   - \( (-\frac{1}{2}, \infty) \)

   Now, test points in each interval to determine where the inequality holds:
   - For \(x = -2\) (in \( (-\infty, -1) \)): \(\frac{-9(-2 + 1)}{2(-2) + 1} = \frac{-9(-1)}{-3} = \frac{9}{-3} = -3\), which is less than 0.
   - For \(x = -0.75\) (in \( (-1, -\frac{1}{2}) \)): \(\frac{-9(-0.75 + 1)}{2(-0.75) + 1} = \frac{-9(0.25)}{-0.5} = \frac{-2.25}{-0.5} = 4.5\), which is greater than 0.
   - For \(x = 0\) (in \( (-\frac{1}{2}, \infty) \)): \(\frac{-9(0 + 1)}{2(0) + 1} = \frac{-9}{1} = -9\), which is less than 0.

5. **Solution in interval notation**:
   The inequality holds when \(x \in (-\infty, -1) \cup \left(-\frac{1}{2}, \infty\right)\). The points \(x = -1\) and \(x = -\frac{1}{2}\) are not included since the inequality is strict (\(<\)).

Thus, the solution is:

\[
x \in (-\infty, -1) \cup \left(-\frac{1}{2}, \infty\right)
\]

---

Do you need further clarification or additional details?

Here are five related questions:
1. How do we solve rational inequalities with a greater than sign (>) instead of a less than sign (<)?
2. How do you handle rational inequalities when the denominator contains more than one term?
3. What changes if we were solving a similar equation but for \(\geq\) instead of <?
4. How does multiplying or dividing by a negative number affect inequality signs?
5. Why is it important to exclude points that make the denominator zero in rational inequalities?

**Tip**: Always check for points that make the denominator zero in rational inequalities to avoid undefined expressions.

Solve the nonlinear inequality: (x - 4) / (2x + 1) < 5. Express the solution using interval notation.

Learn how to solve the nonlinear inequality (x - 4) / (2x + 1) < 5 step-by-step and express the solution using interval notation. This problem involves algebraic concepts like rational inequalities and critical points.

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