https://r2.erweima.ai/mathbot/image/1725928016993ygcrzcr1.jpg

The inequality shown in the image is:

\[
\frac{|3x + 11| |x|}{x - 8} \geq 0
\]

To solve this inequality, we need to determine when the expression is greater than or equal to zero. Here’s a step-by-step breakdown:

### Step 1: Identify Key Components
- The numerator is \( |3x + 11| |x| \), which involves two absolute values.
- The denominator is \( x - 8 \).

The critical points are the values of \(x\) where the expression changes its behavior. These occur when:
1. \( 3x + 11 = 0 \) (because absolute value expressions change sign at their zero point),
2. \( x = 0 \) (the absolute value function changes here), and
3. \( x = 8 \) (the denominator becomes zero here).

### Step 2: Find Critical Points
- Solve \( 3x + 11 = 0 \):
  \[
  3x = -11 \implies x = -\frac{11}{3}
  \]
- Other critical points are \( x = 0 \) and \( x = 8 \).

So the critical points are:
\[
x = -\frac{11}{3}, \, x = 0, \, x = 8
\]

### Step 3: Analyze the Sign of Each Factor in Each Interval
We will now break the real number line into intervals based on these critical points:
- Interval 1: \( (-\infty, -\frac{11}{3}) \)
- Interval 2: \( \left(-\frac{11}{3}, 0\right) \)
- Interval 3: \( (0, 8) \)
- Interval 4: \( (8, \infty) \)

For each interval, determine the sign of \( |3x + 11| \), \( |x| \), and \( x - 8 \).

- **Interval 1: \( (-\infty, -\frac{11}{3}) \)**:
  - \(

Solve the inequality \( \frac{|3x + 11| |x|}{x - 8} \geq 0 \).

Learn how to solve the inequality \( \frac{|3x + 11| |x|}{x - 8} \geq 0 \). This problem covers important algebraic concepts like absolute values, rational functions, and critical points. Suitable for students in Grades 10-12.

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