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.

Would you like to see the graph visualized?

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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.