https://file.aiquickdraw.com/mathbot/file/1726594187840ae4xony1.jpg

The inequality in the image is:

\[
\frac{(x + 5)^3}{(x - 6)^3} \leq 0
\]

To solve this, we can follow these steps:

### 1. **Find the critical points**:
   - Set the numerator and denominator equal to zero to find where the expression equals zero or is undefined.

   - The numerator, \((x + 5)^3 = 0\), gives \(x = -5\).

   - The denominator, \((x - 6)^3 = 0\), gives \(x = 6\). At this point, the expression is undefined.

### 2. **Create a sign chart**:
   These critical points split the number line into three intervals: 
   - \(x < -5\)
   - \(-5 < x < 6\)
   - \(x > 6\)

### 3. **Test the sign in each interval**:
   Pick a test value from each interval to determine the sign of the expression:

   - For \(x < -5\) (e.g., \(x = -6\)):
     \[
     \frac{(x + 5)^3}{(x - 6)^3} = \frac{(-6 + 5)^3}{(-6 - 6)^3} = \frac{(-1)^3}{(-12)^3} = \frac{-1}{-1728} > 0
     \]

   - For \(-5 < x < 6\) (e.g., \(x = 0\)):
     \[
     \frac{(0 + 5)^3}{(0 - 6)^3} = \frac{5^3}{(-6)^3} = \frac{125}{-216} < 0
     \]

   - For \(x > 6\) (e.g., \(x = 7\)):
     \[
     \frac{(7 + 5)^3}{(7 - 6)^3} = \frac{12^3}{1^3} = \frac{1728}{1} > 0
     \]

### 4. **Include the endpoints**:
   - At \(x = -5\), the numerator is zero, so the entire expression is zero.
   - At \(x = 6\), the expression is undefined.

### 5. **Determine the solution**:
   The inequality \(\frac{(x + 5)^3}{(x - 6)^3} \leq 0\) is satisfied where the expression is less than or equal to zero:
   - The interval where it is less than zero is \(-5 < x < 6\).
   - It equals zero at \(x = -5\).

Therefore, the solution is:

\[
x \in [-5, 6)
\]

On the sign chart, plot a filled circle at \(x = -5\) and an open circle at \(x = 6\), and shade the region between \(-5\) and \(6\).

---

Would you like further clarification or details on any step? Here are some related questions to explore:

1. What is the difference between an open and closed circle in a sign chart?
2. How does the behavior of cubic terms affect the sign chart?
3. Can you explain how to handle inequalities when the expression is undefined?
4. How would the solution change if the inequality were strict, i.e., \((x + 5)^3 / (x - 6)^3 < 0\)?
5. How would you solve the inequality if the exponents were even instead of odd?

**Tip:** Always be cautious about undefined points (like division by zero) in inequalities and exclude them from the solution set.

Make a sign chart that represents the expression ((x + 5)^3) / ((x - 6)^3) ≤ 0 and use your sign chart to solve the inequality. Plot x-values for which the expression equals zero or is undefined, then click next.

Solve the rational inequality ((x + 5)^3) / ((x - 6)^3) ≤ 0 by using a sign chart. This step-by-step solution explores critical points, sign testing, and endpoint behavior, ideal for students in Grades 9-11.

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