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The inequality provided is:

\[
x + 3y \geq 9
\]

This is a linear inequality in two variables, \(x\) and \(y\). Let's analyze it:

### 1. Rearranging the Inequality:
We can rearrange this inequality in slope-intercept form (similar to \(y = mx + b\)):

\[
3y \geq 9 - x
\]

Now, divide both sides by 3:

\[
y \geq \frac{9 - x}{3}
\]

This simplifies to:

\[
y \geq 3 - \frac{x}{3}
\]

### 2. Graphical Interpretation:
- The line \(y = 3 - \frac{x}{3}\) represents the boundary.
- The inequality \(y \geq 3 - \frac{x}{3}\) means the solution includes all points above or on this line.
- Since it's a "greater than or equal to" ( \(\geq\) ), the line itself is included in the solution, and the region above the line is shaded.

Would you like further details on solving or graphing inequalities? Here are some related questions to explore further:

1. How do we find specific points that satisfy this inequality?
2. What would change if the inequality was \(<\) instead of \(\geq\)?
3. How do you solve systems of inequalities graphically?
4. What is the slope and y-intercept of the line \(y = 3 - \frac{x}{3}\)?
5. Can this inequality be represented using interval notation?

**Tip**: When solving or graphing inequalities, always pay attention to whether the inequality includes or excludes the boundary (solid vs. dashed line).

Learn how to solve and graph the linear inequality x + 3y ≥ 9. This problem focuses on algebraic concepts such as rearranging equations to slope-intercept form and graphing inequalities. Suitable for students in Grades 9-12.

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