To graph the inequality \( x - 3y > 9 \), follow these steps:

### 1. Solve for \( y \):
Start by rewriting the inequality in slope-intercept form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.

\[
x - 3y > 9
\]

Subtract \( x \) from both sides:

\[
-3y > -x + 9
\]

Now divide by \(-3\) (remember to reverse the inequality sign because you're dividing by a negative number):

\[
y < \frac{1}{3}x - 3
\]

### 2. Graph the boundary line:
The inequality is now \( y < \frac{1}{3}x - 3 \). The boundary line is \( y = \frac{1}{3}x - 3 \). 

- The slope is \( \frac{1}{3} \).
- The y-intercept is \( -3 \) (point \( (0, -3) \)).

To plot the boundary line:
- Start at \( (0, -3) \).
- Use the slope \( \frac{1}{3} \): for every 1 unit you go up on the y-axis, go 3 units to the right on the x-axis.

This boundary line will be **dashed** since the inequality is strictly \( < \), meaning the points on the line are not included in the solution.

### 3. Shade the correct region:
Since the inequality is \( y < \frac{1}{3}x - 3 \), shade the region **below** the line because you're interested in where \( y \) is less than the line.

### 4. Final graph description:
- A dashed line going through points like \( (0, -3) \), \( (3, -2) \), \( (6, -1) \), etc.
- The region below the line is shaded, indicating the solutions to the inequality.

If you'd like an image of this graph, I can create one for you.

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### Related questions:
1. How do you graph inequalities with horizontal or vertical lines?
2. What happens when the inequality includes "≥" or "≤"?
3. How do you find points of intersection between two inequalities?
4. How do you solve and graph a system of inequalities?
5. Can the same process be applied for nonlinear inequalities?

**Tip:** Always check the region by testing a point (like \( (0,0) \)) to ensure it's correctly shaded.

Learn how to graph the linear inequality x - 3y > 9 with a step-by-step breakdown. This tutorial covers key algebraic concepts such as solving for y, graphing the boundary line, and shading the correct region. Perfect for students in Grades 8-10.

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