https://file.aiquickdraw.com/mathbot/file/1727435040983bm3sdaw5.jpg

The inequality shown in the image is:

\[
\frac{x + 9}{x - 3} \leq 0
\]

To solve this inequality, follow these steps:

### Step 1: Analyze the Critical Points
The expression \(\frac{x + 9}{x - 3}\) will change signs at the points where the numerator or denominator is equal to zero.

- **Numerator:** \(x + 9 = 0\) gives \(x = -9\).
- **Denominator:** \(x - 3 = 0\) gives \(x = 3\). However, \(x = 3\) is not included in the solution because division by zero is undefined, so this is a critical point to consider as a boundary.

### Step 2: Determine the Sign of the Expression
We divide the real number line into intervals based on the critical points \(x = -9\) and \(x = 3\).

The intervals are:
- \( (-\infty, -9) \)
- \( (-9, 3) \)
- \( (3, \infty) \)

Now, we will check the sign of the expression \(\frac{x + 9}{x - 3}\) in each of these intervals by choosing a test point from each interval.

- **For \(x \in (-\infty, -9)\):** Choose \(x = -10\).
  \[
  \frac{-10 + 9}{-10 - 3} = \frac{-1}{-13} = \frac{1}{13} > 0
  \]
  The expression is positive in this interval.

- **For \(x \in (-9, 3)\):** Choose \(x = 0\).
  \[
  \frac{0 + 9}{0 - 3} = \frac{9}{-3} = -3 < 0
  \]
  The expression is negative in this interval.

- **For \(x \in (3, \infty)\):** Choose \(x = 4\).
  \[
  \frac{4 + 9}{4 - 3} = \frac{13}{1} = 13 > 0
  \]
  The expression is positive in this interval.

### Step 3: Include the Critical Points
The inequality is \(\frac{x + 9}{x - 3} \leq 0\), so we want the expression to be less than or equal to zero. The expression is equal to zero when \(x = -9\), and it is negative in the interval \((-9, 3)\). At \(x = 3\), the expression is undefined, so this point is excluded.

### Step 4: Write the Solution
The solution is:
\[
x \in [-9, 3)
\]
This means \(x\) can be between \(-9\) and \(3\), including \(-9\) but not including \(3\).

Would you like further details on this process, or do you have any questions?

Here are some related questions you might consider:
1. How do we solve inequalities involving rational expressions?
2. What is the importance of critical points in solving inequalities?
3. Can inequalities with rational expressions have multiple intervals in their solutions?
4. How does dividing by a negative number affect inequality signs?
5. What would happen if the inequality was strict (i.e., \(<\) instead of \(\leq\))?

**Tip:** Always carefully check the domain of rational expressions, as dividing by zero is undefined!

Solve the inequality (x + 9) / (x - 3) ≤ 0

Learn how to solve the rational inequality (x + 9) / (x - 3) ≤ 0 using critical points and sign analysis. This step-by-step solution demonstrates key algebraic concepts such as sign determination in different intervals and handling undefined points. Suitable for students in Grades 9-11.

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