The inequality \( 13x + 11y \geq 0 \) represents a linear inequality in two variables, \(x\) and \(y\). This inequality can be analyzed by examining its boundary and determining the region of the coordinate plane that satisfies the inequality.

### Step 1: Boundary Line
The inequality \( 13x + 11y = 0 \) is the boundary line, which separates the plane into two regions. To plot this line, we find two points by setting either \(x\) or \(y\) to zero and solving for the other variable.

- When \(x = 0\):
  \[
  13(0) + 11y = 0 \implies y = 0
  \]
  So, the point is \( (0, 0) \).
  
- When \(y = 0\):
  \[
  13x + 11(0) = 0 \implies x = 0
  \]
  So, the point is also \( (0, 0) \).

Therefore, the boundary line passes through the origin \( (0, 0) \), and because both intercepts are the origin, it's useful to explore other points to establish the direction of the line.

### Step 2: Testing the Inequality
Next, test a point not on the boundary line to determine which region satisfies the inequality. Let's use the point \( (1, 0) \):

\[
13(1) + 11(0) = 13
\]
Since \( 13 \geq 0 \), the point \( (1, 0) \) satisfies the inequality. Therefore, the region on the side of the line that includes this point (the half-plane) satisfies the inequality.

### Final Answer:
The inequality \( 13x + 11y \geq 0 \) represents the region of the plane where the linear combination of \(x\) and \(y\) is non-negative, including the points on the boundary line \( 13x + 11y = 0 \).

### Questions:
1. How do you graph inequalities like \( 13x + 11y \geq 0 \)?
2. What happens if the inequality is strict, i.e., \( 13x + 11y > 0 \)?
3. How does changing the coefficients of \(x\) and \(y\) affect the inequality?
4. Can inequalities represent regions other than half-planes?
5. What is the difference between solving equations and inequalities in two variables?

### Tip:
When solving or graphing linear inequalities, always test a point to determine which region satisfies the inequality.

Explore how to graph and solve the linear inequality 13x + 11y ≥ 0. This step-by-step solution helps students understand the boundary line and determine the region that satisfies the inequality. Perfect for students in Grades 9-11 studying algebra and coordinate geometry.

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